Coinvariant Hilbert series from homogeneous basic invariants

A positive-degree homogeneous regular sequence generating the invariant algebra generates the actual Hilbert ideal of the action. Consequently the Hilbert series of the coinvariant algebra is the Hilbert series of the symmetric algebra multiplied by ∏ i, (1 - t^(d i)).

For polynomial functions on a finite-dimensional representation of a finite group generated by pseudoreflections, regularity of the basic invariants is supplied by the flatness theorem, under the hypothesis that the group order is invertible in the field. No characteristic-zero or faithfulness hypothesis is needed for this product formula. The coefficients may be taken in ℤ, retaining the dimensions in positive characteristic. The empty basic system is included. This formula does not identify the product with a Coxeter length sum or compute the Hilbert series of the ambient symmetric algebra.

References

• R. P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), §3.
• R. Kane, Reflection Groups and Invariant Theory, Springer (2001), §§17–18.

theorem Representation.mk_finrank_quotientPiece_symmetricAlgebraHilbertIdeal {K : Type u_1} {G : Type u_2} {M : Type u_3} {S : Type u_4} [Field K] [Monoid G] [AddCommGroup M] [Module K M] [Module.Finite K M] [CommRing S] (ρ : Representation K G M) {n : ℕ} (f : Fin n → ↥ρ.symmetricAlgebraInvariants) (d : Fin n → ℕ) (hd : ∀ (i : Fin n), 0 < d i) (hf : ∀ (i : Fin n), ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K M (d i)) (hsurj : Function.Surjective ⇑(MvPolynomial.aeval f)) (hreg : RingTheory.Sequence.IsWeaklyRegular (SymmetricAlgebra K M) (List.ofFn fun (i : Fin n) => ↑(f i))) : (PowerSeries.mk fun (j : ℕ) => ↑(Module.finrank K ↥(TauCeti.GradedAlgebra.quotientPiece (TauCeti.SymmetricAlgebra.homogeneousSubmodule K M) ρ.symmetricAlgebraHilbertIdeal j))) = (PowerSeries.mk fun (j : ℕ) => ↑(Module.finrank K ↥(TauCeti.SymmetricAlgebra.homogeneousSubmodule K M j))) * ∏ i : Fin n, (1 - PowerSeries.X ^ d i)

The coinvariant Hilbert series for a homogeneous weakly regular generating family of invariants. The left side uses the actual Hilbert ideal of the representation, not an independently presented quotient. Surjectivity of evaluation suffices to identify that ideal; algebraic independence is not required here.

theorem Representation.mk_finrank_quotientPiece_symmetricAlgebraHilbertIdeal_dual_of_pseudoreflectionSubgroup_eq_top {K : Type u_1} {G : Type u_2} {V : Type u_3} {S : Type u_4} [Field K] [Group G] [AddCommGroup V] [Module K V] [Module.Finite K V] [Fintype G] [Invertible ↑(Fintype.card G)] [CommRing S] (ρ : Representation K G V) (hG : ρ.pseudoreflectionSubgroup = ⊤) {n : ℕ} (f : Fin n → ↥ρ.dual.symmetricAlgebraInvariants) (d : Fin n → ℕ) (hd : ∀ (i : Fin n), 0 < d i) (hf : ∀ (i : Fin n), ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) (d i)) (hbij : Function.Bijective ⇑(MvPolynomial.aeval f)) : (PowerSeries.mk fun (j : ℕ) => ↑(Module.finrank K ↥(TauCeti.GradedAlgebra.quotientPiece (TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V)) ρ.dual.symmetricAlgebraHilbertIdeal j))) = (PowerSeries.mk fun (j : ℕ) => ↑(Module.finrank K ↥(TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) j))) * ∏ i : Fin n, (1 - PowerSeries.X ^ d i)

The coinvariant Hilbert-series product formula for a group generated by pseudoreflections and any chosen homogeneous polynomial basic invariants. Its regular sequence hypothesis is proved from the finite group's nonmodular flatness theorem. Integer power-series coefficients retain the unreduced graded dimensions.