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/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import algebra.group_ring_action
import algebra.hom.group_action
import data.polynomial.algebra_map
import data.polynomial.monic
import group_theory.group_action.quotient
/-!
# Group action on rings applied to polynomials
This file contains instances and definitions relating `mul_semiring_action` to `polynomial`.
-/
variables (M : Type*) [monoid M]
open_locale polynomial
namespace polynomial
variables (R : Type*) [semiring R]
variables {M}
lemma smul_eq_map [mul_semiring_action M R] (m : M) :
((•) m) = map (mul_semiring_action.to_ring_hom M R m) :=
begin
suffices :
distrib_mul_action.to_add_monoid_hom R[X] m =
(map_ring_hom (mul_semiring_action.to_ring_hom M R m)).to_add_monoid_hom,
{ ext1 r, exact add_monoid_hom.congr_fun this r, },
ext n r : 2,
change m • monomial n r = map (mul_semiring_action.to_ring_hom M R m) (monomial n r),
simpa only [polynomial.map_monomial, polynomial.smul_monomial],
end
variables (M)
noncomputable instance [mul_semiring_action M R] : mul_semiring_action M R[X] :=
{ smul := (•),
smul_one := λ m,
(smul_eq_map R m).symm ▸ polynomial.map_one (mul_semiring_action.to_ring_hom M R m),
smul_mul := λ m p q,
(smul_eq_map R m).symm ▸ polynomial.map_mul (mul_semiring_action.to_ring_hom M R m),
..polynomial.distrib_mul_action }
variables {M R}
variables [mul_semiring_action M R]
@[simp] lemma smul_X (m : M) : (m • X : R[X]) = X :=
(smul_eq_map R m).symm ▸ map_X _
variables (S : Type*) [comm_semiring S] [mul_semiring_action M S]
theorem smul_eval_smul (m : M) (f : S[X]) (x : S) :
(m • f).eval (m • x) = m • f.eval x :=
polynomial.induction_on f
(λ r, by rw [smul_C, eval_C, eval_C])
(λ f g ihf ihg, by rw [smul_add, eval_add, ihf, ihg, eval_add, smul_add])
(λ n r ih, by rw [smul_mul', smul_pow', smul_C, smul_X, eval_mul, eval_C, eval_pow, eval_X,
eval_mul, eval_C, eval_pow, eval_X, smul_mul', smul_pow'])
variables (G : Type*) [group G]
theorem eval_smul' [mul_semiring_action G S] (g : G) (f : S[X]) (x : S) :
f.eval (g • x) = g • (g⁻¹ • f).eval x :=
by rw [← smul_eval_smul, smul_inv_smul]
theorem smul_eval [mul_semiring_action G S] (g : G) (f : S[X]) (x : S) :
(g • f).eval x = g • f.eval (g⁻¹ • x) :=
by rw [← smul_eval_smul, smul_inv_smul]
end polynomial
section comm_ring
variables (G : Type*) [group G] [fintype G]
variables (R : Type*) [comm_ring R] [mul_semiring_action G R]
open mul_action
open_locale classical
/-- the product of `(X - g • x)` over distinct `g • x`. -/
noncomputable def prod_X_sub_smul (x : R) : R[X] :=
(finset.univ : finset (G ⧸ mul_action.stabilizer G x)).prod $
λ g, polynomial.X - polynomial.C (of_quotient_stabilizer G x g)
theorem prod_X_sub_smul.monic (x : R) : (prod_X_sub_smul G R x).monic :=
polynomial.monic_prod_of_monic _ _ $ λ g _, polynomial.monic_X_sub_C _
theorem prod_X_sub_smul.eval (x : R) : (prod_X_sub_smul G R x).eval x = 0 :=
(monoid_hom.map_prod
((polynomial.aeval x).to_ring_hom.to_monoid_hom : R[X] →* R) _ _).trans $
finset.prod_eq_zero (finset.mem_univ $ quotient_group.mk 1) $
by simp
theorem prod_X_sub_smul.smul (x : R) (g : G) :
g • prod_X_sub_smul G R x = prod_X_sub_smul G R x :=
finset.smul_prod.trans $ fintype.prod_bijective _ (mul_action.bijective g) _ _
(λ g', by rw [of_quotient_stabilizer_smul, smul_sub, polynomial.smul_X, polynomial.smul_C])
theorem prod_X_sub_smul.coeff (x : R) (g : G) (n : ℕ) :
g • (prod_X_sub_smul G R x).coeff n =
(prod_X_sub_smul G R x).coeff n :=
by rw [← polynomial.coeff_smul, prod_X_sub_smul.smul]
end comm_ring
namespace mul_semiring_action_hom
variables {M}
variables {P : Type*} [comm_semiring P] [mul_semiring_action M P]
variables {Q : Type*} [comm_semiring Q] [mul_semiring_action M Q]
open polynomial
/-- An equivariant map induces an equivariant map on polynomials. -/
protected noncomputable def polynomial (g : P →+*[M] Q) : P[X] →+*[M] Q[X] :=
{ to_fun := map g,
map_smul' := λ m p, polynomial.induction_on p
(λ b, by rw [smul_C, map_C, coe_fn_coe, g.map_smul, map_C, coe_fn_coe, smul_C])
(λ p q ihp ihq, by rw [smul_add, polynomial.map_add, ihp, ihq, polynomial.map_add, smul_add])
(λ n b ih, by rw [smul_mul', smul_C, smul_pow', smul_X, polynomial.map_mul, map_C,
polynomial.map_pow, map_X, coe_fn_coe, g.map_smul, polynomial.map_mul, map_C,
polynomial.map_pow, map_X, smul_mul', smul_C, smul_pow', smul_X, coe_fn_coe]),
map_zero' := polynomial.map_zero g,
map_add' := λ p q, polynomial.map_add g,
map_one' := polynomial.map_one g,
map_mul' := λ p q, polynomial.map_mul g }
@[simp] theorem coe_polynomial (g : P →+*[M] Q) :
(g.polynomial : P[X] → Q[X]) = map g :=
rfl
end mul_semiring_action_hom
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