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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{A. S. Buch} and \textit{R. Rimányi} [C. R., Math., Acad. Sci. Paris 339, No. 1, 1--4 (2004; Zbl 1051.14062)] proved a formula for a specialization of double Grothendieck polynomials based on the Yang-Baxter equation related to the degenerate Hecke algebra. A geometric proof was found by \textit{A. Woo} and \textit{A. Yong} [Am. J. Math. 134, No. 4, 1089--1137 (2012; Zbl 1262.13044)] by constructing a Gröbner basis for the Kazhdan-Lusztig ideals. In this note, we give an elementary proof for this formula by using only divided difference operators. Buch-Rimányi formula; double Grothendieck polynomial; specialization Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) A note on specializations of Grothendieck polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use the modules introduced by \textit{W. Kraskiewicz} and \textit{P. Pragacz} [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011); Eur. J. Comb. 25, No. 8, 1327--1344 (2004; Zbl 1062.14065)] to show some positivity properties of Schubert polynomials. We give a new proof to the classical fact that the product of two Schubert polynomials is Schubert-positive, and also show a new result that the plethystic composition of a Schur function with a Schubert polynomial is Schubert-positive. The present submission is an extended abstract on these results and the full version of this work will be published elsewhere. Schubert polynomials; Schubert functors; Kraśkiewicz-Pragacz modules; Schubert calculus Classical problems, Schubert calculus, Algebraic combinatorics Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Cohomological or \(K\)-theoretic classes of Schubert subvarieties of Grassmannians play an important role in enumerative geometry and representation theory. Recent works indicate the importance of replacing the Grassmannian with the total space of its cotangent bundle. The natural counterpart of a Schubert variety in this setting is the conormal bundle of a Schubert variety. The present paper considers the classes of conormal bundles of Schubert varieties in equivariant \(K\)-theory. Moreover, and very importantly, not only the K theory class of the conormal bundle, but the representing sheave as well. The authors present a conjectured formula for that sheave, as a rectangular domain partition function. They verify the conjecture when the Schubert variety is smooth. The authors compute that the \(K\)-theory class of their sheave is a partition function of an integrable loop model. Also, when pushed forward to a point, it is a solution to the level-1 Knizhnik-Zamolodchikov equation. The computation of the push-forward to a point is carried out by a simultaneous degeneration of the conormal bundle and of the sheave. The sheave degeneration turns out to follow the combinatorics of analogous processes for plane partitions. As result, the authors obtain a geometric interpretation of the Razumov-Stroganov correspondence. quantum Knizhnik-Zamolodchikov equation; equivariant \(K\)-theory; cotangent bundle of the Grassmannian; loop model Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Exactly solvable models; Bethe ansatz Grassmann-Grassmann conormal varieties, integrability, and plane partitions

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a case of a positivity conjecture of \textit{L. C. Mihalcea} and \textit{R. Singh} [``Mather classes and conormal spaces of Schubert varieties in cominuscule spaces'', Preprint, \url{arXiv:2006.04842}], concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian \(LG(n,2n)\). Combined with work of \textit{P. Aluffi} et al. [``Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells'', Preprint, \url{arXiv:1709.08697}], this further implies the positivity of the Mather classes for Schubert varieties in \(LG(n,2n)\), which Mihalcea-Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan-Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for \(LG(n,2n)\) the Euler obstructions \(e_{y,w}\) may vanish for certain pairs \((y,w)\) with \(y\le w\) in the Bruhat order. Our combinatorial description allows us to classify all the pairs \((y,w)\) for which \(e_{y,w}=0\). Restricting to the big opposite cell in \(LG(n,2n)\), which is naturally identified with the space of \(n\times n\) symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification. local Euler obstructions; Schubert stratification; Lagrangian Grassmannian; tree labelings Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Trees, Local complex singularities, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) Euler obstructions for the Lagrangian Grassmannian

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We make a broad conjecture about the \(k\)-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the \(k\)-Schur expansion of (1) Hall-Littlewood polynomials, proving the \(q = 0\) case of the strengthened Macdonald positivity conjecture from [\textit{L. Lapointe} et al., Duke Math. J. 116, No. 1, 103--146 (2003; Zbl 1020.05069)]; (2) the product of a Schur function and a \(k\)-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties; (3) \(k\)-split polynomials, solving a substantial special case of a problem of \textit{B. Broer} [Prog. Math. 123, 1--19 (1994; Zbl 0855.22015)] and \textit{M. Shimozono} and \textit{J. Weyman} [Eur. J. Comb. 21, No. 2, 257--288 (2000; Zbl 0956.05100)] on parabolic Hall-Littlewood polynomials. In addition, we prove the conjecture that the \(k\)-Schur functions defined via \(k\)-split polynomials [\textit{L. Lapointe} and \textit{J. Morse}, J. Comb. Theory, Ser. A 101, No. 2, 191--224 (2003; Zbl 1018.05101)] agree with those defined in terms of strong tableaux [\textit{T. Lam} et al., Affine insertion and Pieri rules for the affine Grassmannian. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1208.14002)]. Macdonald polynomials; Gromov-Witten invariants; Schubert structure constants; parabolic Hall-Littlewood polynomials; strong tableaux Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations \(k\)-Schur expansions of Catalan functions

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X_w\) be a Schubert subvariety of a cominuscule Grassmannian \(X\), and let \(\mu :T^*X\rightarrow\mathcal{N}\) be the Springer map from the cotangent bundle of \(X\) to the nilpotent cone \(\mathcal{N}\). In this paper, we construct a resolution of singularities for the conormal variety \(T^*_XX_w\) of \(X_w\) in \(X\). Further, for \(X\) the usual or symplectic Grassmannian, we compute a system of equations defining \(T^*_XX_w\) as a subvariety of the cotangent bundle \(T^*X\) set-theoretically. This also yields a system of defining equations for the corresponding orbital varieties \(\mu (T^*_XX_w)\). Inspired by the system of defining equations, we conjecture a type-independent equality, namely \(T^*_XX_w=\pi^{-1}(X_w)\cap\mu^{-1}(\mu (T^*_XX_w))\). The set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian of type A, B, or C. For Part I, see [the author and \textit{V. Lakshmibai}, ``Conormal varieties on the cominuscule Grassmannian'', Preprint, \url{arXiv:1712.06737}]. Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Coadjoint orbits; nilpotent varieties Conormal varieties on the cominuscule Grassmannian. II

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials As a generalization of skew Schur functions, the refined dual stable Grothendieck polynomials \(\tilde{g}_{\lambda/\mu}(x;t)\) can be easily defined using reverse plane partitions. Motivated by the Jacobi-Trudi formula and its dual for Schur functions, Grinberg conjectured a Jacobi-Trudi type formula for \(\tilde{g}_{\lambda/\mu}(x;t)\) for any skew partition \(\lambda/\mu\). The case for \(\mu=\emptyset\) has been confirmed by \textit{D. Yeliussizov} [J. Algebr. Comb. 45, No. 1, 295--344 (2017; Zbl 1355.05263)]. In the paper under review, the author completely proved Grinberg's conjecture. Comparing the elegant proof of the classical Jacobi-Trudi formula, the author's proof of Grinberg's conjecture is highly nontrivial, which relies on two bijections due to \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)] on RSE-tableaux and requires tedious analysis of the properties of these bijections on extended skew RSE-tableaux. As remarked by the author, Grinberg's conjecture was also independently proved by \textit{A. Amanov} and \textit{D. Yeliussizov} [`Determinantal formulas for dual Grothendieck polynomials'', Preprint, \url{arXiv:2003.03907}]. Jacobi-Trudi formula; plane partition; Grothendieck polynomial; Young tableau Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Jacobi-Trudi formula for refined dual stable Grothendieck polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a semisimple adjoint algebraic group \(G\) and a Borel subgroup \(B\), consider the double classes \(BwB\) in \(G\) and their closures in the canonical compactification of \(G\); we call these closures large Schubert varieties. We show that these varieties are normal and Cohen-Macaulay; we describe their Picard group and the spaces of sections of their line bundles. As an application, we construct geometrically a filtration à la van der Kallen of the algebra of regular functions on \(B\). We also construct a degeneration of the flag variety \(G/B\) embedded diagonally in \(G/B\times G/B\), into a union of Schubert varieties. This yields formulae for the class of the diagonal of \(G/B\times G/B\) in \(T\)-equivariant \(K\)-theory, where \(T\) is a maximal torus of \(B\). semisimple adjoint algebraic group; large Schubert varieties; Picard group; filtration; algebra of regular functions; flag variety; equivariant \(K\)-theory Brion, M; Polo, P, Large Schubert varieties, Represent. Theory, 4, 97-126, (2000) Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Representation theory for linear algebraic groups, \(K\)-theory of schemes, Group actions on varieties or schemes (quotients) Large Schubert varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The results of this paper are a generalization of those in the authors' paper [Special functions, Proc. Hayashibara Forum, Okayama/Jap. 1990, ICM-90 Satell. Conf. Proc., 149-168 (1991)]. Let \(k_ q[G]\) be the quantum algebra of functions on a semisimple algebraic group \(G\) of rank \(\ell\) (in [loc. cit.], the considered \(G= \text{SL}(N)\)). Let \(B\) be a Borel subgroup of \(G\) and let \(P\supseteq B\) be a maximal parabolic subgroup of \(G\). Let \(k_ q[B]\) be the quantum Hopf algebra of functions on \(B\). Let \(w\) be an element of the Weyl group and let \(X(w) \subset G/B\) be the corresponding Schubert variety. The authors define the quantum algebras \(k_ q[G/P]\), \(k_ q[G/B]\), \(k_ q[X(w)]\); the first two are subcomodules of \(k_ q[G]\), the last is a quotient of \(k_ q[G/B]\). Each of these algebras has, in the classical case, a basis consisting of standard monomials---compatible with canonical \(\mathbb{Z}\) or \(\mathbb{Z}^ \ell\)-gradations. The authors prove the existence of such basis and gradations in the quantum case and give a presentation for \(k_ q[G/B]\). quantum algebra of functions; semisimple algebraic group; Schubert variety; basis; gradations V. Lakshmibai and N. Reshetikhin. ''Quantum flag and Schubert schemes''. Deformation Theory and Quantum Groups with Applications to Mathematical Physics. Contemp. Math., Vol. 134. American Mathematical Society, 1992, pp. 145--181. Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Quantum flag and Schubert schemes

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We establish algebraically and geometrically a duality between the Iwahori-Hecke algebra of type \( \mathbf D\) and two new quantum algebras arising from the geometry of \(N\)-step isotropic flag varieties of type \(\mathbf D\). This duality is a type \( \mathbf D\) counterpart of the Schur-Jimbo duality of type \( \mathbf A\) and the Schur-like duality of type \( \mathbf B/\mathbf C\) discovered by \textit{H. Bao} and \textit{W. Wang} [A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs. Astérisque 402. Paris: Société Mathématique de France (SMF) (2018; Zbl 1411.17001), Preprint, \url{arXiv:1310.0103}]. The new algebras play a role in the type \( \mathbf D\) duality similar to the modified quantum \( \mathfrak{gl}(N)\) in type \( \mathbf A\), and the modified coideal subalgebras of quantum \( \mathfrak{gl}(N)\) in type \( \mathbf B/\mathbf C\). We construct canonical bases for these two algebras. For Part I see [\textit{H. Bao} et al., Transform. Groups 23, No. 2, 329--389 (2018; Zbl 1440.17009), Preprint, \url{arXiv:1404.4000}]. Iwahori-Hecke algebra of type \(\mathbf D\); flag variety of type \(\mathbf D\); Schur-type duality; canonical basis Z. Fan, Y. Li, \textit{Geometric Schur duality of classical type,} II, Trans. Amer. Math. Soc., Series \textbf{B 2} (2015), 51\(-\)92. Quantum groups (quantized enveloping algebras) and related deformations, Classical groups (algebro-geometric aspects), Schur and \(q\)-Schur algebras Geometric Schur duality of classical type. II

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this survey paper we review recent advances in the calculus of Chern-Schwartz-MacPherson, motivic Chern, and elliptic classes of classical Schubert varieties. These three theories are one-parameter \((\hbar)\) deformations of the notion of fundamental class in their respective extraordinary cohomology theories. Examining these three classes in conjunction is justified by their relation to Okounkov's stable envelope notion. We review formulas for the \(\hbar\)-deformed classes originating from Tarasov-Varchenko weight functions, as well as their orthogonality relations. As a consequence, explicit formulas are obtained for the Littlewood-Richardson type structure constants. Schubert calculus; elliptic cohomology Research exposition (monographs, survey articles) pertaining to algebraic topology, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Classical problems, Schubert calculus, Elliptic cohomology, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Equivariant \(K\)-theory, Elliptic genera \(\hbar\)-deformed Schubert calculus in equivariant cohomology, \(K\)-theory, and elliptic cohomology

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The equivariant quantum cohomology ring \(QH^*_T(Fl({\underline n}))\) of the \(m\)-step partial flag variety in \({\mathbb C}^n\) is studied. This ring is an algebra over \(\Lambda[{\underline q}]={\mathbb Z}[t_1,t_2,\dots,t_n,q_1,q_2,\dots,q_m]\) and can be thought of as a deformation of the usual equivariant cohomology ring which is an algebra over \(\Lambda={\mathbb Z}[t_1,t_2,\dots,t_n]\). Additively \(QH^*_T(Fl({\underline n}))\simeq H^*_T(Fl({\underline n}))\otimes \Lambda[{\underline q}]\). For the parameters \({\underline q}=0\) the algebra structure was computed by Borel. The Schubert classes form a basis of the \(\Lambda[{\underline q}]\)-module \(QH^*_T(Fl({\underline n}))\). The authors give a Giambelli type formula for Schubert classes in terms of the \textit{quantum elementary polynomials}. The resulting \textit{equivariant quantum Schubert polynomials} are specializations of \textit{W. Fulton}'s \textit{universal Schubert polynomials} [Duke Math. J. 96, No. 3, 575--594 (1999; Zbl 0981.14022)] and has already appeared in non-equivariant situation [\textit{I. Ciocan-Fontanine}, Duke Math. J. 98, No. 3, 485--524 (1999; Zbl 0969.14039)]. As a by-product a presentation of \(QH^*_T(Fl({\underline n}))\) was obtained; originally this is a result of [\textit{B. Kim}, Int. Math. Res. Not. 1996, No. 17, 841--851 (1996; Zbl 0881.55007)]. The proof relies on the moving lemma with respect to the mixing group. The paper contains a comprehensive account of the history of recent developments of the Schubert calculus. Schubert calculus; flag varieties; quantum equivariant cohomology; Giambelli formula Anderson, D.; Chen, L.: Equivariant quantum Schubert polynomials. Adv. math. 254, 300-330 (2014) Classical problems, Schubert calculus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Equivariant homology and cohomology in algebraic topology Equivariant quantum Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathfrak g\) be a Kac-Moody algebra and \(Q\) be some orientation of the Dynkin diagram of \(\mathfrak g\). Let \(\widetilde Q\) be the double quiver of \(Q\). The main result of this paper asserts that a certain quiver Grassmannian for \(\widetilde Q\) is homeomorphic to a certain Lagrangian Nakajima quiver variety. The authors also refine this result by finding quiver Grassmannians that are homeomorphic to the Demazure quiver varieties, and others which are homeomorphic to the graded/cyclic quiver varieties. The Demazure quiver Grassmannian allows the authors to describe injective objects in the category of locally nilpotent modules over the preprojective algebra. Motivated by an earlier version of the paper, \textit{I. Shipman} has recently proved [see Math. Res. Lett. 17, No. 5, 969-976 (2010; Zbl 1231.16010)] that Lusztig's bijection between the points of the Lagrangian Nakajima quiver variety and the points of a type of quiver Grassmannian inside a projective object is, in fact, an isomorphism of algebraic varieties. The present paper has an Appendix where the authors explain how this result allows one to conclude that the maps between quiver Grassmannians and Lagrangian Nakajima quiver varieties described in the paper under review are also isomorphisms of algebraic varieties. quivers; preprojective algebras; quiver Grassmannians; quiver varieties; Kac-Moody algebras; Demazure modules Savage, A; Tingley, P, Quiver Grassmannians, quiver varieties and the preprojective algebra, Pacific J. Math., 251, 393-429, (2011) Representations of quivers and partially ordered sets, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Group actions on affine varieties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Quiver Grassmannians, quiver varieties and the preprojective algebra.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review the authors show that the quantized coordinate ring \(\mathbb{K}_q[\mathrm{Gr}(2,n)]\), \(n\geq 3\), of the Grassmannian is a quantum cluster algebra of type \(A_{n-3}\). Using computer-aided calculation the authors also show that \(\mathbb{K}_q[\mathrm{Gr}(3,n)]\), \(n=6,7,8\), are quantum cluster algebras of types \(D_4\), \(E_6\) and \(E_7\), respectively. Using this, the authors obtain quantum cluster algebra structures on quantum Schubert cells of the \(k=2\) Grassmannians. It turns out that the quantum Schubert cells associated to the partition \((t,s)\), where \(t\geq s\) and \(t,s\leq n-2\), is of type \(A_{s-1}\). These cases are precisely those where the quantum cluster algebra is of finite type and the structures the authors describe quantize the structures known in the classical case. cluster algebra; quantum Grassmannian; Schubert cell; finite type; seed Jan E. Grabowski, Stéphane Launois, Lifting quantum cluster algebra structures, 2015, in preparation. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Cluster algebras, Representations of quivers and partially ordered sets Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram \(\delta \subset {\mathbb Z} \times {\mathbb Z}\), written as \(\widetilde H_{\delta}(X;q,t)\) and \(\widetilde H_{\delta}(X;0,t)\), respectively. We then give an explicit Schur expansion of \(\widetilde H_{\delta}(X;0,t)\) as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given by \textit{A. Lascoux} and \textit{M.-P. Schüztenberger} [C. R. Acad. Sci., Paris, Sér. A 286, 323--324 (1978; Zbl 0374.20010)]. We further define the symmetric function \(R_{\gamma,\delta}(X)\) as a refinement of \(\widetilde H_{\delta}(X;0,t)\) and similarly describe its Schur expansion. We then analyze \(R_{\gamma,\delta}(X)\) to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of \(\widetilde H_{\delta}(X;q,t)\). To gain these results, we use a construction from \textit{S. Assaf} [Dual equivalence graphs, ribbon tableaux and Macdonald polynomials. Berkeley, CA: University of California, Berkeley (PhD Thesis) (2007), \url{http://www-bcf.usc.edu/~shassaf/PhinisheD.pdf}] to associate each Macdonald polynomial with a signed colored graph \(\mathcal{H}_\delta\). In the case where a subgraph of \(\mathcal{H}_\delta\) is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries. Hall-Littlewood polynomials; dual equivalence; Schur functions; symmetric functions; Macdonald polynomials Roberts, Austin, On the Schur expansion of Hall-Littlewood and related polynomials via Yamanouchi words, Electron. J. Combin., 24, 1, Paper 1.57, 30 pp., (2017) Symmetric functions and generalizations, Classical problems, Schubert calculus, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) On the Schur expansion of Hall-Littlewood and related polynomials via Yamanouchi words

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce two families of symmetric functions generalizing the factorial Schur \(P\)- and \(Q\)-functions due to Ivanov. We call them \(K\)-theoretic analogues of factorial Schur \(P\)- and \(Q\)-functions. We prove various combinatorial expressions for these functions, e.g.~as a ratio of Pfaffians, a sum over set-valued shifted tableaux, and a sum over excited Young diagrams. As a geometric application, we show that these functions represent the Schubert classes in the \(K\)-theory of torus equivariant coherent sheaves on the maximal isotropic Grassmannians of symplectic and orthogonal types. This generalizes a corresponding result for the equivariant cohomology given by the authors. We also discuss a remarkable property enjoyed by these functions, which we call the \(K\)-theoretic \(Q\)-cancellation property. We prove that the \(K\)-theoretic \(P\)-functions form a (formal) basis of the ring of functions with the \(K\)-theoretic \(Q\)-cancellation property. Schubert class; Schur \(Q\)-functions; isotropic Grassmannians; equivariant \(K\)-theory Ikeda, T.; Naruse, H., \textit{K}-theoretic analogue of Schur \textit{P}-, \textit{Q}-functions, Adv. Math., 243, 22-66, (2013) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Equivariant \(K\)-theory \(K\)-theoretic analogues of factorial Schur \(P\)- and \(Q\)-functions

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Perhaps the most important open problem in the combinatorial-algebraic-geometric study of the classical manifold of flags in \(n\)-space is to find an analog of the Littlewood-Richardson rule. The cohomology ring of the flag manifold has a distinguished integral basis of Schubert classes \(S_u\) which are indexed by permutations \(u\) on \(n\) letters, and the coefficients \(c^w_{u,v}\) express the product \(S_u\cdot S_v\) in terms of the basis elements \(S_w\). These coefficients are known to be positive from geometry and the challenge is to find a manifestly positive combinatorial formula for them. This paper by Kogan contains the best results to date on this problem. His result is for the admittedly special cases when the permutation \(v\) has a single descent at \(k\) (\(v\) is Grassmannian) and \(u\) has no descents at positions greater than \(k\), and it also holds when the partition corresponding to \(v\) has the shape of a hook. Nevertheless, his result is very appealing, as in the special case when \(u\) is also Grassmannian with descent at \(k\), his formula coincides with the Littlewood-Richardson formula. Also, when \(v\) has hook shape, it coincides with the formula obtained by the reviewer [Ann. Inst. Fourier 46, 89-110 (1996; Zbl 0837.14041)]. His formula expresses these numbers \(c^w_{u,v}\) in terms of filling of the partition shape of \(v\) by numbers corresponding to certain chains in the Bruhat order from \(u\) to \(w\). Furthermore, his proof is via an insertion algorithm that generalizes Schensted insertion. The insertion algorithm was introduced by \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, 257-269 (1993; Zbl 0803.05054)] to give a proof of Monk's formula (when \(v\) is a simple reflection). Littlewood-Richardson rule; Schubert classes Kogan, M., Generalisation of Schensted insertion algorithm to the case of hooks and semi-shuffles, J. Combin. Theory Ser. A, 102, 110-135, (2003) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus Generalization of Schensted insertion algorithm to the cases of hooks and semi-shuffles

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in \textit{M. Chan} et al. [Trans. Am. Math. Soc. 370, No. 5, 3405--3439 (2018; Zbl 1380.05007)], and it generalizes a formula of \textit{C. Lenart} [Ann. Comb. 4, No. 1, 67--82 (2000; Zbl 0958.05128)] and a recent result of \textit{V. Reiner} et al. [J. Comb. Theory, Ser. A 158, 66--125 (2018; Zbl 1391.05269)] to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials. Schur functions; Grothendieck polynomials; insertion algorithms; set-valued tableaux; Brill-Noether theory Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Combinatorial relations on skew Schur and skew stable Grothendieck polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a combinatorial expansion of a Schubert homology class in the affine Grassmannian \(\mathrm{Gr}_{\mathrm{SL}_k}\) into Schubert homology classes in \(\mathrm{Gr}_{\mathrm{SL}_{k+1}}\). This is achieved by studying the combinatorics of a new class of partitions called \(k\)-shapes, which interpolates between \(k\)-cores and \(k+1\)-cores. We define a symmetric function for each \(k\)-shape, and show that they expand positively in terms of dual \(k\)-Schur functions. We obtain an explicit combinatorial description of the expansion of an ungraded \(k\)-Schur function into \(k+1\)-Schur functions. As a corollary, we give a formula for the Schur expansion of an ungraded \(k\)-Schur function. symmetric functions; Schur functions; tableaux; Schubert calculus Lam, Thomas; Lapointe, Luc; Morse, Jennifer; Shimozono, Mark, The poset of \(k\)-shapes and branching rules for \(k\)-Schur functions, Mem. Amer. Math. Soc., 223, 1050, vi+101 pp., (2013) Symmetric functions and generalizations, Classical problems, Schubert calculus, Combinatorics of partially ordered sets The poset of \(k\)-shapes and branching rules for \(k\)-Schur functions

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The superb paper under review is quite (and probably unavoidably) technical. That is why the reviewer will not put a special effort to simplify the description of its content. Otherwise this short summary may turn useless for both non experts and specialists: the former because in any case would lack background, and the latter because the essential mathematical content of the paper would be lost behind excess of simplification. In any case, to save a minimum of friendly shape, let us begin this short summary as if it were a tale. Schubert calculus. At the very beginning was the intersection theory of the complex Grassmann manifolds \(G(k,n)\) parametrizing \(k\)-dimensional subspaces of \({\mathbb C}^n\). Grassmannians, however, are a special kind of flag varieties, which are in turn special kind of homogeneous projective varieties, i.e. quotient \(G/B\) of a complex connected semi-simple Lie group modulo the action of a Borel subgroup \(B\). Thus, nowadays, the location Schubert calculus has acquired a much broader meaning. Not only for Grassmannians, but on general flag varieties, and not only classical cohomology, but also quantum, equivariant or quantum-equivariant, up to its \(K\)-theory and its connective \(K\)-theory (a theory interpolating, in a suitable sense, the \(K\)-theory and the intersection theory of a homogeneous space). The paper under review puts itself in this very general framework using in a creative manner a new algebraic tool, what the authors call \textsl{formal root polynomials}, with the purpose of studying the elliptic cohomology of the homogeneous space \(G/B\): this means a cohomology theory where all the odd parts vanish and there is an invertible element \(h\in H^2\) inducing a complex orientation with the same formal group law as that of an elliptic curve. There is a correspondence between generalized cohomology theories and formal group laws, and in particular the authors investigate the \textsl{hyperbolic group law} introduced in Section 2.2. The corresponding Schubert calculus is so called by the authors \textsl{hyperbolic Schubert calculus.} and enables to study the elliptic cohomology of homogeneous spaces, extending previous work by Billey and Graham-Willems letting it to work uniformly in all Lie type. The definition of formal root polynomial is quite technical and is not worth to be recalled in the present review. However the idea is that of replacing, or rather extend, the notion of root polynomials heavily used by \textit{S. C. Billey} [Duke Math. J. 96, No. 1, 205--224 (1999; Zbl 0980.22018)] and by \textit{M. Willems} [Bull. Soc. Math. Fr. 132, No. 4, 569--589 (2004; Zbl 1087.19004)]. After introducing the formal root polynomial, whose definition depends on a reduced word for a Weyl group element, the main Theorem 3.10 states that indeed it does not depend on such a word provided that the formal group law is the hyperbolic one. Section 4 is devoted to applications: in particular the authors show how their techniques provide an efficient method to compute the transition matrix between two natural bases of the formal \textsl{Demazure algebra}, another gadget introduced and explained in Section 2. Section 5 is concerned with localization formulas in cohomology and \(K\)-theory, while section 6 is not only devoted to show further applications of root polynomials to compute Bott-Samelson classes, but also to propose a couple of conjectures in the hyperbolic Schubert calculus, based mainly on analogies and experimental evidence. The paper ends with a comprehensive reference list: among the key ones, the paper by Goresky, Kottwitz and MacPherson [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)], the important 1974 paper by \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér. (4) 7, 53--88 (1974; Zbl 0312.14009)] on desingularization of generalized Schubert varieties, a couple of papers by Graham and Graham-Kumar on equivariant \(K\)-theory, and the papers by Billey and Willems, that inspired the research developed in this amazing step forward a generalized cohomology Schubert calculus. Schubert calculus; equivariant oriented cohomology; flag variety; root polynomial; hyperbolic formal group law ] C. Lenart and K. Zainoulline, Towards generalized cohomology Schubert calculus via formal root polynomials, arXiv:1408.5952. Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Equivariant \(K\)-theory, Generalized (extraordinary) homology and cohomology theories in algebraic topology, Bordism and cobordism theories and formal group laws in algebraic topology, Algebraic combinatorics Towards generalized cohomology Schubert calculus via formal root polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur's \(P\) and \(Q\) functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type \(C\) Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one. integral homology rings; integral cohomology rings [27] Thomas Lam, Anne Schilling &aMark Shimozono, &\(K\)-theory Schubert calculus of the affine Grassmannian&#xCompos. Math.146 (2010) no. 4, p. 811Article | &MR 26 | &Zbl 1256. Symmetric functions and generalizations, Classical problems, Schubert calculus Schubert polynomials for the affine Grassmannian of the symplectic group

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the cohomology or Chow ring of a homogeneous space $G/P$. The classes of Schubert varieties of $G/P$ form a ``geometric'' basis. In Schubert calculus one studies the structure constants of the ring with respect to this basis -- c.f. Littlewood-Richardson coefficients. Much is known about cohomological (Chow) Schubert calculus, and even about its $K$-theory generalization. \par The paper under review studies an even more general cohomology theory, in fact the universal oriented algebraic cohomology theory: algebraic cobordism. In such generality the classes of Schubert varieties are not well defined, they depend on choices. The authors make their choices (Bott-Samelson resolution), and prove a formula for the product of the class of a smooth Schubert variety with an arbitrary Bott-Samelson class -- for type A Grassmannians. The last sections of the paper also establish some results on polynomials (``generalized Schubert polynomials'') representing Schubert varieties for hyperbolic formal group laws. Schubert calculus; cobordism; Grassmannian; generalized cohomology Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Connective \(K\)-theory, cobordism, Bordism and cobordism theories and formal group laws in algebraic topology Smooth Schubert varieties and generalized Schubert polynomials in algebraic cobordism of Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In a recent preprint, \textit{E. Carlsson} and \textit{A. Oblomkov} [``Affine Schubert calculus and double coinvariants'', Preprint, \url{arXiv:1801.09033}] obtain a long sought-after monomial basis for the ring \(\text{DR}_n\) of diagonal coinvariants. Their basis is closely related to the ``schedules'' formula for the Hilbert series of \(\text{DR}_n\) which was conjectured by \textit{J. Haglund} and \textit{N. Loehr} [Discrete Math. 298, No. 1--3, 189--204 (2005; Zbl 1070.05007)] and first proved by \textit{E. Carlsson} and \textit{A. Mellit} [J. Am. Math. Soc. 31, No. 3, 661--697 (2018; Zbl 1387.05265)], as a consequence of their proof of the famous Shuffle Conjecture. In this article, we obtain a schedules formula for the combinatorial side of the Delta Conjecture, a conjecture introduced by \textit{J. Haglund} et al. [Trans. Am. Math. Soc. 370, No. 6, 4029--4057 (2018; Zbl 1383.05308)], which contains the Shuffle Theorem as a special case. Motivated by the Carlsson-Oblomkov basis for \(\text{DR}_n\) and our Delta schedules formula, we introduce a (conjectural) basis for the super-diagonal coinvariant ring \(\text{SDR}_n\), an \(S_n\)-module generalizing \(\text{DR}_n\) introduced recently by \textit{M. Zabrocki} [``A module for the delta conjecture'', Preprint, \url{arXiv:1902.08966}], which conjecturally corresponds to the Delta Conjecture. delta conjecture; parking function; coinvariant ring; super-space Combinatorial aspects of representation theory, Symmetric functions and generalizations, Classical problems, Schubert calculus Schedules and the Delta Conjecture

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives a survey of recent joint work with \textit{A. S. Buch} and \textit{H. Tamvakis} [J. Am. Math. Soc. 16, No. 4, 901--915 (2003; Zbl 1063.53090) and Adv. Math. 185, No. 1, 80--90 (2004; Zbl 1053.05121)] on the quantum cohomology of Grassmannians for classical Lie types. The starting point is a review of both classical and quantum formulae for \(\text{Gr}(k,n)\), including a description of the Gromov-Witten invariants as certain classical structure constants on a partial flag variety \(F(k-d;k+d;n)\). By using a conjectural interpretation of the classical cohomology of a partial flag variety by \textit{A. Knutson, T.C. Tao} and \textit{C. T. Woodward} [J. Am. Math. Soc. 17, 19--48 (2004; Zbl 1043.05111)], the author derives a combinatorial description for genus 0 Gromov-Witten invariants. This conjectural quantum Littlewood-Richardson formula has been proved for \(k\leq 3\) and computer-checked for \(n\leq 16\). Finally, Grassmannians of the other classical types \(B\), \(C\) and \(D\) are considered, and a quantum Pieri formula is announced. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is a survey paper about representations of \(U(\mathfrak b)\) and \(KP\) modules. A good example for \(U(\mathfrak b)\) are Demazure modules. They are given by spaces of sections of line bundles on Schubert varieties in flag manifolds. They were invented by \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér. (4) 7, 53--88 (1974; Zbl 0312.14009)]. On the other hand \(KP\) modules, though related to Schubert varieties, are different. In the beginning of \(1980\)s \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)] discovered Schubert polynomials. They are certain polynomial lifts of cohomology classes of Schubert varieties in flag manifolds. Lascoux conjectured in \(1983\): There should be a functorial version of this construction. It was solved affirmatively by \textit{W. Kraskiewicz} and the author [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)]. The so-obtained modules were called Kraskiewicz-Pragacz modules in short \(KP\) modules by \textit{M. Watanabe} [J. Algebra 443, 422--429 (2015; Zbl 1326.14128)]. KP modules; Demazure modules; highest weight represenations; Schubert varieties; flag manifolds; Schur functors; Schur functions Classical problems, Schubert calculus On a certain family of \(U( \mathfrak{b})\)-modules

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials How to multiply two Schubert polynomials is a notorious open problem in Schubert calculus. The author addresses the special case where a Schubert polynomial is multiplied by a Schur polynomial. His result is a (not very efficient but still beautiful) description of the expansion coefficients in this product as the number of pairs of an RC-graph as introduced by \textit{S. Fomin} and \textit{A. N. Kirillov} [Discrete Math. 153, 123-143 (1996; Zbl 0852.05078)] and a Young tableau, which have to be related in a certain way. The proof of this result is entirely based on the insertion algorithm for RC-graphs due to \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, 257-269 (1993; Zbl 0803.05054)]. Schubert polynomials; Schur functions; Littlewood-Richardson rule; Monk's rule; Pieri's rule; RC-graphs Kogan, M.: RC-graphs and a generalized Littlewood--Richardson rule. Int. Math. Res. Not. 2001(15), 765--782 (2001) Algebraic combinatorics, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus RC-graphs and a generalized Littlewood-Richardson rule

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert varieties appear in a natural way in many parts of mathematics and its applications. They are on the crossroad of representation theory, combinatorics and algebraic, analytic and differential geometry. The structure of the singularities of the Schubert varieties is extremely complicated, but still manageable. This makes Schubert varieties extremely interesting objects and explain why the work on these varieties has contributed so much to mathematics. There is a vast literature on their properties, in particular on their singularities and their cohomology. The literature list of the present book contains nearly 160 items, most of them on singularities, and most of them are written after 1984. The book started out as a survey article. This is reflected in the text which consists mostly of explanation of concepts and statements of results. There are very few proofs, and the background material is limited to a minimum. In spite of this the book is more than 200 pages long. This illustrates well the size of the task that the authors have undertaken. That they have succeeded in writing a readable text is partly due to the fact that they both are experts in the area. About one fifth of the articles in the reference list are written by the authors. The book is closer to a survey of recent work on the singularities of Schubert varieties, than a textbook. What distinguishes it from a survey is that it contains a wealth of examples and calculations. This makes the book extremely valuable to those that want to learn the area. The contents of the book is: Chapter 2: This chapter contains background material on Schubert varieties, Bruhat-Chevalley orders and parabolic groups, weights, algebraic groups, Weyl groups, and representations of semi-simple algebraic groups. It would take several courses, and books, to cover properly this material alone. -- Chapter 3: To compensate for the rapid introduction of concepts in chapter 2 this chapter contains concrete computations of the Bruhat-Chevalley order in the classical groups, the grassmannians, the flag manifolds and their Schubert varieties. -- Chapter 4: This chapter is devoted to the study of the tangent space to a Schubert variety. -- Chapter 5: In this chapter results by P. Polo on the singular locus of a Schubert variety is recalled, and Polo's root system description of the tangent space is described. This is used to obtain a very detailed treatment of the classical groups. -- Chapter 6: Here Kazhdan-Lusztig polynomials and their use for characterizing rational singularities is explained. -- Chapter 7: In this chapter Shrawan Kumars criteria for smoothness, and generic smoothness are discussed. The relation between nil-Hecke rings and characters on the tangent cone is explained. This chapter also contains some proofs. -- Chapter 8: In this chapter combinatorial algorithms for testing smoothness and rational smoothness of Schubert varieties for the classical groups are given. The last four chapters are devoted to the singularities of special varieties. -- Chapter 9: In this chapter the minuscule and cominuscule cases are discussed. The major part is devoted to the small resolutions of A. Zelevinsky and of P. Sankaran and P. Vanchinatan. Also the description of their tangent space made by M. Brion and Polo is given. -- Chapter 10: This chapter is devoted to the rank \(2\) case. -- Chapter 11: In this chapter some results on the relation between smoothness and factorization of the Poincaré polynomial are described. -- Chapter 12: In this chapter determinantal varieties are treated. The book contains a wealth of results, and a large number of very useful and illustrating examples. It treats many central topics. The authors seem to concentrate on the work treating singularities of Schubert varieties only. There is a large body of work on the cohomology of manifold where the singularities of the Schubert varieties enter in an essential way, and where many of the techniques for treating singularities originated. It would have been very interesting to have a short description of this work and to get an explanation of the relations to the work on the singularities of Schubert varieties. Also it would have made the reference list more complete. The book is well written, and the background material clearly presented. It is valuable to both experts and beginners. Beginners will find a way to get an idea about the area before indulging into the large and technically complicated background material. Together with the experts they will find useful detailed examples, together with lists of minimal bad patterns and the singular loci of the special groups. The experts will find a wealth of interesting material, much of it new. Schubert varieties; singular loci; roots; Weyl groups; tangent cone; parabolic groups; Bruhat-Chevalley orders; grassmannian; Kazhdan-Lusztig polynomials; classical groups; nil-Hecke ring; minuscule groups; Poincaré polynomial; determinantal varieties; Dynkin diagrams Billey, Sara; Lakshmibai, V., Singular loci of Schubert varieties, Progress in Mathematics, vol. 182, (2000), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA, MR 1782635 Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Determinantal varieties Singular loci of Schubert varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give an explicit and entirely poset-theoretic way to compute, for any permutation \(v\), all the Kazhdan-Lusztig polynomials \(P(x,y)\) for \(x,y\leq v\), starting from the Bruhat interval \([e,v]\) as an abstract poset. This proves, in particular, that the intersection cohomology of Schubert varieties depends only on the inclusion relations between the closures of its Schubert cells. Kazhdan-Lusztig polynomials; posets; Bruhat order F. Brenti, ''The intersection cohomology of Schubert varieties is a combinatorial invariant,'' Europ. J. Combin. 25 (2004), 1151--1167. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorics of partially ordered sets The intersection cohomology of Schubert varieties is a combinatorial invariant

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A theorem of Witten asserts that the fusion ring \(\mathcal{F}(\widehat{\mathfrak{gl}}(n))_k\) of integrable highest weight \(\widehat{\mathfrak{gl}}(n)\)-modules at level \(k\) is isomorphic to the specialization at \(q=1\) of the quantum cohomology ring \(qH^{\bullet}(\mathrm{Gr}_{k,n+k})_{q=1}\) for the Grassmannian \(\mathrm{Gr}_{k,n+k}\). The main aim of the paper under review is to give a mathematically rigorous realization of \(\mathcal{F}(\widehat{\mathfrak{sl}}(n))_k\) as a quotient of \(qH^{\bullet}(\mathrm{Gr}_{k,n+k})\) with the defining relations explained via Bethe Ansatz equations of a quantum integrable system. Both rings are also described combinatorially in terms of symmetric polynomials (Schur polynomials) in pairwise noncommuting variables. The starting point of authors' analysis is an observation that generating function of the elementary symmetric polynomials turns out to be the transfer matrix for a quantum integrable system. As a consequence, the authors obtain a simple particle formulation of both rings leading to new recursion formulae for the structure constants of the fusion ring as well as for Gromov-Witten invariants. The proposed approach also leads naturally to the Verlinde formula and equips the combinatorial fusion algebra with a \(\mathrm{PSL}(2,\mathbb{Z})\)-action. quantum cohomology; integrable module; plactic algebra; Hall algebra; Bethe Ansatz; fusion ring; Verlinde algebra; symmetric function Korff, C.; Stroppel, C., The \(s l(n)\)-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology, Adv. Math., 225, 200, (2010) Quantum groups (quantized enveloping algebras) and related deformations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Symmetric functions and generalizations, Exactly solvable models; Bethe ansatz, Inverse scattering problems in quantum theory The \(\widehat {\mathfrak {sl}}(n)_k\)-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Q\) be a quiver of extended Dynkin type \(\tilde{D_n}\). In this first of two papers, we show that the quiver Grassmannian \(\text{Gr}_e(M)\) has a decomposition into affine spaces for every dimension vector \(e\) and every indecomposable representation of defect \(-1\) and defect \(0\), with exception of the non-Schurian representations in homogeneous tubes. We characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for \(M\). The method of proof is to exhibit explicit equations for the Schubert cells of and to solve this system of equations successively in linear terms. This leads to an intricate combinatorial problem, for whose solution we develop the theory of Schubert systems. In Part 2 [the authors, ``Quiver Grassmannians of type \(D_n\). Part 2: Schubert decompositions and \(F\)-polynomials'', Preprint, \url{arXiv:1507.00395}] we extend the result of this paper to all indecomposable representations \(M\) of \(Q\) and determine explicit formulae for the \(F\)-polynomial of \(M\). Research exposition (monographs, survey articles) pertaining to commutative algebra, Cluster algebras, Topological properties in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Representations of quivers and partially ordered sets, Combinatorial aspects of representation theory, Homogeneous spaces and generalizations, Representation type (finite, tame, wild, etc.) of associative algebras Quiver Grassmannians of extended Dynkin type \(D\). Part 1: Schubert systems and decompositions into affine spaces

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In these notes, we present some fundamental results concerning flag varieties and their Schubert varieties. By a flag variety, we mean a complex projective algebraic variety \(X\), homogeneous under a complex linear algebraic group. The orbits of a Borel subgroup form a stratification of \(X\) into Schubert cells. These are isomorphic to affine spaces; their closures in \(X\) are the Schubert varieties, generally singular. The classes of the Schubert varieties form an additive basis of the cohomology ring, and one easily shows that the structure constants in this basis are all non-negative. Our main goal is to prove a related, but more hidden, statement in the Grothendieck ring \(K(X)\) of coherent sheaves on \(X\). This ring admits an additive basis formed of structure sheaves of Schubert varieties, and the corresponding structure constants turn out to have alternating signs. These structure constants admit combinatorial expressions in the case of Grassmannians. This displayed their alternation of signs, and Buch conjectured that this property extends to all the flag varieties. In this setting, the structure con- stants of the cohomology ring (a fortiori, those of the Grothendieck ring) are yet combinatorially elusive, and Buch's conjecture was proved in [the author, J. Algebra 258, No. 1, 137--159 (2002; Zbl 1052.14054)] by purely algebro- geometric methods. Here we have endeavored to give a self-contained exposition of this proof. The main ingredients are geometric properties of Schubert varieties (e.g., their normality), and vanishing theorems for cohomology of line bundles on these varieties (these are deduced from the Kawamata-Viehweg theorem, a powerful generalization of the Kodaira vanishing theorem in complex geometry). Of importance are also the intersections of Schubert varieties with opposite Schubert varieties. These ``Richardson varieties'' are systematically used in these notes to provide geometric explanations for many formulae in the cohomology or Grothendieck ring of flag varieties. Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes Lectures on the geometry of flag varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a new description of the Pieri rule for \(k\)-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and \(k\)-Schur functions. \(k\)-Schur functions; Pieri rule; Bruhat order; Macdonald polynomials; Hall-Littlewood polynomials; \(k\)-tableaux Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorial aspects of representation theory, Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds The ABC's of affine Grassmannians and Hall-Littlewood polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We present a partial generalization of the classical Littlewood-Richardson rule (in its version based on Schützenberger's jeu de taquin) to Schubert calculus on flag varieties. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin's growth diagrams (for chains in Young's lattice of partitions) to chains of permutations in the so-called \(k\)-Bruhat order. Our work is based on the recent thesis of Beligan [\textit{M. Beligan}, ``Insertion for tableaux of transpositions. A generalization of Schensted's algorithm,'' PhD thesis, York University, Canada (2007)], in which he generalizes the classical plactic structure on words to chains in certain intervals in \(k\)-Bruhat order. Potential applications of our work include the generalization of the \(S_{3}\)-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams. Schubert calculus; flag variety; Littlewood-Richardson rule; plactic relation; jeu de taquin; growth diagram; \(k\)-Bruhat order Lam, T., Lauve, A., Sottile, F.: Skew Littlewood-Richardson rules for Hopf algebras. IMRN (2010). arXiv:0908.3714 Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Growth diagrams for the Schubert multiplication

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider \(Gr\), the Grassmannian of \(n\)-dimensional subspaces of the space of degree at most \(2n\) polynomials in \(z\). The Wronskian map \(Wr\) maps \(Gr\) to \(P(V)\) where \(V\) is the space of degree \(n(n+1)\) polynomials in \(z\). If \(h\in V\) has only real roots, then there is a well defined bijection between \(Wr^{-1}(h)\) and a certain set of Young tableaux. Young tableaux are combinatorial objects relevant in the combinatorial description of the ring structure of Grassmannians. This bijection is a consequence of the proof of Mukhin-Tarasov-Varchenko of the Shapiro-Shapiro conjecutre. The paper under review considers a subset \(OGr\) of \(Gr\), the orthogonal Grassmannian. The \(Wr\)-value of points in \(OGr\) are perfect squares. The main theorem of the paper is the following: Suppose \(Wr(x)\) is a perfect square with only real roots. Then \(x\in OGr\) if and only if the tableau corresponding to \(x\) has a certain combinatorial symmetry. Since the combinatorics of the symmetric tableaux is similar to the combinatorics of the earlier studied ``standard shifted tableaux'', the author can provide an elegant geometric proof of the Littlewood-Richardson rule for the orthogonal Grassmannian. Schubert calculus; Wronski map; orthogonal Grassmannian; symmetric tableaux Purbhoo, K., The Wronski map and shifted tableau theory, Int. math. res. not. IMRN, 24, 5706-5719, (2011) Classical problems, Schubert calculus The Wronski map and shifted tableau theory

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Two conjectures about the characters of the Hecke algebra \(H_ n(q)\) of type \(A_{n-1}\), evaluated at elements of its Kazhdan-Lusztig basis are announced and some supporting evidence for the two conjectures is provided. One of the two conjectures asserts that certain integral linear combinations of irreducible characters take values on the Kazhdan-Lusztig basis which are polynomials in \(q\) with nonnegative, symmetric, and unimodal integer coefficients. Certain permutations, called codominant, whose corresponding Schubert varieties are smooth and very simple to describe, can be picked. The second conjecture states that for each Kazhdan-Lusztig basis element \(C_ w'\) there is a sum \(C_{w_ 1}'+ \cdots + C_{w_ k}'\) of basis elements with \(w_ j\) codominant, such that \(\chi(C_ w') = \chi(C_{w_ 1}' + \cdots + C_{w_ k}')\) for every Hecke algebra character \(\chi\). A conjectured immanant inequality for Jacobi-Trudi matrices defined in this paper is proved and it is shown how the two conjectures mentioned above would imply stronger inequalities of a similar kind. codominant permutations; Hecke algebra; Kazhdan-Lusztig basis; irreducible characters; Schubert varieties; immanant inequality for Jacobi-Trudi matrices M. Haiman. ''Hecke algebra characters and immanant conjectures''. J. Amer. Math. Soc. 6 (1993), pp. 569--595.DOI. Representation theory for linear algebraic groups, Linear algebraic groups over finite fields, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Modular representations and characters, Grassmannians, Schubert varieties, flag manifolds, Other matrix groups over fields Hecke algebra characters and immanant conjectures

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{V. Kreiman} and \textit{V. Lakshmibai} [in: Algebra, arithmetic and geometry with applications (Berlin: Springer) (2003; Zbl 1092.14060)] described two conjectures on the multiplicity at a singular point \(P\) on a Schubert variety \(X\) in the Grassmannian. While conjecture 1 describes the Hilbert series of the tangent cone at \(P\), conjecture 2 gives a formula for the multiplicity of \(X\) at \(P\). In this paper, the author gives a combinatorial proof of conjecture 2. The author also describes an equivalent combinatorial formulation of conjecture 1, i.e., the Hilbert series of the tangent cone at \(P\). Together with the recent work of \textit{V. Kodiyalam} and \textit{K. N. Raghavan} [J. Algebra 270, No. 1, 28--54 (2003; Zbl 1083.14056)] wherein the authors give a proof of the validity of conjecture 1, the author's formulation (in this paper) gives a nice combinatorial description of the Hilbert series of tangent cones. This paper makes a nice contribution to the singularities of Schubert varieties. For part I, cf. [Sémin. Lothar. Comb. 45, B45c, 11 p. (2000; Zbl 0965.14023)]. Hilbert series C. Krattenthaler, On multiplicities of points on Schubert varieties in Graßmannians. II, J. Algebraic Combin. 22 (2005), no. 3, 273 -- 288. Grassmannians, Schubert varieties, flag manifolds, Exact enumeration problems, generating functions, Singularities of curves, local rings On multiplicities of points on Schubert varieties in Grassmannians. II

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study a category of Whittaker modules over a complex semisimple Lie algebra by realizing it as a category of twisted \(\mathcal{D}\)-modules on the associated flag variety using Beilinson-Bernstein localization. The main result of this paper is the development of a geometric algorithm for computing the composition multiplicities of standard Whittaker modules. This algorithm establishes that these multiplicities are determined by a collection of polynomials we refer to as Whittaker Kazhdan-Lusztig polynomials. In the case of trivial nilpotent character, this algorithm specializes to the usual algorithm for computing multiplicities of composition factors of Verma modules using Kazhdan-Lusztig polynomials. Whittaker modules; D-modules; localization of representations; Kazhdan-Lusztig polynomials Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials A Kazhdan-Lusztig algorithm for Whittaker modules

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathbf G}{\mathbf r}\) be the affine Grassmannian for a connected complex reductive group \(G\). Let \({\mathcal C}_G\) be the complex vector space spanned by (equivalence classes of) Mirkovic-Vilonen cycles in \({\mathbf G}{\mathbf r}\). The Beilinson-Drinfeld Grassmannian can be used to define a convolution product on MV-cycles, making \({\mathcal C}_G\) into a commutative algebra. We show, in type A, that \({\mathcal C}_G\) is isomorphic to \(\mathbb{C}[N]\), the algebra of functions on the unipotent radical \(N\) of a Borel subgroup of \(G\): then each MV-cycle defines a polynomial in \(\mathbb{C}[N]\). which we call an MV-polynomial. We conjecture that those MV-polynomials which are cluster monomials for a Fomin-Zelevinsky cluster algebra structure on \(\mathbb{C}[N]\) are naturally expressible as determinants, and we conjecture a formula for many of them. J. Anderson and M. Kogan, ''The algebra of Mirković-Vilonen cycles in type A,'' Pure Appl. Math. Q., vol. 2, iss. 4, part 2, pp. 1187-1215, 2006. Classical groups (algebro-geometric aspects) The algebra of Mirković-Vilonen cycles in type A

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper takes into consideration that ``the saturation theorem of \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] concerns the nonvanishing of Littlewood-Richardson coefficients''. Further, ``in combination with work of \textit{A. A. Klyachko} [Sel. Math., New Ser. 4, No. 3, 419--445 (1998; Zbl 0915.14010)], it implies \textit{A. Horn}'s conjecture [Pac. J. Math. 12, 225--241 (1962; Zbl 0112.01501)] about eigenvalues of sums of Hermitian matrices''. Then ``the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians'' are illustrated. The main result of this paper deals with an extension of Schubert calculus presenting a Schubert calculus interpretation of Friedland's problem, via equivariant cohomology of Grassmannians, and it derives a saturation theorem for this setting. eigenvalue problem; equivariant cohomology; Schubert calculus; Littlewood-Richardson coefficients; Hermitian matrices; Grassmannians Inequalities involving eigenvalues and eigenvectors, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Equivariant algebraic topology of manifolds Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give bijective proofs of Monk's rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams. Schubert polynomials; bumpless pipe dreams Symmetric functions and generalizations, Classical problems, Schubert calculus Bijective proofs of Monk's rule for Schubert and double Schubert polynomials with bumpless pipe dreams

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials While genus-zero Gromov-Witten invariants (quantum cohomology) of toric varieties are relatively well understood the same can not be said about other varieties. \textit{A. Bertram} and two of the current authors conjectured [J. Algebr. Geom. 17, No. 2, 275--294 (2008; Zbl 1166.14035)] a correspondence between Gromov-Witten potentials of nonabelian and abelian GIT quotients \(X//G\) and \(X//T\) of a projective variety \(X\). Here \(G\) is a reductive Lie group acting holomorphically on \(X\) with a maximal torus \(T\). This paper extends the conjecture to the equivariant setting and restates it in terms of Frobenius structures on big quantum cohomology rings \(QH^*(X//G)\) and \(QH^*(X//T)\). The authors also describe a formulation of the correspondence in terms of big \(J\)-functions of Givental and show that it can often be reduced to a relation between small \(J\)-functions via the Gromov-Witten reconstruction theorems. They give a proof of the conjecture for partial flag manifolds \(X//G\) based on this reduction. The main idea is as follows. It is known that any \(\sigma\in H^*(X//G)\) admits a (non-unique) lift \(\widetilde{\sigma}\) to the Weyl invariant subspace of \(H^*(X//T)\) and this lifting respects cup-products with the Weyl anti-invariant class \(\omega\), i.e. \[ \widetilde{(\sigma\cup_{X//G}\sigma')}\cup\omega =\widetilde{\sigma}\cup_{X//T}(\widetilde{\sigma}'\cup\omega). \] This fails for big quantum products \(*\), but the authors conjecture that the equality can be saved by replacing \(\widetilde{\sigma}\) with \(\xi\) satifying \(\xi\,*_{X//T}\,\omega=\widetilde{\sigma}\,\cup\,\omega\). Thus we get \[ (\widetilde{(\sigma*_{X//G}\sigma')}\cup\omega)(t) =(\xi*_{X//T}(\widetilde{\sigma}'\cup\omega))(\widetilde{t}), \] after an explicit change of coordinates \(\widetilde{t}(t)\). At the level of Gromov-Witten invariants this means that the naive expression for \(\langle\sigma_1,\dots,\sigma_n\rangle_{0,n,\beta}\) receives correction terms from sums of products of invariants of \(X//T\) of the same type. The appearence of \(\xi\) is most naturally explaind in terms of Frobenius structures. The above lifting provides a canonical Frobenius structure on the Novikov ring \(N(X//G)\). However, the new flat coordinates are not the same as for the standard structure and therefore \(\widetilde{\sigma}\) are not horizontal. The horizontal fields are exactly \(\xi\) and \(\widetilde{t}(t)\) is the corresponding change of coordinates. I. Ciocan-Fontanine, B. Kim, C. Sabbah, The abelian/nonabelian correspondence and Frobenius manifolds. \textit{Invent. Math}. \textbf{171} (2008), 301-343. MR2367022 Zbl 1164.14012 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Group actions on varieties or schemes (quotients) The abelian/nonabelian correspondence and Frobenius manifolds

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors give a criterion for a Schubert variety \(X_ w\) in the flag variety SL(n)/B to be smooth. The Schubert varieties \(X_ w\) in SL(n)/B are indexed by the elements \(w\in W\) of the Weyl group of SL(n). Each \(w\in W\) can be represented by a permutation on n symbols, \(w=(a_ 1,...,a_ n)\). The authors prove: \(X_ w\) is singular iff there exist \(1\leq i<j<k<\ell \leq n\) such that either \(a_ k<a_{\ell}<a_ i<a_ j\) or \(a_{\ell}<a_ j<a_ k<a_ i\). This theorem is based on the authors' previous work describing a natural basis for \(H^ 0(X,L)\) where L is an ample line bundle on SL(n)/B (``standard monomial theory'') [see, e.g., \textit{V. Lakshmibai}, \textit{C. Musili} and \textit{C. S. Seshadri}, Proc. Indian Acad. Sci., Sect. A, Part III 88, No.4, 279-362 (1979; Zbl 0447.14013)]. Schubert variety; flag variety; standard monomial Lakshmibai, V.; Sandhya, B., Criterion for smoothness of Schubert varieties in \(\operatorname{Sl}(n) / B\), Proc. Indian Acad. Sci. Math. Sci., 100, 1, 45-52, (1990), MR 1051089 Grassmannians, Schubert varieties, flag manifolds Criterion for smoothness of Schubert varieties in Sl(n)/B

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We state a precise conjectural isomorphism between localizations of the equivariant quantum \(K\)-theory ring of a flag variety and the equivariant \(K\)-homology ring of the affine Grassmannian, in particular relating their Schubert bases and structure constants. This generalizes Peterson's isomorphism in (co)homology. We prove a formula for the Pontryagin structure constants in the \(K\)-homology ring, and we use it to check our conjecture in few situations. flag manifolds; affine Grassmannian; quantum \(K\)-theory; \(K\)-homology of the affine Grassmannian Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Equivariant \(K\)-theory A conjectural Peterson isomorphism in \(K\)-theory

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Wir betrachten graduierte Bimoduln über der symmetrischen Algebra \(S\) des Dualraums der Cartanschen Unteralgebra einer Kac-Moody-Algebra, spezieller diejenigen, die man als direkte Summanden iterierter Tensorprodukte von \(S\) über den Invarianten (in \(S\)) von einfachen Spiegelungen erhält. Es wird gezeigt, daß sich die Heckealgebra der Weylgruppe von \(\mathfrak g\) so in den zerfällenden Grothendieckring der Kategorie dieser Bimoduln einbetten lässt, daß die Elemente ihrer Kazhdan-Lusztig-Basis genau den unzerlegbaren unter den betrachteten Bimoduln entsprechen. Dazu wird der rekursive Algorithmus zur Berechnung der Kazhdan-Lusztig-Basis auf die Geometrie der Fahnenmannigfaltigkeit der zugehörigen Kac-Moody-Gruppe übertragen. Die unzerlegbaren Bimoduln entstehen dann als äquivariante Schnittkohomologie von Schubertvarietäten in dieser Fahnenmannigfaltigkeit. Die Arbeit verallgemeinert Ergebnisse von W. Soergel, mit deren Hilfe sich für eine halbeinfache Liealgebra \(\mathfrak g\) die Struktur der Kategorie \(\mathcal O\) von Bernstein, Gelfand und Gelfand beschreiben lässt. Equivariant intersection cohomology; flag variety; Schubert variety; Kazhdan-Lusztig basis M. Härterich, Kazhdan-Lusztig-Basen, unzerlegbare Bimoduln und die Topologie der Fahnenmannigfaltigkeit einer Kac-Moody-Gruppe, 1999. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Loop groups and related constructions, group-theoretic treatment, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Hecke algebras and their representations Kazhdan-Lusztig bases, indecomposable bimodules and the topology of flag manifolds of a Kac-Moody group

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider a complex vector space \(V\) of dimension \(N\) equipped with a nondegenerate symmetric form. Choose an integer \(m < N/2\) and consider the Grassmannian \(\mathrm{OG}=\mathrm{OG}(m; N)\) parametrizing isotropic \(m\)-dimensional subspaces of \(V\). In this paper the authors prove a Giambelli formula expressing the Schubert classes on OG as polynomials in certain special Schubert classes that generate the cohomology ring \(H^* (\mathrm{OGF};\mathbb Z)\) for the even \(N\). Giambelli formula; orthohogonal Grassmannians; Schubert classes; Dynkin diagrams Buch, A.; Kresch, A.; Tamvakis, H., \textit{A Giambelli formula for even orthogonal Grassmannians}, J. Reine Angew. Math., 708, 17-48, (2015) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Equivariant algebraic topology of manifolds A Giambelli formula for even orthogonal Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce edge labeled Young tableaux. Our main results provide a corresponding analogue of Schützenberger's theory of jeu de taquin. These are applied to the equivariant Schubert calculus of Grassmannians. Reinterpreting, we present new (semi)standard tableaux to study factorial Schur polynomials, after Biedenharn-Louck, Macdonald, Goulden-Greene, and others. Consequently, we obtain new combinatorial rules for the Schubert structure coefficients, complementing work of \textit{A. I. Molev} and \textit{B. E. Sagan} [Trans. Am. Math. Soc. 351, No. 11, 4429--4443 (1999; Zbl 0972.05053)], \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)], \textit{A. I. Molev} [J. Algebra 321, No. 11, 3450--3468 (2009; Zbl 1169.05050)], and \textit{V. Kreiman} [Trans. Am. Math. Soc. 362, No. 5, 2589--2617 (2010; Zbl 1205.05244)]. We also describe a conjectural generalization of one of our rules to the equivariant \(K\)-theory of Grassmannians, extending our previous work on non-equivariant \(K\)-theory. This conjecture concretely realizes the ``positivity'' known to exist by a result of \textit{D. Anderson} et al. [J. Eur. Math. Soc. (JEMS) 13, No. 1, 57--84 (2011; Zbl 1213.19003)]. It provides an alternative to the conjectural rule of Knutson-Vakil reported in [\textit{I. Coskun} and \textit{R. Vakil}, Proc. Symp. Pure Math. 80, 77--124 (2009; Zbl 1184.14080)]. Schubert calculus; equivariant cohomology; Grassmannians; jeu de taquin Thomas, H.; Yong, A.: Equivariant Schubert calculus and jeu de taquin. Ann. inst. Fourier (2015) Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Symmetric functions and generalizations, Equivariant algebraic topology of manifolds Equivariant Schubert calculus and jeu de taquin

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The survey is devoted to the concept of ``standard monomials'', introduced in the 1940's by Hodge to study the Schubert varieties in the Grassmannian/flag varieties, and developed in the 1970's into a ``theory'' by Seshadri, in collaboration with Musili, Lakshmibai, Littelmann, etc., to study the Schubert varieties in the generalized flag varieties \(G/Q\), where \(G\) is a semisimple group and \(Q\) is a parabolic subgroup of \(G\). The author begins with the classical results on the Grassmannians and the Schubert varieties, such as the Plücker coordinates, the quadratic relations and the standard monomials in the Plücker coordinates that form a basis in the algebra of regular functions on the affine cones over the Grassmannians. This nice basis, parametrized by the standard Young tableaux, allows to prove many important geometric properties of the Grassmannians and the Schubert varieties -- the projective normality, the projective factoriality (for the Grassmannians), the projective Cohen-Macaulay property. In the next sections possible generalizations to the cases of the flag variaties corresponding to irreducible \(G\)-modules with 1) a minuscule highest weight; 2) a quasi-minuscule highest weight; 3) a classical type highest weight are given. Some applications of the theory to the study of determinantal varieties, ladder determinantal varieties, varieties of idempotents, varieties of complexes, quiver varieties and singular loci of the Schubert varieties are discussed. For the later developments of the standard monomial theory see the preceding review [\textit{V.Lakshmibai}, in: A tribute to C. S. Seshadri. Birkhäuser, Trends in Mathematics, 283--309 (2003; Zbl 1056.14065)], which may be found in the same volume. This report may be recommended both to beginners who are looking for main definitions, problems and results in the area, and to specialists who would like to have a systematic historical review of the subject. Grassmannians; semisimple algebraic groups; flag varieties; Schubert varieties; minuscule weights Musili, C.: The development of standard monomial theory. I. In: A tribute to C. S. Seshadri, Chennai, 2002. Trends Math., pp. 385--420. Birkhäuser, Basel (2003) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), History of mathematics in the 20th century The development of standard monomial theory. I

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(w_0,\dots,w_n\) be strictly positive integers. The author proves the following mirror theorem: the Frobenius manifold associated to the orbifold quantum cohomology of the weighted projective space \({\mathbb P}(w_0,\dots,w_n)\) is isomorphic to the one attached to the Laurent polynomial \(f(u_0,\dots,u_n)=u_0+\cdots+u_n\) restricted to the region \(U=\{(u_0,\dots,u_n)\in {\mathbb C}^{n+1}\,| \, \prod_iu_i^{w_i}=1\}\). This generalizes Barannikov's isomorphism between the quantum cohomology of \({\mathbb P}^n({\mathbb C})\) and the Frobenius manifold associated to the Laurent polynomial \(x_1+x_2+\cdots+x_n+(x_1\cdots x_n)^{-1}\) [\textit{S.~Barannikov}, Semi-infinite Hodge structures and mirror symmetry for projective spaces, \url{arXiv:math/0010157}]. In particular, at the classical level, one has an isomorphism of Frobenius algebras \[ \left(H^{2\bullet}_{\text{ orb}}({\mathbb P}(w_0,\dots,w_n);{\mathbb C}),\cup,\langle\cdot,\cdot\rangle\right)\simeq\left(\text{ gr}^{\mathcal N}_\bullet\left(\Omega^n(U)/df\wedge\Omega^{n-1}(U)\right),\cup,[\![g]\!](\cdot,\cdot)\right). \] On the left hand side, \(\cup\) denotes the orbifold cup product, and \(\langle\cdot,\cdot\rangle\) the orbifold Poincaré duality [\textit{W.~Chen, Y.~Ruan}, Commun. Math. Phys. 248, No. 1, 1--31 (2004; Zbl 1063.53091)]. On the right hand side, \({\mathcal N}\) denotes the Newton filtration on the vector space \(\Omega^n(U)/df\wedge\Omega^{n-1}(U)\); the associative product \(\cup\) and the pairing \([\![g]\!](\cdot,\cdot)\) come from the Jacobian algebra of \(f\) via the linear isomorphism with \(\Omega^n(U)/df\wedge\Omega^{n-1}(U)\) induced by a volume form on \(U\) [\textit{A.~Douai, C.~ Sabbah}, Ann. Inst. Fourier 53, No. 4, 1055--1116 (2003; Zbl 1079.32016)]. Moreover, the author proves a reconstruction theorem. Namely, that the full genus 0 Gromov-Witten potential of \({\mathbb P}(w_0,\dots,w_n)\) can be reconstructed from the 3-point invariants. Frobenius manifolds; quantum cohomology; orbifolds [18] Étienne Mann, &Orbifold quantum cohomology of weighted projective spaces&#xJ. Algebraic Geom.17 (2008) no. 1, p.~137Article | &MR~23 | &Zbl~1146. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Orbifold quantum cohomology of weighted projective spaces

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This interesting article is a survey of recent results about a conjecture of Boris and Michael Shapiro on real solutions of certain enumerative problems. Various formulations of this conjecture exist, regarding Grassmannians or more general flag manifolds. For the Grassmannians the conjecture has been proved to be true by \textit{E. Mukhin, V. Tarasov} and \textit{A. Varchenko} [Ann. Math. (2) 170, No. 2, 863--881 (2009; Zbl 1213.14101); Schubert calculus and representations of the general linear group, J. Am. Math. Soc. 22, No. 4, 909--940 (2009), cf. also \url{arXiv:0711.4079}], while it is in general false for other flag varieties. Let \(P\) be a subspace of dimension \(n+1\) of the \((d+1)\)-dimensional vector space \(\mathbb C_d[t]\) of polynomials of degree at most \(d\). \(P\) is a point of the Grassmannian \(\mathbb G(n,d)\). The Wronski map \(Wr\), from \(\mathbb G(n,d)\) to the complex projective space of dimension \((n+1)(d-n)\), takes \(P\) to the Wronskian of a basis of \(P\). The simplest version of the theorem of Mukhin, Tarashov and Varchenko (Theorem 1 in this paper) says that if \(Wr(P)\) has only real roots, then \(P\) has a basis of real polynomials. A stronger form is Theorem 1.11 and can be formulated as follows. For \(s\in \mathbb C\), let \(F_\bullet(s)\) denote the complete flag of subspaces of \(\mathbb C_d[t]\): \[ \mathbb C(t-s)^d\subset \mathbb C_1[t] (t-s)^{d-1}\subset\ldots \subset \mathbb C_{d-1}[t] (t-s)\subset \mathbb C_d[t]. \] For \(0\leq a_0<a_1<\ldots a_n\leq d\), let \(\Omega_aF_\bullet(s)\) be the corresponding Schubert variety in \(\mathbb G(n,d)\). If we fix now a list of sequences \(a^{(i)}=(a_0^{(i)},\ldots , a_n^{(i)})\), \(i=1,\ldots ,m\), such that \(\Sigma_{i,j}(a_j^{(i)}-j)=(n+1)(d-n)\), and \(m\) distinct real numbers \(s_1,\ldots , s_m\), then the intersection of Schubert varieties \[ \Omega_{a^{(1)}}F_\bullet(s_1)\cap \dots \cap\Omega_{a^{(m)}}F_\bullet(s_m) \] is transverse and consists solely of real points. In this article, it is shown how these two theorems are related to each other and to some beautiful geometry of rational normal curves. The proofs of the theorems are described: the first one uses the Bethe ansatz for the periodic Gaudin model on certain representations of the Lie algebra \(\mathfrak{sl}_{n+1}\mathbb C\), the second one also passes through integrable systems and representations theory and provides a deep connection of them with Schubert calculus. Many applications of these theorems are given: to maximally inflected curves, to rational functions with real critical points, to tableaux combinatorics, to the degree of the Wronski map. Other cases of the Shapiro conjecture are discussed and some new conjectures and generalizations are proposed, also for general flag manifolds, for instance the monotone conjecture, the secant conjecture, the discriminant conjecture. Evidence to all these conjectures is given by a huge amount of experiments made on a supercomputer. Schubert calculus; Bethe ansatz; Wronskian; Calogero-Moser space Mukhin, E., Tarasov, V. & Varchenko, A., Bethe eigenvectors of higher transfer matrices. \textit{J. Stat. Mech. Theory Exp}., 8 (2006), P08002, 44~pp. Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Frontiers of reality in Schubert calculus

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this paper is to present certain symmetric properties of the Gromov-Witten invariants for type A complex flag manifolds. Several related results were found by \textit{N. Bergeron} and \textit{F. Sottile} [Duke Math. J. 95, 373--423 (1998; Zbl 0939.05084)] and \textit{S. Agnihotri} and \textit{C. Woodward} [Math. Res.Lett. 5, No. 6, 817--836 (1998; Zbl 1004.14013)]. Denoting by \(*\) the quantum product of the Schubert classes in the (small) quantum cohomology ring of the flag manifold, the quantum product being defined as \( \sigma_u * \sigma_v = \sum_{ w \in S_n} C_{u,v,w} \sigma_{w_0 w}\) where \(w_0\) is the longest permutation in the symmetric group of \(n\) letters, \(S_n \). After defining precisely the Gromov-Witten invariants the author states five basic properties for them. The author states a formula (proposition 1) for \(C_{u,v,w}\) which he claims follows from propositions 3 and 4 of those five basic ones. Proposition 2 is stated and taken from a previous paper by \textit{S. Fomin, S. Gelfand} and \textit{A. Postnikov} [J. Am. Math. Soc. 10, No. 3, 565--596 (1997; Zbl 0912.14018)]. Let \( o = (1,2, \dots, n) \) be the cyclic permutation of \(S_n\) given by \(o(i) = i + 1 \) for \( i= 1, \ldots, n-1\), \(o(n) = 1\) and \( q_{ij} = q_i q_{i+1} \dots q_{j-1}\) for \( i <j \). Define \( q_{ij} = q_{ji}^{-1}\) for \(i > j \) and \( q_{ii}= 1\). The author proves: For any \( u,v,w \in S_n \), \( C_{u,v,w} = q_{ij} C_{u,o^{-1}v, ow} \) where \( i= v^{-1}(1)\) and \( j = w^{-1}(n) \) (theorem 4). The proof of this theorem is given in section 4. type A complex flag manifolds; Schubert classes; quantum cohomology; symmetries of Gromov-Witten invariants Alexander Postnikov, Symmetries of Gromov-Witten invariants, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001, pp. 251 -- 258. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Symmetries of Gromov-Witten invariants

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One of the equivalent forms of the classical Littlewood-Richardson rule describes the structure constants obtained when the cup product of two Schubert classes in the cohomology ring of a complex Grassmannian is written as a linear combination of Schubert classes. In the paper under review the authors give a short and self-contained argument which shows that this rule is a direct consequence of the Pieri formula [\textit{M. Pieri}, Lomb. Ist. Rend. (2) XXVI. 534--546 (1893), XXVII. 258--273 (1894; JFM 25.1038.02)] for the product of a Schubert class with a special Schubert class. The Littlewood-Richardson rule has a generalization [\textit{J. R. Stembridge}, Adv. Math. 74, No. 1, 87--134 (1989; Zbl 0677.20012)] for the Grassmannians which parametrize maximal isotropic subspaces of \({\mathbb C}^n\) equipped with a symplectic or orthogonal form. The same arguments work equally well in more difficult cases and give a simple derivation of the rule of Stembridge from the analogues of the Pieri formula in [\textit{H. Hiller} and \textit{B. Boe}, Adv. Math. 62, 49--67 (1986; Zbl 0611.14036)]. Littlewood-Richardson rule; Grassmannian; Schubert class; Schur functions; Pieri formula Buch, AS; Kresch, A; Tamvakis, H, Littlewood-Richardson rules for Grassmannians, Adv. Math., 185, 80-90, (2004) Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Littlewood-Richardson rules for Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus \(g\) to \(\mathrm{GL}_{n}(\mathbb{C})\) with fixed generic semisimple conjugacy classes at \(k\) punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper. The first is a formula for the E-polynomial of these character varieties which we obtain using the character table of \(\mathrm{GL}_{n}(\mathbb {F}_{q})\). We use this formula to compute the Euler characteristic of character varieties. The second formula gives the Poincaré polynomial of certain associated quiver varieties which we obtain using the character table of \({\mathfrak{g}\mathfrak{l}}_{n}(\mathbb {F}_{q})\). In the last main result we prove that the Poincaré polynomials of the quiver varieties equal certain multiplicities in the tensor product of irreducible characters of \(\mathrm{GL}_{n}(\mathbb {F}_{q})\). As a consequence we find a curious connection between Kac-Moody algebras associated with comet-shaped, and typically wild, quivers and the representation theory of \(\mathrm{GL}_{n}(\mathbb {F}_{q})\). \textsc{T.~Hausel}, \textsc{E.~Letellier} and \textsc{F.~Rodriguez-Villegas}, {Arithmetic Harmonic Analysis on Character and Quiver Varieties}, \emph{Duke Math. Jour.}~\textbf{160} (2011), 323--400. DOI 10.1215/00127094-1444258; zbl 1246.14063; MR2852119; arxiv 0810.2076 [math.RT] Homogeneous spaces and generalizations, Representations of finite groups of Lie type Arithmetic harmonic analysis on character and quiver varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a generalization of the classic work of Beilinson, Lusztig and MacPherson [\textit{A. A. Beilinson} et al., Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)]. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in the setting of partial flag varieties of type \texttt{B}/\texttt{C} are two (modified) coideal subalgebras of the quantum general linear Lie algebra, \( \dot{\mathbf{U}}^j \) and \( \dot{\mathbf{U}}^i \). We provide a geometric realization of the Schur-type duality of Bao-Wang [\textit{H. Bao} and \textit{W. Wang}, A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs. Astérisque 402. Paris: Société Mathématique de France (SMF) (2018; Zbl 1411.17001)] between such a coideal algebra and Iwahori-Hecke algebra of type \texttt{B}. The monomial bases and canonical bases of the Schur algebras and the modified coideal algebra \( \dot{\mathbf{U}}^j \) are constructed. In an Appendix by three authors (Bao, Li and Wang), a more subtle 2-step stabilization procedure leading to \( \dot{\mathbf{U}}^i \) is developed, and then monomial and canonical bases of \( \dot{\mathbf{U}}^i \) are constructed. It is shown that \( \dot{\mathbf{U}}^i \) is a subquotient of \( \dot{\mathbf{U}}^j \) with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established. For Part II, see \textit{Z. Fang} and \textit{Y. Li}, Trans. Am. Math. Soc., Ser. B 2, 51--92 (2015; Zbl 1339.17012). quantum algebras; coideal subalgebras; quantum general linear Lie algebra; geometric realization of the Schur-type duality; Iwahori-Hecke algebra of type B Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Geometric Schur duality of classical type

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0728.00006.] This article gives an informative survey of the combinatorial theory of Schubert polynomials developed by A. Lascoux and M.-P. Schützenberger. Its topic may be described by the titles of the sections and subsections: Permutations (Bruhat order, diagrams and codes, vexillary permutations); divided differences; multi-Schur functions (duality); Schubert polynomials; orthogonality; double Schubert polynomials. In many cases proofs have been omitted. Bruhat order; Schur functions; permutations; divided differences; Schubert polynomials Macdonald, I. G., Notes on Schubert polynomials, (1991), Publications du Laboratoire de Combinatoire et D'informatique Mathématique, Dép. de Mathématiques et D'informatique, Universitédu Québec à Montréal, available at Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Orthogonal polynomials [See also 33C45, 33C50, 33D45], Grassmannians, Schubert varieties, flag manifolds Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author surveys applications of the localization technique to enumerative geometry, focussing on the cases of Schubert calculus on the flag manifold and Gromov-Witten invariants of rational curves. First, equivariant cohomology theory and the Atiyah-Bott localization theorem are reviewed. The next section gives \textit{J. Kong}'s [Schubert calculus on flag manifolds via localization, Ph.D. dissertation, Univ. of Utah, Salt Lake City, 2000] results on the Schubert calculus of the partial flag manifolds. Finally, it is described how formulae for genus zero Gromov-Witten invariants of a smooth projective variety \(X\) can be obtained by localization even if there is no group action on \(X\). By applying this method, the quantum Lefschetz hyperplane theorem and the genus zero reconstruction theorem are derived. Bertram, A.: Some applications of localization to enumerative problems. Michigan Math. J. \textbf{48}, 65-75 (2000) \textbf{(Dedicated to William Fulton on the occasion of his 60th birthday)} Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus Some applications of localization to enumerative problems.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The very interesting paper under review can be seen as a finite type version of a celebrated result in representation theory, due to \textit{E. Date} et al. [J. Phys. Soc. Japan 50, 3806--3812 (1981; Zbl 0571.35099)]. The latter supplied a precise description of the ring \(\mathbf{C}[x_1,x_2,\ldots]\) in infinitely many indeterminates as a representation of the Lie algebra of the complex valued matrices of infinite size with finitely many non-zero diagonals. The DJKM representation is based on the fact that the so-called \textit{Fermionic Fock Space} can be seen as the fundamental representation of the infinite Lie algebra \(gl_\infty(\mathbb{C})\). From a representation theoretical point of view, the \textit{Fermionic Fock space} can be roughly thought of an infinite wedge power of the vector space \(\mathbb{C}[X^{-1}, X]\) of Laurent polynomials in the indeterminate \(X\). The natural question the authors asked themselves is how the DJKM picture can be detected already for polynomial rings in finitely many indeterminates. To explain what and how they do let us walk a few steps backward, to render more precisely the feeling of their result. To begin with, le \(K[X]\) be the ring of polynomials in one indeterminate over a field \(K\) of characteristic zero. Let \(B_r:=K[x_1,\ldots,x_r]\) be the \(K\)-algebra of polynomials in the indeterminates \(\mathbf{x}:=(x_1,\ldots,x_r)\). It is easy to convince oneself that \(B_r\) is isomorphic to the \(r\)-th exterior power \(\bigwedge^rK[X]\). Although a number of mathematicians like to explain this fact as a special case of some sophisticated \textit{Geometric Satake Correspondence}, the naive reason is that both spaces possess a basis parametrized by all the partitions of length at most \(r\). It is also easy to see that, as the \(r\)-th exterior power of \(K^n\) is a representation of the Lie algebra \(gl_n(K)\) of the \(K\)-valued \(n\times n\) square matrices, then \(\bigwedge^rK[X]\) turns into a representation of the Lie algebra \(gl_\infty(K)\) of the \(K\)-valued matrices \((a_{ij})_{i,j\geq 0}\) whose entries are all zero but finitely many. The isomorphism \(B_r\rightarrow \bigwedge^rK[X]\) then makes \(B_r\) itself into a representation of \(gl_\infty(K)\). The main result of the paper under review, Theorem 4.11, consists in determining what the authors call the \textit{generating formal power series} \(\mathcal{E}(z,w^{-1}, t_1,\ldots, t_r)\), which describes \(B_r\) as a representation of the Lie algebra \(gl_\infty(K)\). This means the following. Recall that the elementary matrices \(E_{i,j}\) with all entries zero but \(1\) in position \((i,j)\) form a basis of \(gl_\infty(K)\) and that, interpreting \(B_r\) as the ring of symmetric polynomials in \(r\) indeterminates, it possesse a basis of Schur determinants \(\Delta_{\lambda}\) constructed out of the complete symmetric polynomials, parametrized by partitions of length at most \(r\). If \(\mathbf{t}_r:=(t_1,\ldots,t_r)\) is an \(r\)-tuple of formal variables, denote by \(s_{\lambda}(\mathbf{t}_r)\) the Schur polynomials in the indeterminates \(\mathbf{t}_r\), like at p. 40 of the book by \textit{I. G. Macdonald} [Symmetric functions and Hall polynomials. With contributions by A. V. Zelevinsky. Reprint of the 1998 2nd edition. Oxford: Oxford University Press (2015; Zbl 1332.05002)]. The determination of the generating formal power series of the aforementioned Theorem 4.11 heavily relies on the techniques introduced in the 2005 reviewer's paper [\textit{L. Gatto}, Asian J. Math. 9, No. 3, 315--322 (2005; Zbl 1099.14045)] and substantially improves results by \textit{L. Gatto} and \textit{P. Salehyan} [Commun. Algebra 48, No. 1, 274--290 (2020; Zbl 1442.14156)]. How does this relate with DJKM work? This is widely discussed in the last section of the paper under review. In a nutshell, taking a suitable limit \(r\to\infty\), one obtains an isomorphism from the ring \(B:=B_\infty\) to the charge zero vector subspace \(\mathcal{F}_0\) of the \textit{fermionic Fock space} \(\mathcal{F}:=\bigwedge^{\infty/2}K[X^{-1}, X]\). There is a vector space isomorphism of \(\mathcal{F}_0\) with the ring \(B\) of polynomials in infinitely many indeterminates \((x_1,x_2,\ldots)\) (the \textit{bosonic Fock space}), because both spaces possess a basis parametrized by partitions. The literature (see, e.g., [\textit{V. G. Kac} et al., Highest weight representations of infinite dimensional Lie algebras. World Scientific (2013)]) often refers to this isomorphism as the \textit{boson-fermion correspondence.} As a consequence, \(B\) is made into a representation of the Lie algebra \(gl(\infty)\), induced by the natural one living on the infinite wedge power. Moreover the vertex operators occurring in the DJKM representation are nothing but an infinite dimensional version of what the authors call \textit{Schubert derivations}, borrowing the own reviewer terminology, which, as the name suggest, are devices useful to cope with Schubert Calculus. The paper concludes itself with an essential but comprehensive reference list. Hasse-Schmidt derivations; vertex operators on exterior algebras; representation of Lie algebras of matrices; bosonic and fermionic representations by Date-Jimbo-Kashiwara-Miwa; symmetric functions Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Vertex operators; vertex operator algebras and related structures, Exterior algebra, Grassmann algebras, Symmetric functions and generalizations On the vertex operator representation of Lie algebras of matrices

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G(k,n)\) be the Grassmannian of \(k\)-planes in \(\mathbb C^n\). A Schubert problem in \(G(k,n)\) asks to determine the (number of) \(k\)-planes satisying some fixed incidence conditions, with respect to general fixed flags of subspaces of \(\mathbb C^n\), such that this number is finite. Every incidence condition defines a Schubert variety, therefore the solution of a Schubert problem can be represented by a transverse intersection of Schubert varieties. In particular, by definition, a simple Schubert problem involves only two Schubert conditions of codimension higher than one, it enjoys the property of being a complete intersection. To a given Schubert problem one naturally associates a Galois group; if it is the full symmetric group, it means that the problem has no underlying structures. The article under review tackles the problem of finding the Galois group of some simple Schubert problems, using the method of homotopy computation, recently developed in the new field of numerical algebraic geometry by Sommese and Wampler (see for instance [\textit{A. J. Sommese} and \textit{C. W. Wampler, II}, The numerical solution of systems of polynomials. Arising in engineering and science. River Edge, NJ: World Scientific. (2005; Zbl 1091.65049)]). This method, initially conceived for applications of mathematics, has proved here to be very efficient also to solve purely theoretic problems. As an example, the following numerical theorem is stated: The Galois group of the Schubert problem of \(3\)-planes in \(\mathbb C^8\) meeting \(15\) fixed \(5\)-planes non-trivially is the full symmetric group on \(6006\) letters. This kind of problems are intractable with the usual symbolic methods, so it appears that in the next future the numerical algorithms will be further developed. The article includes a precise description of the software used. The lines of future research in this field are also indicated. The article is clearly written and very pleasant to read. polynomial homotopy continuation; Schubert problem; Galois group Leykin, A; Sottile, F, Galois groups of Schubert problems via homotopy computation, Math. Comp., 78, 1749-1765, (2009) Classical problems, Schubert calculus, Global methods, including homotopy approaches to the numerical solution of nonlinear equations Galois groups of Schubert problems via homotopy computation

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In J. Differ. Geom. 20, 389--431 (1984; Zbl 0565.17007), \textit{S. Kumar} described the Schubert classes which are the dual to the closures of the Bruhat cells in the flag varieties of the Kac-Moody groups associated to the infinite dimensional Kac-Moody algebras. These classes are indexed by affine Weyl groups and can be chosen as elements of integral cohomologies of the homogeneous space \(\widehat{L}_{\text{pol}} G_{\mathbb C}/ \widehat{B}\) for any compact simply connected semi-simple Lie group \(G\). Later, \textit{S. Kumar} and \textit{B. Kostant} [Adv. Math. 62, 187--237 (1986; Zbl 0641.17008)] gave explicit cup product formulas of these classes in the cohomology algebras by using the relation between the invariant-theoretic relative Lie algebra cohomology theory (using the representation module of the nilpotent part) with the purely nil-Hecke rings. These explicit product formulas involve some BGG-type operators \(A^i\) and reflections. Using some homotopy equivalences, we determine cohomology ring structures of \(LG/T\) where \(LG\) is the smooth loop space on \(G\). Here, as an example we calculate the products and explicit ring structure of \(LSU_2/T\) using these ideas. Note that these results grew out of a chapter of the author's thesis [On the complex cobordism of flag varieties associated to loop groups, PhD thesis, University of Glasgow (1998)]. divided-power algebras; Schubert classes; Kac-Moody groups; Kac-Moody algebras; BGG-type operators; homotopy equivalences; cohomology ring C. Özel, ``On the cohomology ring of the infinite flag manifold LG/T,'' Turkish Journal of Mathematics, vol. 22, no. 4, pp. 415-448, 1998. Cohomology of Lie (super)algebras, Homology and cohomology of homogeneous spaces of Lie groups, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Loop groups and related constructions, group-theoretic treatment On the cohomology ring of the infinite flag manifold \(LG/T\)

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We define a polynomial representative of the Schubert class in the cohomology of an affine flag variety associated to \( SL(n)\), called an affine Schubert polynomial. Affine Schubert polynomials are defined by using divided difference operators, generalizing those operators used to define Schubert polynomials, so that Schubert polynomials are special cases of affine Schubert polynomials. Also, affine Stanley symmetric functions can be obtained from affine Schubert polynomials by setting certain variables to zero. We study affine Schubert polynomials and divided difference operators by constructing an affine analogue of the Fomin-Kirillov algebra called an affine Fomin-Kirillov algebra. We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine Fomin-Kirillov algebra to describe the cohomology of the affine flag variety and affine Schubert polynomials, and by doing so we also obtain a Murnaghan-Nakayama rule for the affine Schubert polynomials. Classical problems, Schubert calculus, Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Combinatorial description of the cohomology of the affine flag variety

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Involution Schubert polynomials represent cohomology classes of \(K\)-orbits in the complete flag variety, where \(K\) is the orthogonal or symplectic group. We show that they also represent \(T\)-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables \(x_i + x_j\), and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. In Knutson and Miller's approach to matrix Schubert varieties [\textit{A. Knutson} and \textit{E. Miller}, Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)], pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting. Schubert polynomials; pipe dreams; spherical orbits Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Involution pipe dreams

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The vanishing problem for Schubert polynomials asks if a coefficient of a Schubert polynomial is zero. We give a tableau criterion to solve this problem, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the \(n \times n\) grid. In contrast, we show that computing these coefficients explicitly is \#\textsf{P-complete}. Schubert polynomial; generalized permutahedra; computational complexity Classical problems, Schubert calculus, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) An efficient algorithm for deciding vanishing of Schubert polynomial coefficients

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Kostant polynomials play a crucial role in the Schubert calculus on \(G/B\) for a semisimple Lie group \(G\) and its Borel subgroup \(B\). These polynomials which are characterized by vanishing properties on the orbits of a regular point under the action of the Weyl group have nonzero values on the corresponding certain elements of higher Bruhat order. The author succeeded in giving explicit forms of these values. It should be emphasized that his description is very minute. Kostant polynomials; Schubert calculus; semisimple Lie group S. C. Billey, ''Kostant Polynomials and the Cohomology Ring for G/B,'' Duke Math. J. 96(1), 205--224 (1999). Semisimple Lie groups and their representations, Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds Kostant polynomials and the cohomology ring for \(G/B\)

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The direct sum map Gr\((a, \mathbb C^n) \times \text{Gr}(b,\mathbb C^m) \to \text{Gr}(a+b,\mathbb C^{m+n})\) on Grassmannians induces a \(K\)-theory pullback that defines the splitting coefficients. We geometrically explain an identity from \textit{A. S. Buch} [``Grothendieck classes of quiver varieties'', Duke Math. J. 115, No. 1, 75--103 (2002; Zbl 1052.14056)] between the splitting coefficients and the Schubert structure constants for products of Schubert structure sheaves. This is related to the topic of product and splitting coefficients for Schubert boundary ideal sheaves. Our main results extend jeu de taquin for increasing tableaux [\textit{H. Thomas} and \textit{A. Yong}, ``A jeu de taquin theory for increasing tableaux, with applications to \(K\)-theoretic Schubert calculus'', Algebra Number Theory 3, No. 2, 121--148 (2009; Zbl 1229.05285)] by proving transparent analogues of \textit{M.-P. Schützenberger}'s [``La Correspondance de Robinson'', Lect. Notes Math. 579, 59--113 (1977; Zbl 0398.05011)] fundamental theorems on well definedness of rectification. We then establish that jeu de taquin gives rules for each of these four kinds of coefficients. Thomas, Hugh; Yong, Alexander, The direct sum map on Grassmannians and jeu de taquin for increasing tableaux, Int. Math. Res. Not., 2011, 12, 2766-2793, (2011) Symmetric functions and generalizations, Combinatorial aspects of partitions of integers, Classical problems, Schubert calculus The direct sum map on Grassmannians and jeu de taquin for increasing tableaux

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives an introduction to the theory of bialgebras and applies it to conics, quartics, and sextics of projective planes over an algebraically closed field \(K\) of characteristic \(0\). Let \(W\) be the \(K\)-vector space of \(n\)-ary linear forms, \(W^*\) the dual space, and \(\langle\;,\;\rangle\) the two natural pairings. A {bimultiplication} consists of two bilinear forms \([\;,\;]:W\times\,W\rightarrow\,W^*\) and \([\;,\;]:W^*\times\,W^*\rightarrow\,W\). The bialgebra \((W,[\;,\;])\) is called {symmetric} (respectively {skew symmetric}), if both the trilinear functions \(\Phi(a,b,c)=\langle [a,b],c\rangle \), \(a,b,c\in\,W\) and \(\Psi(p,q,r)=\langle [p,q],r\rangle \), \(p,q,r\in\,W^*\) are symmetric (respectively alternative). If \((W,[\;,\;])\) is skew symmetric with PGL(\(W\))-equivariant bimultiplication and \(m\) a natural then there exists a unique PGL(\(W\))-equivariant bimultiplication on the symmetric powers \(S^m(W)\) such that \([a^m,b^m]=[a,b]^m\) and \([p^m,q^m]=[p,q]^m\); for even \(m\) the bimultiplication on \(S^m(W)\) is symmetric. Two elements \(A,B\in\,S^m(W)\) are {apolar}, if \(\langle A,\hat B\rangle =\langle B,\hat A\rangle =0\) where \(\hat A=[A,A]/2\). Put \(a^{2m}=:s_1\), \(b^{2m}=:s_2\), \(c^{2m}=:s_3\), \((bc)^m=:s_4\), \((ca)^m=:s_5\), \((ab)^m=:s_6\), i.e., \(s_k\in\,S^{2m}(W)\), and \([b,c]^{2m}=:t_1\), \([c,a]^{2m}=:t_2\), \([a,b]^{2m}=:t_3\), \(([c,a][a,b])^m=:t_4\), \(([a,b][b,c])^m=:t_5\), \(([b,c][c,a])^m=:t_6\), i.e., \(t_k\in\,S^{2m}(W^*)\). Let \(U_{2m}(a,b,c)\) and \(V_{2m}(a,b,c)\) be the set of all linear combinations of \(s_1,\dots,s_6\) respectively \(t_1,\dots,t_6\), then \((U_{2m}(a,b,c),V_{2m}(a,b,c))\) is a subalgebra of \((S^{2m}(W),[\;,\;])\) called by the author {Salmon subalgebra}. Let \(U'_{2m}(a,b,c)\) and \(V'_{2m}(a,b,c)\) be the pencil spanned by \(s_1+s_2+s_3\) and \(s_4+s_5+s_6\) respectively \(t_1+t_2+t_3\) and \(t_4+t_5+t_6\), then \((U'_{2m}(a,b,c),V'_{2m}(a,b,c))\) is a subalgebra of \((S^{2m}(W),[\;,\;])\) called by the author {Wiman subalgebra} or {Wiman pencil}. Assume \(n=3\), then there exists a (projectively) unique skew symmetric, special bialgebra \((W,[\;,\;])\) with non-zero and PGL(\(W\))-equivariant bimultiplication. The bialgebra \(S^2(W)\) of conics is semisimple since there exist six mutually apolar conics [cf. \textit{F. Gerbaldi}, Torino Atti 17, 358--371, 566--580 (1882; JFM 14.0537.02) and \textit{P. Gordan}, Math. Ann. 61, 453--562 (1906; JFM 37.0142.01)]. The author proves: If \(A\), \(A'\) are two conics of a set \(G\) of six mutually apolar conics, then each of the eight points of contact determined by the four bitangents of \(A\), \(A'\) belongs to a conic of \(G\setminus\{A,A'\}\). A quartic is an idempotent of \(S^4(W)\), iff it is either a double conic or a Fermat quartic or a Klein quartic; this is a reformulation of \textit{E. Ciani} [Palermo Rend. 14, 16--21 (1900; JFM 31.0113.01)]. The Salmon quartic subalgebra is semisimple since it contains six mutually apolar idempotent quartics. The author gives also a list of twelve mutually apolar idempotent quartics, but it is an open question whether \(S^4(W)\) is semisimple or not. The author proves that \(S^4(W)\) is weakly semisimple. Using Gerbaldi's conics the author constructs 15 quartic idempotent pencils \((U''_j,V''_j)\), (\(j=1,2\ldots,15\)). Each pair \((U''_j,V''_j)\) is projectively isomorphic to Wiman's quartic subalgebra. Each of the 15 pencils \(U''_j\) contains two Klein quartics, one Fermat quartic, one Capolari quartic, one Clebsch quartic, and one Bernoulli lemniscate. For the bialgebra \(S^6(W)\) the author gives examples of 2- and 4-dimensional semisimple subalgebras. Reviewer's remark: In [34] of the references the name is written wrong, correct is \textit{S. Crass} [Exp. Math. 8, No. 3, 209--240 (1999; Zbl 1060.14530)]. symmetric bialgebra; skew symmetric bialgebra; bialgebra of binary forms; bialgebra of ternary forms; semisimplicity; weak semisimplicity; apolarity; Salmon subalgebra; Wiman subalgebra M. Gizatullin, ``Bialgebra and geometry of plane quartics'', Asian J. Math., 5:3 (2001), 387 -- 432 Questions of classical algebraic geometry, Projective techniques in algebraic geometry, Plane and space curves Bialgebra and geometry of plane quartics.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review is concerned with the quantum \(T\)-equivariant \(K\)-theory of a homogeneous variety \(X:=G/P\), where \(G\) is a connected semisimple complex linear algebraic group, \(P\) a parabolic subgroup and \(T\) any maximal subtorus. Since \(X\) is obviously acted on by \(T\), it is natural to wonder about the \(T\)-equivariant Grothendieck ring \(K_T(X)\) of \(T\)-equivariant vector bundles on the homogeneous variety \(X\), which is an algebra over \(\Gamma:=K_T(pt)\), the equivariant Grotendieck ring of a point. The beautiful main result of the paper under review is the determination of the quantum equivariant product of the class of a Schubert variety with its \textit{opposite} in the Grothendieck ring of the generalised flag varieties. To be more precise, for the convenience of the potential reader's audience, recall that there are two ways to define Schubert varieties in \(X\) starting by elements of the Weyl group of \(G\). If \(B\) is a Borel subgroup such that \(T\subseteq B\subseteq P\), the opposite Borel subgroup \(B^{-}\) is characterised by the equality \(B\cap B^{-}=T\). Each element \(w\) of the Weyl group of \(G\) hence determines a \(B\)-invariant Schubert variety \(X_w\) and a \(B^{-}\)--invariant one \(X^w\), which are defined to be one the \textit{opposite} of the other. Another piece of information is that the second cohomology group \(H^2(X,Z)\) possesses a basis \([X_{s_\beta}]\), where \(s_\beta\) is the element of the Weil group of \(G\) corresponding to \(\beta\in \Delta\setminus \Delta_P\), where \(\Delta\) is the set of the simple positive roots of \(G\) and \(\Delta_P\) those of \(P\). Given a formal variable \(q\), one attaches a formal variable \(q^\beta\) to each simple root \(\beta\) and define \(\Gamma[[q]]\) as the ring of formal power series in the variables \(q^\beta\). Let \(q^d:=\prod_{\beta\in \Delta\setminus\Delta_P}q^{d_\beta}_\beta\). One defines, à la \textit{A. Givental} [Mich. Math. J. 48, 295--304 (2000; Zbl 1081.14523)], the small quantum Grothendieck ring \(QK_T(X)\) as being the tensor product of \(K_T(X)\) with \(\Gamma[[q]]\) with respect to a product which deforms the classical one via corrections coming from \(K\)-theoretical Gromov-Witen invariants. This is very well explained already in the article's introduction. The main character of the paper is a suitable quantum extension of the \textit{sheaf Euler characteristic} \(\chi_X:K_T(X)\rightarrow \Gamma\), which maps a class \([E]\) of equivariant vector bundles to its generalised Euler characteristic \(\chi_X([E])\), namely to the formal sum \(\sum_{i\geq 0}(-1)^i[H^i(X,E)]\). The latter can however be defined as the unique map such that \(\chi_X([\mathcal{O}_{X_u}])=\chi_X([\mathcal{O}_{X^v}])=1\), for all \(u,v\in W\). Consider the \(\Gamma[[q]]\)-linear extension \(\chi\) of \(\chi_X\). The central goal of the paper is then to prove that such sheaf Euler characteristic \(\chi\) of the product of a Schubert class and an opposite Schubert class in the quantum \(K\)-theory ring of a (generalized) flag variety \(G/P\) is equal to \(q^d\), where \(d\) is the smallest degree \(\sum d_\alpha X_{s_\alpha}\) of a rational curve joining the two Schubert varieties. This implies that the sum of the structure constants of any product of Schubert classes is equal to \(1\). Along the way, a description of the smallest degree \(d\) in terms of its projections to flag varieties defined by maximal parabolic subgroups is provided. This result was already proved by \textit{A. S. Buch} and \textit{S. Chung} [J. Lond. Math. Soc., II. Ser. 97, No. 2, 145--148 (2018; Zbl 1390.14169)] for cominuscule Flag varieties. The paper consists of just three sections. Section one is a very well written and explicative introduction: the reader gets immedately aware about what the authors are concerned with. Section 2 recall and explain the interesting notion of distance between two Schubert varieties and Section 3, finally, is devted to provide the last missing preliminaries in order to prove the main Theorem 8, we have mentioned above. The essential reference list conclude this very inspiring paper. equivariant \(K\)-theory of generalised flag varieties; opposite Schubert varieties; Euler characteristic sheaf; quantum cohomology; Gromov Witten invariants; distances between Schubert varieties Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), \(K\)-theory of schemes, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Euler characteristics in the quantum \(K\)-theory of flag varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood-Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier result of B. Sagan and the author [\textit{A.I. Molev} and \textit{B.E. Sagan},''A Littlewood-Richardson rule for factorial Schur functions,'' Trans. Am. Math. Soc. 351, No.\,11, 4429--4443 (1999; Zbl 0972.05053)]. The new rule provides a formula for these polynomials which is positive in the sense of \textit{W. Graham} [''Positivity in equivariant Schubert calculus,'' Duke Math. J. 109, No.\,3, 599--614 (2001; Zbl 1069.14055)]. We apply this formula for the calculation of the product of equivariant Schubert classes on Grassmannians which implies a stability property of the structure coefficients. The first manifestly positive formula for such an expansion was given by \textit{A. Knutson} and \textit{T. Tao} [''Puzzles and (equivariant) cohomology of Grassmannians,'' Duke Math. J. 119, No.\,2, 221--260 (2003).] by using combinatorics of puzzles while the stability property was not apparent from that formula. We also use the Littlewood-Richardson polynomials to describe the multiplication rule in the algebra of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants constructed by \textit{A. Okounkov} and \textit{G. Olshanski} [''Shifted Schur functions,'' St. Petersbg. Math. J. 9, No.\,2, 239--300 (1998); translation from Algebra Anal. 9, No.\,2, 73--146 (1997; Zbl 0894.05053).]. Littlewood-Richardson rule; double symmetric functions; equivariant Schubert classes; Grassmannians; quantum immanants; Schur functions; Littlewood-Richardson polynomials; combinatorics of puzzles; Casimir elements; general linear Lie algebra Alexander I. Molev, ``Littlewood-Richardson polynomials'', J. Algebra321 (2009) no. 11, p. 3450-3468 Symmetric functions and generalizations, Classical problems, Schubert calculus Littlewood-Richardson polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be the symplectic group defined over an algebraically closed field, and \(G/P\) be the Grassmannian of maximal isotropic subspaces. In particular, \(P\) is a maximal parabolic subgroup of \(G\). For any Schubert variety \(X_w\) in the symplectic Grassmannian \(G/P\) the authors give an explicit combinatorial description of the multiplicity, and more generally, the Hilbert function of the tangent cone at any point in \(X_w\). Moreover, they formulate and prove the two conjectures of Kreiman and Lakshmibai in this case. A reformulation of the main result in terms of Gröbner bases is given. A key ingredient in the proofs is the standard monomial theory for symplectic Grassmannians and their Schubert subvarieties developed by De Concini and Lakshmibai-Musili-Seshadri. This allows to translate the problem from geometry to combinatorics. In particular, the multiplicity is interpreted as the number of certain non-intersecting lattice paths. Relations between Schubert varieties in symplectic Grasmannians and symmetric determinantal varieties are also discussed. singular points; multiplicities; tangent cone; standard monomial theory S. R. Ghorpade, K. N. Raghavan, Hilbert functions of points on Schubert varieties in the symplectic Grassmannian, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5401--5423. Grassmannians, Schubert varieties, flag manifolds, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Rings with straightening laws, Hodge algebras Hilbert functions of points on Schubert varieties in the symplectic Grassmannian

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a complex semisimple Lie group and let \(P\subset Q\) be a pair of parabolic subgroups of \(G\). There is an induced sequence of flag varieties \(Q/P \rightarrow G/P\rightarrow G/Q\). The author shows that certain structure constants in \(H^*(G/P)\) with respect to the Schubert basis can be written as a product of structure constants coming from \(H^*(G/Q)\) and \(H^*(Q/P)\) in a very natural way: they can be read on the Weyl groups of these flag varieties. Secondly, he uses this result to compute Levi-movable structure constants defined by \textit{P. Belkale} and \textit{S. Kumar} in [Invent. Math. 166, No. 1, 185--228 (2006; Zbl 1106.14037)]. He also give a generalization of this product formula in the branching Schubert calculus setting. flag varieties; Levi-movable structure Richmond, E, A multiplicative formula for structure constants in the cohomology of flag varieties, Michigan Math. J., 61, 3-17, (2012) Classical problems, Schubert calculus, Homogeneous spaces and generalizations A multiplicative formula for structure constants in the cohomology of flag varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We relate a certain category of sheaves of \(k\)-vector spaces on a complex affine Schubert variety to modules over the \(k\)-Lie algebra (for \(\text{char\,}k>0\)) or to modules over the small quantum group (for \(k=0\)) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig's conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation-theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig's modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity-one case for all relevant primes. One of the fundamental problems in representation theory is the calculation of the simple characters of a given group. This problem often turns out to be difficult and there is an abundance of situations in which a solution is out of reach. In the case of algebraic groups over fields of positive characteristic we have a partial, but not yet a full answer. In 1979, George Lusztig conjectured a formula for the simple characters of a reductive algebraic group defined over a field of characteristic greater than the associated Coxeter number; [cf. \textit{G. Lusztig}, Proc. Symp. Pure Math. 37, 313-317 (1980; Zbl 0453.20005)]. Lusztig outlined in 1990 a program that led, in a combined effort of several authors, to a proof of the conjecture for almost all characteristics. This means that for a given root system \(R\) there exists a number \(N=N(R)\) such that the conjecture holds for all algebraic groups associated to the root system \(R\) if the underlying field is of characteristic greater than \(N\). This number, however, is unknown in all but low rank cases. One of the essential steps in Lusztig's program was the construction of a functor between the category of intersection cohomology sheaves with complex coefficients on an affine flag manifold and the category of representations of a quantum group (this combines results of \textit{M. Kashiwara} and \textit{T. Tanisaki} [Duke Math. J. 77, No. 1, 21-62 (1995; Zbl 0829.17020)], and \textit{D. Kazhdan} and \textit{G. Lusztig} [J. Am. Math. Soc. 6, No. 4, 905-947, 949-1011 (1993; Zbl 0786.17017); ibid. 7, No. 2, 335-381, 383-453 (1994; Zbl 0802.17007, Zbl 0802.17008)]). This led to a proof of the quantum (i.e. characteristic 0) analog of the conjecture. \textit{H. H. Andersen, J. C. Jantzen} and \textit{W. Soergel} then showed that the characteristic zero case implies the characteristic \(p\) case for almost all \(p\) [cf. Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Astérisque 220 (1994; Zbl 0802.17009)]. One of the principal functors utilized in Lusztig's program was the affine version of the Beilinson-Bernstein localization functor. It amounts to realizing an affine Kac-Moody algebra inside the space of global differential operators on an affine flag manifold. A characteristic \(p\) version of this functor is a fundamental ingredient in Bezrukavnikov's program for modular representation theory [cf. \textit{R. Bezrukavnikov, I. Mirković} and \textit{D. Rumynin}, Ann. Math. (2) 167, No. 3, 945-991 (2008; Zbl 1220.17009)], and recently Frenkel and Gaitsgory used the Beilinson-Bernstein localization idea in order to study the critical level representations of an affine Kac-Moody algebra [cf. \textit{P. Fiebig}, Duke Math. J. 153, No. 3, 551-571 (2010; Zbl 1207.20040)]. There is, however, an alternative approach that links the geometry of an algebraic variety to representation theory. It was originally developed in the case of finite-dimensional complex simple Lie algebras by \textit{W. Soergel} [J. Am. Math. Soc. 3, No. 2, 421-445 (1990; Zbl 0747.17008)]. The idea was to give a ``combinatorial description'' of both the topological and the representation-theoretic categories in terms of the underlying root system using Jantzen's translation functors. This approach gives a new proof of the Kazhdan-Lusztig conjecture, but it is also important in its own right: when taken together with the Beilinson-Bernstein localization it establishes the celebrated Koszul duality for simple finite-dimensional complex Lie algebras [cf. \textit{W. Soergel}, loc. cit., and \textit{A. Beilinson, V. Ginzburg, W. Soergel}, J. Am. Math. Soc. 9, No. 2, 473-527 (1996; Zbl 0864.17006)]. In this paper we develop the combinatorial approach for quantum and modular representations. We relate a certain category of sheaves of \(k\)-vector spaces on an affine flag manifold to representations of the \(k\)-Lie algebra or the quantum group associated to Langlands' dual root datum (the occurrence of Langlands' duality is typical for this type of approach). As a corollary we obtain Lusztig's conjecture for quantum groups and for modular representations for large enough characteristics. The main tool that we use is the theory of sheaves on moment graphs, which originally appeared in the work on the localization theorem for equivariant sheaves on topological spaces by \textit{M. Goresky, R. Kottwitz} and \textit{R. MacPherson} [Invent. Math. 131, No. 1, 25-83 (1998; Zbl 0897.22009)] and \textit{T. Braden} and \textit{R. MacPherson} [Math. Ann. 321, No. 3, 533-551 (2001; Zbl 1077.14522)]. In particular, we state a conjecture in terms of moment graphs that implies Lusztig's quantum and modular conjectures for all relevant characteristics. Although there is no general proof of this moment graph conjecture yet, some important instances are known: The smooth locus of a moment graph is determined by \textit{P. Fiebig} [loc. cit.], which yields the multiplicity-one case of Lusztig's conjecture in full generality. Moreover, by developing a Lefschetz theory on a moment graph we obtain in [\textit{P. Fiebig}, J. Reine Angew. Math. 673, 1-31 (2012; Zbl 1266.20059)] an upper bound on the exceptional primes, i.e. an upper bound for the number \(N\) referred to above. Although this bound is huge (in particular, much greater than the Coxeter number), it can be calculated by an explicit formula in terms of the underlying root system. Kazhdan-Lusztig polynomials; irreducible characters; highest weight modules; simple Lie algebras; quantized enveloping algebras; reductive algebraic groups; positive characteristic; root systems; intersection cohomology sheaves; Schubert varieties; character formulae; Coxeter numbers; Lusztig conjecture; affine flag manifolds; affine Kac-Moody algebras; moment graphs Fiebig, Peter, Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture, J. Amer. Math. Soc., 0894-0347, 24, 1, 133\textendash 181 pp., (2011) Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Modular representations and characters, Sheaf cohomology in algebraic topology Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review studies equivariant Schubert calculus for general (oriented) cohomology theories. The classical (cohomology, \(K\) theory) level of such questions is in the crossroads of geometry, representation theory, and algebraic combinatorics. It turns out that for more general cohomology theories key geometric objects are not necessarily the classical Schubert varieties, but their Bott-Samelson resolutions. The main results of the paper can be interpreted as a GKM description of the general cohomology of these Bott-Samelson resolutions. Namely, the authors describe the torus fixed point restrictions of a distinguished basis. Their formulas imply the injectivity of the fixed-point localization map. Moreover, they describe the image of that map in the expected GKM style. The authors also provide connection to more standard point-of-views: they provide formulas for the ``Schubert classes'' as opposed to the ``Bott-Samelson classes'', that is they calculate the push-forward images of the distinguished basis, that now live in the general cohomology of flag varieties. The results promise applications in the emerging branch of geometric representation theory that studies general cohomologies. Bott-Samelson variety; flag variety; equivariant oriented cohomology; restriction formula Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Connective \(K\)-theory, cobordism, Bordism and cobordism theories and formal group laws in algebraic topology, Homology and cohomology of homogeneous spaces of Lie groups, Equivariant cobordism On equivariant oriented cohomology of Bott-Samelson varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert calculus allows to count geometric objects subject to certain incidence relations. Its rigorous foundation was established by intersection theory. For example, consider \(2d-2\) lines in general position in \(\mathbb{P}^d(\mathbb{C})\). The Schubert Calculus allows to prove that the number of \(2\)-subspaces of \(\mathbb{P}^d(\mathbb{C})\) meeting all \(2d-2\) lines is exactly the \(d\)-th Catalan number. From the point of view of intersection theory, such a problem is solved by counting intersection multiplicities of Schubert varieties in the Grassmannian. If all the \(2d-2\) lines are reals one can ask how many of the previous \(2\)-subspaces are real. Problem like this are considerably more complicated. For this particular problem, in [\textit{F. Sottile}, Exp. Math. 9, No. 2, 161--182 (2000; Zbl 0997.14016)] was proved that for some choice of the \(2d-2\) lines all these 2-codimension subspaces are real. Boris and Michael Shapiro conjectured that this happen when all the given lines are tangent to the rational normal curve \(\gamma(t)=[1,t,t^{2},\dots,t^d]\) at distinct real points. This conjecture has been proved in [\textit{A. Eremenko} and \textit{A. Gabrielov}, Ann. Math. (2) 155, No. 1, 105--129 (2002; Zbl 0997.14015)]. In [\textit{E. Mukhin, V. Tarasov} and \textit{A. Varchenko}, Ann. Math. (2) 170, No. 2, 863--881 (2009; Zbl 1213.14101)], a more general version of the previous conjecture for higher dimensional subspaces was proved. Then B. and M. Shapiro suggested an extension of their conjecture to flags, but this last conjecture fails (see the previous work of Sottile for some counterexample). However, experiments suggest that the conjecture might hold whenever a certain monotonicity condition is met (see [\textit{J. Ruffo, Y. Sivan, E. Soprunova} and \textit{F. Sottile}, Exp. Math. 15, No. 2, 199--221 (2006; Zbl 1111.14049)]). The authors are interested in a special case of the previous generalized conjecture, which (assuming the monotonicity condition) was proved in [\textit{A. Eremenko, A. Gabrielov, M. Shapiro} and \textit{A. Vainshtein}, Proc. Am. Math. Soc. 134, No. 4, 949--957 (2006; Zbl 1110.14052)]. More precisely, they are interested to the following problem: Let \(y_1,\dots,y_{2d-3},r,s\) be distinct real points. For \(1\leq i\leq 2d-3\), let \(T_i\) be the line tangent to \(\gamma\) at \(y_i\) and let \(T_{2d-2}\) be the line through the points \(\gamma(r)\) and \(\gamma(s)\). Among the codimension 2-subspaces of \(\mathbb{P}^d(\mathbb{C})\) meeting all the previous lines, how many are real? In this case the monotonicity condition means that the interval \(I\) with endpoints \(r\) and \(s\) contains all or none of the \(y_1,\dots,y_{2d-3}\). In this case (or more generally when the special case of the conjecture holds) all the points are real. By the cited result of Sottile there are cases where none points are real. In this article, a lower bound to the number of real solutions of the previous problem is given (when the monotonicity condition does not hold). The authors use a point of view similar to the cited Eremenko and Gabrielov proof of the first B. and M. Shapiro conjecture. Eremenko and Gabrielov have proved that, if the critical points of a rational function are all real, then the function is a real rational function up to composing with a fractional linear transformation. The authors of this work study the following problem (equivalent to the previous one): Let \(y_1,\dots,y_{2d-3},r,s,I\) as before and suppose that the monotonicity condition is not satisfied, i.e. \(I\) contains some but not all the \(y_i\). How many equivalence classes of real rational functions \(f\) of degree \(d\) having critical points at \(y_1,\dots,y_{2d-3}\) and satisfying \(f(r)=f(s)\) are there? Schubert calculus; real enumerative geometry; rational functions; Wronski map DOI: 10.1007/s00454--010--9314--8. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Real algebraic and real-analytic geometry Some lower bounds in the B. and M. Shapiro conjecture for flag varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(R=\mathbb{Z}[x_{1},\dots,x_{n}]\) be a polynomial ring in \(n\) variables over the integers \(\mathbb{Z}\). A symmetric function is a polynomial in \(R\) that is fixed by the natural action of the permutation group \(S_{n}\) on the variables \(\{x_{1},\dots,x_{n}\}\). These functions are fundamental objects in many areas of mathematics, including combinatorics, topology and algebraic geometry. In the paper under review the authors study truncated symmetric functions, i.e., polynomials in a proper subset of \(\{x_{1},\dots,x_{n}\}\) that are symmetric under the corresponding permutation subgroup. They show combinatorial identities relating two different kinds of truncated symmetric polynomials, namely elementary and complete symmetric polynomials. Furthermore they use truncated elementary symmetric functions to build a family of ideals \(I_{h}\) in \(R\) that are parameterized by Hessenberg functions \(h\) (equivalently Dyck paths or ample partitions). The ideals \(I_{h}\) generalize algebraically a family of ideals, commonly known as Tanisaki ideals, which is used in the Springer theory. This theory is an example of geometric representation theory that constructs representations of the symmetric group \(S_{n}\) on the cohomology of a family of varieties parameterized by partitions. The authors prove several properties of \(I_{h}\), including that if \(h>h'\) in the natural partial order on Dyck paths, then \(I_{h} \subset I_{h'}\), and construct a Gröbner basis for \(I_{h}\). They consider a second family of ideals \(J_{h}\) and prove that \(I_{h}=J_{h}\). Finally they show that the ideals \(I_{h}=J_{h}\) generalize the Tanisaki ideals both algebraically and geometrically, from Springer varieties to a family of nilpotent Hessenberg varieties. symmetric functions; Tanisaki ideal; Springer variety; Hessenberg variety; Gröbner basis Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Generalizing Tanisaki's ideal via ideals of truncated symmetric functions

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected semisimple algebraic group, \(B\) a Borel subgroup, \(T\) a maximal torus in \(B\) with Weyl group \(W\), and \(Q\) a subgroup containing \(B\). For \(w\in W\), let \(X_{wQ}\) denote the Schubert variety \(\overline {BwQ}/Q\). For \(y\in W\) such that \(X_{yQ} \subseteq X_{wQ}\), one knows that \(ByQ/Q\) admits a \(T\)-stable transversal in \(X_{wQ}\), which we denote by \({\mathcal N}_{yQ,wQ}\). We prove that, under certain hypotheses, \({\mathcal N}_{yQ,wQ}\) is isomorphic to the orbit closure of a highest weight vector in a certain Weyl module. We also obtain a generalisation of this result under slightly weaker hypotheses. Further, we prove that our hypotheses are satisfied when \(Q\) is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight, and \(X_{yQ}\) is an irreducible component of the boundary of \(X_{wQ}\) (that is, the complement of the open orbit of the stabiliser in \(G\) of \(X_{wQ})\). As a consequence, we describe the singularity of \(X_{wQ}\) along \(ByQ/Q\) and obtain that the boundary of \(X_{wQ}\) equals its singular locus. Borel subgroup; Weyl group; Schubert variety; weight vector; singular locus Brion M., Polo P.: Generic singularities of certain Schubert varieties. Math. Z. 231, 301--324 (1999) Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Linear algebraic groups over arbitrary fields Generic singularities of certain Schubert varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials are refined by the key polynomials of Lascoux-Schützenberger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles [\textit{S. Assaf} and \textit{D. Searles}, Adv. Math. 306, 89--122 (2017; Zbl 1356.14039)]. In this paper we determine which fundamental slide polynomial refinements of key polynomials, indexed by strong compositions, are multiplicity free. We also give a recursive algorithm to determine all terms in the fundamental slide polynomial refinement of a key polynomial indexed by a strong composition. From here, we apply our results to begin to classify which fundamental slide polynomial refinements, indexed by weak compositions, are multiplicity free. We completely resolve the cases when the weak composition has at most two nonzero parts or the sum has at most two nonzero terms. Schubert polynomials; Lascoux-Schützenberger key polynomials Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Slide multiplicity free key polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\Omega_w\) be a Schubert variety in the symplectic flag variety, and let \(e_v \in \Omega_w\) be a torus fixed point. We give a combinatorial formula for the Hilbert-Samuel multiplicity of \(\Omega_w\) at the point \(e_v\), in the case where is a vexillary signed permutation. Our formula is phrased in terms of excited Young diagrams, extending results by Ghorpade-Raghavan and Ikeda-Naruse for Grassmannians [\textit{S. R. Ghorpade} and \textit{K. N. Raghavan}, Trans. Am. Math. Soc. 358, No. 12, 5401--5423 (2006; Zbl 1111.14046)], as well as \textit{L. Li} and \textit{A. Yong} [Adv. Math. 229, No. 1, 633--667 (2012; Zbl 1232.14033)] for vexillary Schubert varieties in type A flag manifolds. Schubert variety; multiplicity; vexillary permutation Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorial aspects of groups and algebras, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Multiplicities of Schubert varieties in the symplectic flag variety

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article we construct Laurent polynomial Landau-Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau-Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety. Laurent polynomial Landau-Ginzburg models for cominuscule homogeneous spaces

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{G. Lusztig} [Adv. Math. 37, 121--164 (1980; Zbl 0448.20039)] introduced the periodic Kazhdan-Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those of an affine Kac-Moody algebra at the critical level. The periodic Kazhdan-Lusztig polynomials can be computed by using another family of polynomials, called the periodic \(R\)-polynomials. In this paper, we prove a (closed) combinatorial formula expressing periodic \(R\)-polynomials in terms of the ``doubled'' Bruhat graph associated to a finite Weyl group and a finite root system. periodic Kazhdan-Lusztig polynomials; periodic \(R\)-polynomials; generic Bruhat order Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Group actions on varieties or schemes (quotients), Reflection and Coxeter groups (group-theoretic aspects), Linear algebraic groups over arbitrary fields, Representation theory for linear algebraic groups A combinatorial formula expressing periodic \(R\)-polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the variety of complete flags Flag\((\mathbb C^n)\), Schubert varieties are defined via the intersection properties with a reference flag \(\mathcal E\). A Schubert variety \(X_w\) is always locally Cohen-Macaulay, although often singular. The authors study conditions under which \(X_w\) is locally Gorenstein. In this setting, \(X_w\) is locally Gorenstein exactly when its dualizing sheaf \(\omega_{X_w}\) is locally free. Using the characterization of the dualizing sheaf of \(X_w\) given by Ramanathan, the authors find an equation, in the group of Cartier divisors of \(X_w\), which is satisfied exactly when \(X_w\) is locally Gorenstein. The existence of solutions for the equation is then translated in necessary and sufficient combinatorial conditions on the permutation \(w\) associated with \(X_w\). These condition are expressed in terms of pattern avoidance and alignment of inner corners. It turns out that \(X_w\) is locally Gorenstein if and only if it is such along its maximal singular locus. As a consequence, the authors are able to give a precise description of the dualizing line bundle of locally Gorenstein Schubert varieties. Woo A. and Yong A., When is a Schubert variety Gorenstein?, Adv. Math. 207 (2006), no. 1, 205-220. Grassmannians, Schubert varieties, flag manifolds When is a Schubert variety Gorenstein?

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We collect \textit{Atiyah-Bott Combinatorial Dreams} (A-B-C-Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron and F. Sottile, of S. Billey and M. Haiman, of P. Pragacz, and of T. Ikeda, L. Mihalcea, and I. Naruse. The relation complements a theorem of A. Kresch and H. Tamvakis in quantum cohomology. Results of A. Buch and V. Ravikumar rule out a similar correspondence in \(K\)-theory. Classical problems, Schubert calculus The A-B-C-Ds of Schubert calculus

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author announces several results on the Schubert varieties \(X_w = \overline {BwB/B} \subseteq G/B\) associated to a semi-simple, simply connected, complex algebraic group \(G\), where \(B\) is a Borel group and \(w\) an element in the Weyl group \(W\). Firstly, the author announces a sharpening of a result by \textit{Rossman}. The result concerns the formal \(T\)-characters of the ring of functions on the scheme theoretic cone \(T_v(X_w)\), where \(T\) is a maximal torus in \(B\) and \(v\) is a \(T\)-fixed point of \(X_w\). These characters are interpreted in terms of the elements of a certain matrix with entries in the \(W\)-field of rational functions on \(T\), and the involution \(e^\lambda \to e^{-\lambda}\) on that field. Secondly the author announces a criterion for smoothness of the points of \(X_w\) in terms of the coordinates of another matrix with elements in the \(W\)-field of rational functions on a Cartan subalgebra. The criterion is applicable, uniformly, to all groups \(G\) of the above type, in contrast to the criteria given by Lashmibai-Seshadri, Ryan, Kazhdan-Lusztig, Carrell-Peterson, Jantzen, and others. nil Hecke ring; singularities; smoothness of points; Schubert varieties Kumar, S.: The nil Hecke ring and singularity of Schubert varieties. In: Lie Theory and Geometry (in honor of Bertram Kostant), J.-L. Brylinski et. al. (eds.), Progress in Math. vol. 123, Birkhäuser (1994) 497--507 Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions The nil Hecke ring and singularity of Schubert varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article a unified description of the structure of the small cohomology rings for all projective homogeneous spaces \(SL_n(\mathbb C)/P\) (with \(P\) a parabolic subgroup) is given. First the results on the classical cohomology rings are recalled. Then the algebraic structure of the quantum cohomology ring is studied. Important results are the general quantum versions of the Giambelli and Pieri formulas of the classical cohomology (classical Schubert calculus). They are obtained via geometric computations of certain Gromov-Witten invariants, which are realized as intersection numbers on hyperquot schemes. quantum cohomology; Gromov-Witten invariants; Schubert calculus; small cohomology rings; homogeneous spaces; Pieri formulas Ionuţ Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485 -- 524. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Homogeneous spaces and generalizations, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On quantum cohomology rings of partial flag varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [Eur. J. Comb. 25, No. 8, 1327--1344 (2004; Zbl 1062.14065); C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)], \textit{W. Kraśkiewicz} and \textit{P. Pragacz} introduced representations of the upper-triangular Lie algebra \(\mathfrak{b}\) whose characters are Schubert polynomials. In [Eur. J. Comb. 58, 17--33 (2016; Zbl 1343.05168)], the author studied the properties of Kraśkiewicz-Pragacz modules using the theory of highest weight categories. From the results there, in particular we obtain a certain highest weight category whose standard modules are KP modules. In this paper we show that this highest weight category is self Ringel-dual. This leads to an interesting symmetry relation on Ext groups between KP modules. We also show that the tensor product operation on \(\mathfrak{b}\)-modules is compatible with Ringel duality functor. Schubert polynomials; Kraśkiewicz-Pragacz modules; highest weight categories; ringel duality; B-modules Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Classical problems, Schubert calculus Kraśkiewicz-Pragacz modules and ringel duality

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is well known that some cycles on Grassmannian manifolds have a close connection with analytic functions called polylogarithms (they generalize the classical dilogarithm). This was discovered by \textit{I. M. Gelfand} and \textit{R. D. MacPherson} [Adv. Math. 44, 279-312 (1982; Zbl 0504.57021)]. Later several new constructions have appeared. The paper under review is one of them. The authors consider an arbitrary real polygon \(P\) in an Euclidean space. Fix a complex projective space \(\mathbb{P}^ n\). A \(P\)-figure is an assignment of a linear subspace of \(\mathbb{P}^ n\) to any face of \(P\) with reasonable properties. The authors show how to attach to any \(P\)-figure a cocycle on a Grassmannian represented by a differential form. The integrals of these forms will be (roughly speaking) the desired polylogarithms. The paper is written very nicely. The author explains the trivial case of \(n=1\) (usual logarithm), then more the complicated case of \(n = 2\) (dilogarithm) and concentrate their efforts on the really difficult case of \(n = 3\). They hope to extend their construction to arbitrary \(n\) in the future. Grassmannian; polylogarithms; real polygon Hanamura (M.), MacPherson (R.).â Geometric construction of polylogarithms, Duke Math. J. 70, p. 481-516 (1993). Grassmannians, Schubert varieties, flag manifolds, Topology of real algebraic varieties, Generalizations (algebraic spaces, stacks) Geometric construction of polylogarithms

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper presents an algebro-combinatorial proof of a quantum Pieri's formula in the (small) quantum cohomology ring \(QH^\ast(Fl_n,\mathbb Z)\) of the flag manifold. \(QH^\ast(Fl_n,\mathbb Z)\) is canonically isomorphic to the quotient \[ \mathbb Z[x_1,\dots,x_n;q_1,\dots,q_{n-1}]/ \langle E_1,E_2,\dots,E_n\rangle. \] where the \(E_k\) are certain \(q\)-deformations of the elementary symmetric polynomials. The isomorphism is given by specifying \(x_1+x_2+\dots x_m\mapsto\sigma_{s_m}\) where \(\sigma_{s_m}\) is the Schubert class corresponding to a transposition from the permutation group, \(s_m\in S_n\), \(1\leq m\leq n-1\). The quantum Pieri formula is expressed in terms of certain \(\mathbb Z[q]\)-linear operators \(t_{ij}\), \(1\leq i<j\leq n\), acting on \(QH^\ast(Fl_n,\mathbb Z)\). Particularly the quantum Monk formula for a product of Schubert classes, with \(w\in S_n\), can be written in the form \( \sigma_{s_m}\ast\sigma_w=\sum_{a\leq m<b}t_{ab}(\sigma_w)\). This result is generalized to products \(\sigma_{c(k,m)}\ast\sigma_w\) and \(\sigma_{r(k,m)}\ast\sigma_w\) with cyclic permutations \(c(k,m)=s_{m-k+1}s_{m-k+2}\dots s_m\) and \(r(k,m)=s_{m+k-1}s_{m+k-2}\dots s_m\). In fact, the formula is first proven in a more abstract form in the framework of a quadratic algebra such that the operators \(t_{ij}\) satisfy the defining relations of that algebra. Furthermore, several corollaries of the formula are derived and discussed. flag manifold; Monk's formula; Pieri's formula; quantum cohomology Alexander Postnikov, On a quantum version of Pieri's formula, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 371 -- 383. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Enumerative problems (combinatorial problems) in algebraic geometry On a quantum version of Pieri's formula

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper gives a sufficient criterion for Schubert intersection numbers to vanish which can be executed in polynomial time. Schubert intersection numbers arise from Schubert varieties \(X_{w}\), which are certain subvarieties of the flag variety \(X\) indexed by permutations \(w \in S_{n}\). The Poincaré duals \(\sigma_{w} = [X_{w}]\) of Schubert varieties form a basis of the cohomology ring \(H^\ast(X)\) of the flag variety. A Schubert problem is a \(k\)-tuple \((w^{(1)}, w^{(2)}, \dots, w^{(k)})\) of permutations in \(S_{n}\) such that the sum of the lengths is \(\binom{n}{2}\). The Schubert intersection number \(C_{w^{(1)}, w^{(2)}, \dots, w^{(k)}}\) is equivalently either \begin{itemize} \item the multiplicity of \(\sigma_{w_{0}}\) in \(\prod_{i = 1}^k \sigma_{w^{(i)}}\), or \item the number of points in \(\bigcap_{i = 1}^k g_{i}X_{w^{(i)}}\), where each \(g_{i}\) lies in some dense open subset of \(GL_{n}\). \end{itemize} Finding a combinatorial counting rule for Schubert intersection numbers is a famous open problem. Many algorithms exist for computing them. As stated previously, the aim of the paper is to give an algorithm to decide whether or not they vanish in a given case. The algorithm which the paper introduces proceeds roughly as follows. Given \(w \in S_{n}\), one can define the Rothe diagram \(D(w)\) as a certain subset of the boxes of the \(n \times n\) grid. Given a Schubert problem \((w^{(1)}, w^{(2)}, \dots, w^{(k)})\), one concatenates the Rothe diagrams \(D(w^{(i)})\) to obtain a diagram \(D\). One then considers the fillings of \(D\) which obey certain rules. If there are no such fillings, then the Schubert intersection number vanishes. There is then a polynomial-time algorithm to determine whether there are indeed no such fillings. The arguments of the paper use generalised permutahedra, which are obtained from the standard permutahedron by degeneration. The particular generalised permutahedra that the authors are interested in are known as ``Schubitopes''. These are constructed from fillings of rectangular grids. In the case of the Rothe diagram \(D(w)\) mentioned above, the Schubitope obtained is the Newton polytope of the Schubert polynomial \(\mathfrak{S}_{w}\) of \(w\) by [\textit{A. Fink} et al., Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)]. The crux is that the emptiness of the set of fillings from the previous paragraph is equivalent to the presence of a certain point in a Schubitope by \textit{A. Adve} et al. [Sémin. Lothar. Comb. 82B, Paper No. 52, 12 p. (2019; Zbl 1436.05115)]. This can then be decided in polynomial time by standard linear programming methods, given the algorithm mentioned above. The paper finishes by comparing the test for vanishing of Schubert intersection numbers proven in the paper to three other tests from the literature. In each instance, there are some cases in which the test from the paper can decide and which the test from the literature cannot, and also some cases where the opposite is true. We mention finally that the paper contains a couple of variations on its main result. Schubitopes; Newton polytopes; Schubert polynomials; Schubert intersection numbers Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Linear programming Generalized permutahedra and Schubert calculus

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Schubert varieties on a flag manifold \(G/P\) give rise to a cell decomposition on \(G/P\) whose Kronecker duals, known as the Schubert classes on \(G/P\), form an additive base of the integral cohomology \(H^*(G/P)\). The Schubert's problem of characteristics asks to express a monomial in the Schubert classes as a linear combination in the Schubert basis. We present a unified formula expressing the characteristics of a flag manifold G/P as polynomials in the Cartan numbers of the group \(G\). As application we develop a direct approach to our recent works on the Schubert presentation of the cohomology rings of flag manifolds \(G/P\). Lie group; flag manifold; Schubert variety; cohomology Grassmannians, Schubert varieties, flag manifolds, Serre spectral sequences On Schubert's problem of characteristics

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We define a new family \(\widetilde{F}_w(X)\) of generating functions for \(w \in \widetilde{S}_n\) which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine Schur functions. As applications, we show how affine Stanley symmetric functions generalize the (dual of the) \(k\)-Schur functions of \textit{L. Lapointe, A. Lascoux} and \textit{J. Morse} [Duke Math. J. 116, 103--146 (2003; Zbl 1020.05069)] as well as the cylindric Schur functions of \textit{A. Postnikov} [Duke Math. J. 128, 473--509 (2005; Zbl 1081.14070)]. Conjecturally, affine Stanley symmetric functions should be related to the cohomology of the affine flag variety. T. Lam. ''Affine Stanley symmetric functions''. Amer. J. Math. 128 (2006), pp. 1553--1586. DOI. Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Affine Stanley symmetric functions

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that for any permutation \(w\) that avoids a certain set of 13 patterns of length 5 and 6, the Schubert polynomial \(\mathfrak{S}_w\) can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner similar to the Jacobi-Trudi identity. For such \(w\), this determinantal formula is equivalent to a (signed) subtraction-free expansion of \(\mathfrak{S}_w\) in the basis of standard elementary monomials. symmetric polynomials; Jacobi-Trudi identity Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices, Combinatorial identities, bijective combinatorics Determinantal formulas for SEM expansions of Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce coplactic raising and lowering operators \(E^{\prime}_i\), \(F^{\prime}_i\), \(E_i\), and \(F_i\) on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of ``doubled crystal'' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are the (skew) Schur \(Q\)-functions. We also give a new criterion for such tableaux to be ballot. combinatorial crystals; shifted Young tableaux; symmetric function theory; orthogonal Grassmannian Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus A crystal-like structure on shifted tableaux

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a root system of type \( A\), a certain extension of the quadratic algebra invented by S. Fomin and the first author is introduced and studied, which makes it possible to construct a model for the equivariant cohomology ring of the corresponding flag variety. As an application, a generalization of the equivariant Pieri rule for double Schubert polynomials is described. For a general finite Coxeter system, an extension of the corresponding Nichols-Woronowicz algebra is constructed. In the case of finite crystallographic Coxeter systems, a construction of an extended Nichols-Woronowicz algebra model for the equivariant cohomology of the corresponding flag variety is presented. root system of type \(A\); equivariant Pieri rule; Nichols-Woronowicz algebra Kirillov, A. N. and Maeno, T., Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties, St. Petersburg Math. J., 22, 3, 447-462, (2011) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be an \(N\)-dimensional complex vector space equipped with a symmetric or skew-symmetric bilinear form \(\omega\), which can be either trivial or non-degenerate. The Grassmannians \(I G_{\omega}(m, N)\) of classical Lie type parameterize \(m\)-dimensional isotropic vector subspaces of \(V\). The cohomology ring of an isotropic Grassmannian \(X =I G_{\omega}(m, N)\), or more generally of a homogeneous variety, has an additive basis of Schubert classes represented by Schubert subvarieties \(X_{\lambda}\). One of the central problems of Schubert calculus is to find a manifestly positive formula for the structure constants of the cup product of two Schubert cohomology classes, or equivalently, for the triple intersection numbers of three Schubert subvarieties in general position. Such a positive formula, called a Littlewood-Richardson rule, has deep connections to various subjects, including geometry, combinatorics and representation theory. An isotropic Grassmannian \(X\) can be written as a quotient of a classical complex simple Lie group \(G\) by a maximal parabolic subgroup \(P\) (with two notable exceptions of Lie type \(D_n\)). Fix a choice of maximal complex torus \(T\) and a Borel subgroup \(B\) with \(T \subset B \subset P\). The Schubert varieties \(X_{\lambda}\) are closures of \(B\)-orbits, and hence are \(T\)-stable. They give a basis \([X_{\lambda}]^T\) for the \(T\)-equivariant cohomology \({H^*}_T (X)\) as a \({H^*}_T (pt)\)-module. The structure coefficients \({N^{\nu}}_{\lambda,\mu}\) in the equivariant product, \[ [X_{\lambda}]^T \cdot [X_{\mu}]^T=\sum\limits_{\nu}{N^{\nu}}_{\lambda,\mu} [X_{\nu}]^T, \] are homogeneous polynomials which satisfy a positivity condition conjectured by \textit{D. Peterson} [Lectures on quantum cohomology of \(G/B\), MIT (1996)] and proved by \textit{W. Graham} [Duke Math. J. 109, No. 3, 599--614 (2001; Zbl 1069.14055)]. In particular, they are Graham-positive, meaning they are polynomials in the negative simple roots, with non-negative integer coefficients. These equivariant structure coefficients carry much more information than the triple intersection numbers of Schubert varieties, and are more challenging to study. In the present paper, the authors give for the first time an equivariant Pieri rule for Grassmannians of Lie types \(B, C\), and \(D\), as well as a new proof of the Pieri rule in type \(A\). Such a rule concerns products with the special Schubert classes \([X_{p}]^T\) , which are related to the equivariant Chern classes of the tautological quotient bundle, and generate the \(T\) -equivariant cohomology ring. Using geometric methods, they give a manifestly positive formula for the structure coefficients \({N^{\mu}}_{\lambda,p}\) of the equivariant multiplication \([X_{\lambda}]^T \cdot [X_{p}]^T\). isotropic Grassmannian; Schubert calculus; Littlewood-Richardson rule; \(T\)-equivariant cohomology; Graham-positive; equivariant Pieri rule Li, C.; Ravikumar, V.: Equivariant Pieri rules for isotropic grassmannians Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Equivariant homology and cohomology in algebraic topology Equivariant Pieri rules for isotropic Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show a new neutral-fermionic presentation of Ikeda-Naruse's \(K\)-theoretic \(Q\)-functions, which represent a Schubert class in the \(K\)-theory of coherent sheaves on the Lagrangian Grassmannian. Our presentation provides a simple description and yields a straightforward proof of two types of Pfaffian formulas for them. We present a dual space of \(G \varGamma \), the vector space generated by all \(K\)-theoretic \(Q\)-functions, by constructing a non-degenerate bilinear form which is compatible with the neutral-fermionic presentation. We give a new family of dual \(K\)-theoretic \(Q\)-functions, their neutral-fermionic presentations, and Pfaffian formulas. \(K\)-theoretic \(Q\)-functions; boson-fermion correspondence; neutral fermions; Pfaffian formulas Symmetric functions and generalizations, Polynomials and finite commutative rings, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Particle exchange symmetries in quantum theory (general) Neutral-fermionic presentation of the \(K\)-theoretic \(Q\)-function

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for \(\mathbb{Z}[x_1,x_2,\dots]\). We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gröbner geometry of matrix Schubert varieties. Schubert polynomials; Gröbner geometry; Young tableau Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) The prism tableau model for Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0625.00015.] The paper is a short exposé of some results of ``Formules de caractère pour les algèbres de Kac-Moody (to appear in Astérisque); in the author's announcement in C. R. Acad. Sci., Paris, Sér. I 303, 391-394 (1986; Zbl 0602.17008) the title was longer. Let S be a Schubert variety in the flag space G/B associated to some Kac-Moody data. Let L be an effective line bundle of G/B. Then it is proved that the restriction map \(\Gamma\) (G/B,L)\(\to \Gamma (S,L)\) is onto, and that H \(q(S,L)=0\) for \(q>0\). The author deduces a generalisation of Demazure-Weyl formulas [also proved independently in characteristic zero par \textit{S. Kumar} in Invent. Math. 89, 395-432 (1987; Zbl 0635.14023)]. He also deduces a generalisation of the Borel-Weil-Bott-Kempf theorem. The main technique of the proof is a refinement of Frobenius splitting (a technique due to Metha, Ramanan and Ramanathan). O. Mathieu : Fibrés en droite sur les variétés de Schubert associées aux algèbres de Kac-Moody . Proceeding de ''Symposium on topological methods in field theory'' (Helsinki-Juin 1986) Scientific World. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Homological methods in Lie (super)algebras, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds Fibrés en droite sur les variétés de Schubert associées aux algèbres de Kac-Moody. (Line bundles on Schubert varieties associated with Kac-Moody algebras)

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is known that the usual Schur \(S\)- and \(P\)-polynomials can be described via the Gysin homomorphisms for flag bundles in the ordinary cohomology theory. Recently, \textit{P. Pragacz} [Proc. Am. Math. Soc. 143, No. 11, 4705--4711 (2015; Zbl 1327.14034); corrigendum ibid. 144, No. 7, 3197 (2016; Zbl 1333.14009)] generalized these Gysin formulas to the Hall-Littlewood polynomials. In this paper, we introduce a universal analogue of the Hall-Littlewood polynomials, which we call the universal Hall-Littlewood functions, and give Gysin formulas for various flag bundles in the complex cobordism theory. Furthermore, we give two kinds of the universal analogue of the Schur polynomials, and some Gysin formulas for these functions are established. Schur \(S\)-, \(p\)-, and \(Q\)-functions; Hall-Littlewood function; complex-oriented generalized cohomology theory; Gysin map Nakagawa, M.; Naruse, H., The universal Gysin formulas for the universal Hall-Littlewood functions, (Contemp. Math., vol. 708, (2018), Amer Math. Soc.), 201-244 Symmetric functions and generalizations, Generalized (extraordinary) homology and cohomology theories in algebraic topology, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Bordism and cobordism theories and formal group laws in algebraic topology, Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism) Universal Gysin formulas for the universal Hall-Littlewood functions

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Denote by \(W_{\vec{a}}\) Schubert varieties associated to \(\vec{a}\) and by \(\sigma_{\vec{a}}\in\text{H}^{2|\vec{a}|}(G,\mathbb{C})\) the corresponding elements in cohomology where \(G\) is the Grassmannian. The symbol \(W_a\) stands for a special Schubert variety associated to \((a,0,\dots,0)\). Choose general points \(p_1,\dots,p_N\in\mathbb{P}^1\) and general translates of the \(W_{\vec{a}}\). The Gromov-Witten intersection number \(\langle W_{\vec{a}_1},\dots,W_{\vec{a}_N}\rangle_d\) is, by a naive definition, the number of holomorphic maps \(f:\mathbb{P}^1\to G\) of degree \(d\) with the property that \(f(p_i)\in W_{\vec{a}_i}\) for all \(i=1,\dots,N\). For \(d=0\) one gets the original intersection number. The small quantum ring is the vector space \(\text{H}^\ast(G,\mathbb{C})[q]\) over \(C[q]\) with an associative product which obeys \[ \sigma_{\vec{a}_1}\ast\dots\ast\sigma_{\vec{a}_N}= \sum_{d\geq 0}q^d\left(\sum_{\vec{a}} \langle W_{\vec{a}},W_{\vec{a}_1},\dots,W_{\vec{a}_N}\rangle_d \sigma_{\vec{a}}\right). \] The paper generalizes Giambelli's formula and Pieri's formula to the small quantum ring. In the quantum Giambelli formula that reads \(\sigma_{\vec{a}}=\Delta_{\vec{a}}(\sigma_\ast)\) no higher terms in \(q\) arise. The Giambelli determinant in cohomology classes corresponding to special Schubert varieties is evaluated in \(\text{H}^\ast(G,\mathbb{C})[q]\). On the other hand, the quantum Pieri formula has a correction term, \[ \sigma_a\ast\sigma_{\vec{a}}= p_{a,\vec{a}}(\sigma_{\vec\ast})+ q\left(\sum_{\vec c}\sigma_{\vec c}\right), \] with an appropriate range of \(\vec c\). Before giving the proofs the Gromov-Witten number is defined rigorously by considering intersections of Schubert varieties on the moduli space \(\mathcal M_d\) of holomorphic maps of degree \(d\) from \(\mathbb{P}^1\) to \(G\) with \(\mathcal M_d\) being an open subscheme in the Grothendieck quoted scheme. As a corollary of the quantum Giambelli's formula the author also shows a Vafa and Intriligator formula for the Gromov-Witten intersection number of special Schubert varieties. Gromov-Witten intersection number; small quantum ring; Giambelli's formula; Pieri's formula A. Bertram. ''Quantum Schubert calculus''. Adv. Math. 128 (1997), pp. 289--305.DOI. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Grassmannians, Schubert varieties, flag manifolds, Relationships between surfaces, higher-dimensional varieties, and physics, Quantum field theory on curved space or space-time backgrounds Quantum Schubert calculus

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials To each polynomial \(P\) with integral non-negative coefficients and constant term equal to 1, of degree \(d\), we associate a pair of elements \((y,w)\) in the symmetric group \(S_n\), where \(n=1+d+P(1)\), for which we prove that the Kazhdan-Lusztig polynomial \(P_{y,w}\) equals \(P\). This pair satisfies \(\ell(w)-\ell(y)=2d+P(1)-1\), where \(\ell(w)\) denotes the number of inversions of \(w\). -- For details see the author's paper: \textit{P. Polo}, Represent. Theory 3, No. 4, 90-104 (1999; Zbl 0968.14029). Kazhdan-Lusztig polynomials; Schubert varieties; intersection cohomology Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Representations of finite symmetric groups, Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Construction of arbitrary Kazhdan-Lusztig polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for \(\mathbb{Z}[x_1,x_2,\ldots]\). We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gröbner geometry of matrix Schubert varieties. Schubert polynomials; bi-Grassmannian permutations; semistandard tableaux A. Weigandt and A. Yong, The Prism tableau model for Schubert polynomials, J. Combin. Theory Ser. A 154 (2018), 551--582. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds The Prism tableau model for Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Young tableaux are well studied in representation theory and combinatorics [see the classic books: \textit{G. James} and \textit{A. Kerber}, ``The representation theory of the symmetric group'' (1981; Zbl 0491.20010) or \textit{G. E. Andrews}, ``The theory of partition'' (1976; Zbl 0371.10001)]. The well-known author gives the definitions and main theorems in these fields. But his main emphasis is on the application of Young tableaux in (algebraic) geometry. Defining equations for Grassmannians and flag varieties are given. We find Schubert varieties in flag manifolds, Chern classes are introduced, and the geometry of flag varieties is used to construct the Schubert polynomials of Lascoux and Schützenberger. The basic facts about intersection theory on Grassmannians are given, proofs at some times are delegated to the standard books of \textit{R. Hartshorne} [``Algebraic geometry'' (3rd edition 1983; Zbl 0531.14001)] and \textit{I. R. Shafarevich} [``Basic algebraic geometry. I and II'' (2nd edition 1994; Zbl 0797.14001 and Zbl 0797.14002)] on algebraic geometry. The wealth of presented material makes this excusable. There are numerous exercises with answers in each chapter and a lot of references to be found at the end of the book. Young tableau; Grassmannian; Schubert polynomials W. Fulton, \textit{Young tableaux: With Applications to Representation Theory and Geometry}, vol. 35 of \textit{London Mathematical Society Student Texts}. Cambridge University Press (1996). Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to group theory, Research exposition (monographs, survey articles) pertaining to combinatorics, Representations of finite symmetric groups, Representations of finite groups of Lie type, Representations of Lie and linear algebraic groups over real fields: analytic methods Young tableaux. With applications to representation theory and geometry

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given a vector bundle \(V\) of rank \(n\), the Grothendieck ring of classes of vector bundles of the relative flag manifold \({\mathcal F}(V)\) is generated by the classes \(a_1,\dots,a_n\) of the so-called tautological line bundles on \({\mathcal F}(V)\). The structure sheaf of a Schubert variety in \({\mathcal F}(V)\), having a finite resolution by vector bundles, can be expressed as a (Laurent) polynomial in the \(a_i\), \(1/a_i\). Explicit representatives \(G_\sigma\), \(\sigma\in{\mathfrak S}_n\) were defined by the author and \textit{M.-P. Schützenberger} [in: Invariant theory. Lect. Notes Math. 996, 118--144 (1983; Zbl 0542.14031) and C. R. Acad. Sci., Paris, 295, 629--633 (1982; Zbl 0542.14030)] under the name ``Grothendieck polynomials''. We describe how general Grothendieck of polynomials are related to those for Grassmann manifolds, which themselves are deformations of Schur functions. The geometry of Grassmann varieties is well understood thanks to Schubert, Giambelli and their successors. It is hoped that relating the Schubert subvarieties of a flag manifold to those of a Grassmannian will be of some help to understand those varieties, the singularities of which we still do not know how to describe. Schubert polynomials; 0-Hecke algebra; Grassmann manifolds A. Lascoux, Transition on Grothendieck polynomials, Physics and combinatorics, 2000 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, pp. 164 -- 179. Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial aspects of representation theory Transition of Grothendieck polynomials.