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feat: renaming symmetric MvPolynomials (#11251)
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Co-authored-by: Junyan Xu <[email protected]>
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Ruben-VandeVelde and alreadydone committed Mar 24, 2024
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20 changes: 20 additions & 0 deletions Mathlib/RingTheory/MvPolynomial/Symmetric.lean
Original file line number Diff line number Diff line change
Expand Up @@ -138,6 +138,16 @@ theorem map (hφ : IsSymmetric φ) (f : R →+* S) : IsSymmetric (map f φ) := f
rw [← map_rename, hφ]
#align mv_polynomial.is_symmetric.map MvPolynomial.IsSymmetric.map

protected theorem rename (hφ : φ.IsSymmetric) (e : σ ≃ τ) : (rename e φ).IsSymmetric := fun _ => by
apply rename_injective _ e.symm.injective
simp_rw [rename_rename, ← Equiv.coe_trans, Equiv.self_trans_symm, Equiv.coe_refl, rename_id]
rw [hφ]

@[simp]
theorem _root_.MvPolynomial.isSymmetric_rename {e : σ ≃ τ} :
(MvPolynomial.rename e φ).IsSymmetric ↔ φ.IsSymmetric :=
fun h => by simpa using (IsSymmetric.rename (R := R) h e.symm), (IsSymmetric.rename · e)⟩

end CommSemiring

section CommRing
Expand All @@ -156,6 +166,16 @@ end CommRing

end IsSymmetric

/-- `MvPolynomial.rename` induces an isomorphism between the symmetric subalgebras. -/
@[simps!]
def renameSymmetricSubalgebra [CommSemiring R] (e : σ ≃ τ) :
symmetricSubalgebra σ R ≃ₐ[R] symmetricSubalgebra τ R :=
AlgEquiv.ofAlgHom
(((rename e).comp (symmetricSubalgebra σ R).val).codRestrict _ <| fun x => x.2.rename e)
(((rename e.symm).comp <| Subalgebra.val _).codRestrict _ <| fun x => x.2.rename e.symm)
(AlgHom.ext <| fun p => Subtype.ext <| by simp)
(AlgHom.ext <| fun p => Subtype.ext <| by simp)

section ElementarySymmetric

open Finset
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