Polynomial invariants from regularity at the vertex

Let a finite group G act linearly on a finite-dimensional vector space M over a field k, and so on the symmetric algebra S = Sym M, with invariant subalgebra A = Sym(M)^G. Write A₊ = S₊ ∩ A for the ideal of invariants with zero constant term, the vertex of Spec A.

If the local ring A_{A₊} is regular, then A is a polynomial algebra on finrank k M homogeneous invariants of positive degree. The invariant subalgebra is homogeneous, hence graded by the degrees of Sym M, connected since Sym⁰ M = k, and its irrelevant ideal is A₊; the general statement for connected graded algebras (TauCeti.GradedAlgebra.exists_bijective_aeval_of_isRegularLocalRing) applies, and the number of generators is the dimension of A_{A₊}, which is finrank k M (Representation.ringKrullDim_atPrime_under_irrelevant).

For M = V*, polynomial functions on V, this is the final step of the homological proof of the Chevalley–Shephard–Todd theorem: for a group generated by pseudoreflections whose order is invertible in k, Sym(V*) is faithfully flat over A (Representation.faithfullyFlat_symmetricAlgebra_dual_of_finrank_quotient_eq_one), so regularity of the polynomial ring descends to A_{A₊}, and then A is polynomial on dim V basic invariants.

Main results

• Representation.exists_bijective_aeval_symmetricAlgebraInvariants_of_isRegularLocalRing: if the invariant ring is regular at its vertex, it is a polynomial algebra on finrank k M homogeneous invariants of positive degree.

References

• R. Kane, Reflection Groups and Invariant Theory, Springer (2001), § 18.
• L. Smith, Polynomial Invariants of Finite Groups, A K Peters (1995), § 7.2.

theorem Representation.exists_bijective_aeval_symmetricAlgebraInvariants_of_isRegularLocalRing {k : Type u_1} {G : Type u_2} {M : Type u_3} [Field k] [Group G] [Finite G] [AddCommGroup M] [Module k M] [FiniteDimensional k M] (ρ : Representation k G M) [IsRegularLocalRing (Localization.AtPrime (Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule k M)).toIdeal))] : ∃ (f : Fin (Module.finrank k M) → ↥ρ.symmetricAlgebraInvariants), (∀ (i : Fin (Module.finrank k M)), ∃ (d : ℕ), 0 < d ∧ ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule k M d) ∧ Function.Bijective ⇑(MvPolynomial.aeval f)

Polynomial invariants from regularity at the vertex. For a finite group acting linearly on a finite-dimensional vector space M, if the invariant subalgebra A = Sym(M)^G is regular at the ideal A₊ of invariants with zero constant term, then there are homogeneous invariants f₁, …, fₙ of positive degree, with n = finrank k M, such that k[X₁, …, Xₙ] → A, Xᵢ ↦ fᵢ, is an isomorphism.