Generators of polynomial invariants from the vertex

Let a group G act linearly on a module M over a commutative semiring R, and so on the symmetric algebra S = Sym M, with invariant subalgebra A = Sym(M)^G. Write A₊ for the ideal of invariants with zero constant term, the contraction to A of the irrelevant ideal S₊.

The algebra A is graded by the homogeneous components it inherits from S (Subalgebra.gradedAlgebra), its irrelevant ideal for this grading is A₊, and its degree-zero part is R. So the graded Nakayama lemma TauCeti.GradedAlgebra.adjoin_eq_top_of_irrelevant_le says that homogeneous invariants of positive degree f₁, f₂, … whose classes span the cotangent space A₊ ⧸ A₊², that is, with A₊ ≤ (f₁, f₂, …) + A₊², generate A as an R-algebra (Representation.adjoin_range_eq_top_of_le_span_sup_sq).

Over a field k, for a finite group and finite-dimensional M, the invariant algebra has Krull dimension dim M (Representation.ringKrullDim_symmetricAlgebraInvariants). A surjection onto it from a polynomial ring in at most dim M variables is then injective, since a nonzero kernel element would lower the dimension (RingHom.injective_of_surjective_of_natCard_le_ringKrullDim_mvPolynomial). So if at most dim M homogeneous invariants of positive degree span A₊ ⧸ A₊², they are algebraically independent (Representation.algebraicIndependent_of_le_span_sup_sq) and there are exactly dim M of them (Representation.natCard_eq_finrank_of_le_span_sup_sq): A is a polynomial algebra on dim M homogeneous generators. The case dim M = 0 is included, with no generators.

This is the last step of the homological proof that, over a field whose characteristic does not divide the group order (for example ℂ), the invariants of a finite group generated by pseudoreflections form a polynomial algebra (Chevalley, Shephard–Todd); in modular characteristic this implication can fail. Regularity of A at its vertex bounds the dimension of A₊ ⧸ A₊² by dim M, and lifting a homogeneous basis gives such a family.

Main results

• Representation.adjoin_range_eq_top_of_le_span_sup_sq: homogeneous invariants of positive degree spanning A₊ modulo A₊² generate the invariant algebra.
• Representation.algebraicIndependent_of_le_span_sup_sq: over a field, for a finite group, at most dim M such invariants are algebraically independent.
• Representation.natCard_eq_finrank_of_le_span_sup_sq: there are then exactly dim M of them.

References

• 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.

theorem Representation.adjoin_range_eq_top_of_le_span_sup_sq {R : Type u_1} {G : Type u_3} {M : Type u_4} {ι : Type u_5} [CommSemiring R] [Monoid G] [AddCommMonoid M] [Module R M] (ρ : Representation R G M) (f : ι → ↥ρ.symmetricAlgebraInvariants) (d : ι → ℕ) (hd : ∀ (i : ι), 0 < d i) (hf : ∀ (i : ι), ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule R M (d i)) (hspan : Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule R M)).toIdeal ≤ Ideal.span (Set.range f) ⊔ Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule R M)).toIdeal ^ 2) : Algebra.adjoin R (Set.range f) = ⊤

Graded Nakayama lemma for polynomial invariants. Homogeneous invariants of positive degree whose classes span A₊ ⧸ A₊², where A₊ is the ideal of invariants with zero constant term, generate the invariant algebra Sym(M)^G as an R-algebra.

theorem Representation.algebraicIndependent_of_le_span_sup_sq {k : Type u_2} {G : Type u_3} {M : Type u_4} {ι : Type u_5} [Field k] [Group G] [Finite G] [AddCommGroup M] [Module k M] [FiniteDimensional k M] (ρ : Representation k G M) [Finite ι] (f : ι → ↥ρ.symmetricAlgebraInvariants) (d : ι → ℕ) (hd : ∀ (i : ι), 0 < d i) (hf : ∀ (i : ι), ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule k M (d i)) (hspan : Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule k M)).toIdeal ≤ Ideal.span (Set.range f) ⊔ Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule k M)).toIdeal ^ 2) (hcard : Nat.card ι ≤ Module.finrank k M) : AlgebraicIndependent k f

Polynomial invariants from the vertex. Over a field, for a finite group acting on a finite-dimensional space M, at most dim M homogeneous invariants of positive degree whose classes span A₊ ⧸ A₊² are algebraically independent. Since they also generate the invariant algebra (Representation.adjoin_range_eq_top_of_le_span_sup_sq), Sym(M)^G is the polynomial algebra on them.

theorem Representation.natCard_eq_finrank_of_le_span_sup_sq {k : Type u_2} {G : Type u_3} {M : Type u_4} {ι : Type u_5} [Field k] [Group G] [Finite G] [AddCommGroup M] [Module k M] [FiniteDimensional k M] (ρ : Representation k G M) [Finite ι] (f : ι → ↥ρ.symmetricAlgebraInvariants) (d : ι → ℕ) (hd : ∀ (i : ι), 0 < d i) (hf : ∀ (i : ι), ↑(f i) ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule k M (d i)) (hspan : Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule k M)).toIdeal ≤ Ideal.span (Set.range f) ⊔ Ideal.under (↥ρ.symmetricAlgebraInvariants) (HomogeneousIdeal.irrelevant (TauCeti.SymmetricAlgebra.homogeneousSubmodule k M)).toIdeal ^ 2) (hcard : Nat.card ι ≤ Module.finrank k M) : Nat.card ι = Module.finrank k M

Over a field, for a finite group acting on a finite-dimensional space M, a family of at most dim M homogeneous invariants of positive degree whose classes span A₊ ⧸ A₊² has exactly dim M members. So the polynomial presentation of Sym(M)^G by such a family, from Representation.algebraicIndependent_of_le_span_sup_sq, has exactly dim M variables.