The Chevalley–Shephard–Todd theorem

Let a finite group G act linearly on a finite-dimensional vector space V over a field K in which #G is invertible, and so on the polynomial functions Sym(V*) through the contragredient action. If G is generated by pseudoreflections, the elements whose fixed space is a hyperplane, then the invariant algebra Sym(V*)^G is a polynomial algebra on dim V homogeneous invariants of positive degree. Neither faithfulness of the action nor algebraic closedness of K is needed; over ℂ this is the forward direction of the Shephard–Todd theorem, and over ℝ it is Chevalley's theorem for finite reflection groups.

The proof is homological, and runs through regularity at the vertex rather than through an explicit system of basic invariants.

• The divided-difference operators of the pseudoreflections, together with the Reynolds operator, kill Tor'₁ over A = Sym(V*)^G with coefficients in Sym(V*), so Sym(V*) is faithfully flat over A (Representation.faithfullyFlat_symmetricAlgebra_dual_of_finrank_quotient_eq_one).
• For any finite group acting on a finite-dimensional space M for which Sym M is faithfully flat over its invariants A, the ring A is regular: Sym M is a regular ring of dimension dim M (TauCeti.SymmetricAlgebra.isRegularRing), so every finite A-module has finite projective dimension, and Serre's criterion applies (TauCeti.IsRegularRing.of_faithfullyFlat). In particular A is regular at its vertex, the ideal A₊ of invariants with zero constant term, and lifting a homogeneous basis of A₊ ⧸ A₊² gives polynomial generators (Representation.exists_bijective_aeval_symmetricAlgebraInvariants_of_isRegularLocalRing).

It suffices that the faithful image G ⧸ ker ρ is finite and has invertible order; the kernel acts trivially on all polynomial functions, and quotienting by it preserves the invariant algebra. The order of this effective group must still be invertible: in modular characteristic there are groups generated by pseudoreflections whose invariants are not polynomial.

Conversely, over a field of characteristic zero, if G acts faithfully and Sym(V*)^G is a polynomial algebra on homogeneous invariants, then G is generated by its pseudoreflections. So for a faithful action in characteristic zero, G is generated by pseudoreflections exactly when its polynomial invariants form a polynomial algebra. The proof compares degrees. Let H be the subgroup generated by the pseudoreflections; by the forward direction, Sym(V*)^H is a polynomial algebra too, say on homogeneous invariants of degrees e₁, …, eₙ, while Sym(V*)^G is polynomial on homogeneous invariants of degrees d₁, …, dₙ.

• Since Sym(V*)^G ⊆ Sym(V*)^H, the basic invariants of G of degree at most m are algebraically independent polynomials in the basic invariants of H of degree at most m, so there are at most as many of them (TauCeti.GradedAlgebra.card_filter_le_le_card_filter_le).
• Comparing Molien's formula with the Hilbert series of the two polynomial algebras gives ∑ dᵢ = dim V + N = ∑ eᵢ, where N is the number of pseudoreflections, which is the same for G and for H (Representation.sum_eq_finrank_add_card_of_mk_finrank_mul_prod_eq_one).
• These two facts force the multisets of degrees to coincide (Multiset.eq_of_forall_countP_le_le_of_sum_le), and then #G = ∏ dᵢ = ∏ eᵢ = #H (Representation.prod_eq_card_of_mk_finrank_mul_prod_eq_one).

This argument establishes the converse without the geometry of the quotient map V → V/G: it does not identify the ramification of V/H → V/G in codimension one, nor show that this map is étale.

Faithfulness cannot be dropped from the converse: a nontrivial group acting trivially has polynomial invariants Sym(V*) but no pseudoreflections.

Main results

• Representation.exists_bijective_aeval_symmetricAlgebraInvariants_of_faithfullyFlat: if Sym M is faithfully flat over its invariants, they form a polynomial algebra on finrank k M homogeneous invariants of positive degree.

• exists_bijective_aeval_symmetricAlgebraInvariants_dual_of_pseudoreflectionSubgroup_eq_top, in the namespace Representation: the Chevalley–Shephard–Todd theorem, for a group generated by pseudoreflections whose order is invertible in the field.

• eq_top_of_bijective_aeval_symmetricAlgebraInvariants_dual_of_pseudoreflectionSubgroup_le, in the namespace Representation: in characteristic zero, for a faithful action with polynomial invariants, a subgroup containing every pseudoreflection and with polynomial invariants is the whole group.

• pseudoreflectionSubgroup_eq_top_of_bijective_aeval_symmetricAlgebraInvariants_dual, in the namespace Representation: the converse of the Chevalley–Shephard–Todd theorem, for a faithful action in characteristic zero.

• pseudoreflectionSubgroup_eq_top_iff_exists_bijective_aeval_symmetricAlgebraInvariants_dual, in the namespace Representation: the two directions together.

• exists_bijective_aeval_symmetricAlgebraInvariants_dual_of_pseudoreflectionSubgroup_sup_ker_eq_top in the namespace Representation: the finite-faithful-image variant, when only that image is finite with invertible order and pseudoreflections together with the kernel generate the group.

References

• C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955).
• G. C. Shephard, J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954).
• R. Kane, Reflection Groups and Invariant Theory, CMS Books in Mathematics, Springer (2001), §§ 17–18.
• L. Smith, Polynomial Invariants of Finite Groups, A K Peters (1995), § 7.2.
• R. P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 475–511, Section 4.

theorem Representation.exists_bijective_aeval_symmetricAlgebraInvariants_of_faithfullyFlat {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) [Module.FaithfullyFlat (↥ρ.symmetricAlgebraInvariants) (SymmetricAlgebra k M)] : ∃ (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 faithful flatness. For a finite group acting linearly on a finite-dimensional vector space M, if Sym M is faithfully flat over the invariant subalgebra A = Sym(M)^G, 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.

theorem Representation.exists_bijective_aeval_symmetricAlgebraInvariants_dual_of_pseudoreflectionSubgroup_eq_top {K : Type u_1} {G : Type u_2} {V : Type u_3} [Field K] [Group G] [Fintype G] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (ρ : Representation K G V) [Invertible ↑(Fintype.card G)] (hG : ρ.pseudoreflectionSubgroup = ⊤) : ∃ (f : Fin (Module.finrank K V) → ↥ρ.dual.symmetricAlgebraInvariants), (∀ (i : Fin (Module.finrank K V)), ∃ (d : ℕ), 0 < d ∧ ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) d) ∧ Function.Bijective ⇑(MvPolynomial.aeval f)

The Chevalley–Shephard–Todd theorem. Let a finite group G, whose order is invertible in the field K, act linearly on a finite-dimensional vector space V, and suppose that G is generated by pseudoreflections, the elements whose fixed space is a hyperplane. Then the polynomial invariants Sym(V*)^G form a polynomial algebra on dim V homogeneous invariants f₁, …, fₙ of positive degree: K[X₁, …, Xₙ] → Sym(V*)^G, Xᵢ ↦ fᵢ, is an isomorphism.

theorem Representation.exists_bijective_aeval_symmetricAlgebraInvariants_dual_of_pseudoreflectionSubgroup_sup_ker_eq_top {K : Type u_1} {G : Type u_2} {V : Type u_3} [Field K] [Group G] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (ρ : Representation K G V) [Finite (G ⧸ MonoidHom.ker ρ)] [Invertible ↑(Nat.card (G ⧸ MonoidHom.ker ρ))] (hG : ρ.pseudoreflectionSubgroup ⊔ MonoidHom.ker ρ = ⊤) : ∃ (f : Fin (Module.finrank K V) → ↥ρ.dual.symmetricAlgebraInvariants), (∀ (i : Fin (Module.finrank K V)), ∃ (d : ℕ), 0 < d ∧ ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) d) ∧ Function.Bijective ⇑(MvPolynomial.aeval f)

The Chevalley–Shephard–Todd theorem for a finite faithful image. Only the effective group G ⧸ ker ρ must be finite and have invertible order in K; the original group may be infinite or have order divisible by the characteristic. If pseudoreflections together with the kernel generate G, then its polynomial invariant algebra has dim V homogeneous algebraically independent generators of positive degree.

theorem Representation.eq_top_of_bijective_aeval_symmetricAlgebraInvariants_dual_of_pseudoreflectionSubgroup_le {K : Type u_1} {G : Type u_2} {V : Type u_3} [Field K] [CharZero K] [Group G] [Finite G] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (ρ : Representation K G V) (hρ : Function.Injective ⇑ρ) {H : Subgroup G} (hH : ρ.pseudoreflectionSubgroup ≤ H) {ι : Type u_4} {κ : Type u_5} [Finite ι] [Finite κ] {f : ι → ↥ρ.dual.symmetricAlgebraInvariants} {d : ι → ℕ} (hf : ∀ (i : ι), ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) (d i)) (hbij : Function.Bijective ⇑(MvPolynomial.aeval f)) {g : κ → ↥(dual (MonoidHom.comp ρ H.subtype)).symmetricAlgebraInvariants} {e : κ → ℕ} (hg : ∀ (j : κ), ↑(g j) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) (e j)) (hgbij : Function.Bijective ⇑(MvPolynomial.aeval g)) : H = ⊤

Comparing the degrees of G and of a subgroup containing its pseudoreflections. Let a finite group G act faithfully and linearly on a finite-dimensional vector space V over a field K of characteristic zero, and let H be a subgroup containing every pseudoreflection of G. If the polynomial invariants of both G and H form polynomial algebras on homogeneous invariants, then H = G.

theorem Representation.pseudoreflectionSubgroup_eq_top_of_bijective_aeval_symmetricAlgebraInvariants_dual {K : Type u_1} {G : Type u_2} {V : Type u_3} [Field K] [CharZero K] [Group G] [Finite G] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (ρ : Representation K G V) (hρ : Function.Injective ⇑ρ) {ι : Type u_4} [Finite ι] {f : ι → ↥ρ.dual.symmetricAlgebraInvariants} {d : ι → ℕ} (hf : ∀ (i : ι), ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) (d i)) (hbij : Function.Bijective ⇑(MvPolynomial.aeval f)) : ρ.pseudoreflectionSubgroup = ⊤

The converse of the Chevalley–Shephard–Todd theorem. Let a finite group G act faithfully and linearly on a finite-dimensional vector space V over a field K of characteristic zero. If the polynomial invariants Sym(V*)^G form a polynomial algebra on homogeneous invariants, then G is generated by its pseudoreflections, the elements whose fixed space is a hyperplane.

theorem Representation.pseudoreflectionSubgroup_eq_top_iff_exists_bijective_aeval_symmetricAlgebraInvariants_dual {K : Type u_1} {G : Type u_2} {V : Type u_3} [Field K] [CharZero K] [Group G] [Finite G] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (ρ : Representation K G V) (hρ : Function.Injective ⇑ρ) : ρ.pseudoreflectionSubgroup = ⊤ ↔ ∃ (f : Fin (Module.finrank K V) → ↥ρ.dual.symmetricAlgebraInvariants), (∀ (i : Fin (Module.finrank K V)), ∃ (d : ℕ), 0 < d ∧ ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) d) ∧ Function.Bijective ⇑(MvPolynomial.aeval f)

The Chevalley–Shephard–Todd theorem. Let a finite group G act faithfully and linearly on a finite-dimensional vector space V over a field K of characteristic zero. Then G is generated by pseudoreflections, the elements whose fixed space is a hyperplane, if and only if the polynomial invariants Sym(V*)^G form a polynomial algebra on homogeneous invariants f₁, …, fₙ of positive degree, where necessarily n = dim V.