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TauCeti.RepresentationTheory.SymmetricAlgebra.Free

Polynomial functions are free over the invariants of a pseudoreflection group #

Let a group G act linearly on a module M, and so on the symmetric algebra S = Sym M, with invariant subalgebra A = Sym(M)^G, the ideal A₊ of invariants with zero constant term, and Hilbert ideal A₊ · S. The quotient S ⧸ A₊ · S is the coinvariant algebra.

Homogeneous elements of S whose images span the coinvariant algebra generate S as an A-module; this is the graded Nakayama lemma, proved by induction on the degree. Over a field k, for a finite group acting on a finite-dimensional space M such that S is flat over A, homogeneous lifts of a k-basis of the coinvariant algebra are moreover linearly independent over A: a relation among them has all its coefficients in A₊, flatness then puts the module of relations inside A₊ times itself, and the Nakayama lemma, in the domain A, makes it zero. So S is a free A-module, of rank the dimension of the coinvariant algebra.

When the action is faithful, the rank of S over A is |G|, the degree of the extension of fraction fields (Representation.finrank_symmetricAlgebraInvariants), so the coinvariant algebra has dimension |G|.

For a finite group generated by pseudoreflections, whose order is invertible in the field, the polynomial functions Sym(V*) are faithfully flat over their invariants (Representation.faithfullyFlat_symmetricAlgebra_dual_of_pseudoreflectionSubgroup_eq_top). So Sym(V*) is a free module over Sym(V*)^G, with a basis of homogeneous lifts of any basis of the coinvariant algebra; for a faithful action its rank and the dimension of the coinvariant algebra are |G|.

Main results #

References #

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theorem Representation.span_range_eq_top_of_span_range_mk_eq_top {R : Type u_1} {G : Type u_2} {M : Type u_3} [CommRing R] [Monoid G] [AddCommMonoid M] [Module R M] (ρ : Representation R G M) {ι : Type u_4} (b : ι → SymmetricAlgebra R M) (hb : ∀ (i : ι), ∃ (d : ℕ), b i ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule R M d) (hspan : Submodule.span R (Set.range (⇑(Ideal.Quotient.mkₐ R ρ.symmetricAlgebraHilbertIdeal) ∘ b)) = ⊤) :
Submodule.span (↥ρ.symmetricAlgebraInvariants) (Set.range b) = ⊤

Graded Nakayama lemma for the coinvariant algebra. Homogeneous elements of Sym M whose images span the coinvariant algebra Sym M ⧸ A₊ · Sym M over R generate Sym M as a module over the invariant subalgebra A = Sym(M)^G.

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theorem Representation.linearIndependent_of_linearIndependent_mk {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.Flat (↥ρ.symmetricAlgebraInvariants) (SymmetricAlgebra k M)] {ι : Type u_4} (b : ι → SymmetricAlgebra k M) (hb : ∀ (i : ι), ∃ (d : ℕ), b i ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule k M d) (hli : LinearIndependent k (⇑(Ideal.Quotient.mkₐ k ρ.symmetricAlgebraHilbertIdeal) ∘ b)) (hspan : Submodule.span k (Set.range (⇑(Ideal.Quotient.mkₐ k ρ.symmetricAlgebraHilbertIdeal) ∘ b)) = ⊤) :
LinearIndependent (↥ρ.symmetricAlgebraInvariants) b

Homogeneous lifts of a basis of the coinvariant algebra are independent. For a finite group acting on a finite-dimensional space M such that Sym M is flat over its invariants A, homogeneous elements of Sym M whose images form a k-basis of the coinvariant algebra Sym M ⧸ A₊ · Sym M are linearly independent over A. With Representation.span_range_eq_top_of_span_range_mk_eq_top, they form an A-basis of Sym M.

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theorem Representation.exists_basis_quotient_mk_eq_of_flat {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.Flat (↥ρ.symmetricAlgebraInvariants) (SymmetricAlgebra k M)] :
∃ (t : Set (SymmetricAlgebra k M ⧸ ρ.symmetricAlgebraHilbertIdeal)) (b : Module.Basis (↑t) (↥ρ.symmetricAlgebraInvariants) (SymmetricAlgebra k M)) (v : Module.Basis (↑t) k (SymmetricAlgebra k M ⧸ ρ.symmetricAlgebraHilbertIdeal)), (∀ (i : ↑t), ∃ (n : ℕ), b i ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule k M n) ∧ ∀ (i : ↑t), (Ideal.Quotient.mk ρ.symmetricAlgebraHilbertIdeal) (b i) = v i

Compatible bases. If Sym M is flat over the invariants A of a finite group acting on a finite-dimensional space M, it has an A-basis b of homogeneous elements whose images form a k-basis v of the coinvariant algebra Sym M ⧸ A₊ · Sym M.

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theorem Representation.free_symmetricAlgebra_of_flat {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.Flat (↥ρ.symmetricAlgebraInvariants) (SymmetricAlgebra k M)] :
Module.Free (↥ρ.symmetricAlgebraInvariants) (SymmetricAlgebra k M)

If Sym M is flat over the invariants A of a finite group acting on a finite-dimensional space M, then it is a free A-module.

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theorem Representation.finrank_symmetricAlgebraInvariants_eq_finrank_quotient {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.Flat (↥ρ.symmetricAlgebraInvariants) (SymmetricAlgebra k M)] :
Module.finrank (↥ρ.symmetricAlgebraInvariants) (SymmetricAlgebra k M) = Module.finrank k (SymmetricAlgebra k M ⧸ ρ.symmetricAlgebraHilbertIdeal)

If Sym M is flat over the invariants A of a finite group acting on a finite-dimensional space M, then its rank over A is the dimension of the coinvariant algebra Sym M ⧸ A₊ · Sym M.

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theorem Representation.finrank_quotient_symmetricAlgebraHilbertIdeal_of_flat {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.Flat (↥ρ.symmetricAlgebraInvariants) (SymmetricAlgebra k M)] (hρ : Function.Injective ⇑ρ) :
Module.finrank k (SymmetricAlgebra k M ⧸ ρ.symmetricAlgebraHilbertIdeal) = Nat.card G

The coinvariant algebra has dimension |G|. If a finite group G acts faithfully on a finite-dimensional space M and Sym M is flat over its invariants A, then the coinvariant algebra Sym M ⧸ A₊ · Sym M has dimension |G|.

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theorem Representation.free_symmetricAlgebra_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 = ⊤) :
Module.Free (↥ρ.dual.symmetricAlgebraInvariants) (SymmetricAlgebra K (Module.Dual K V))

Polynomial functions are free over reflection invariants. If a finite group G, whose order is invertible in the field K, acts linearly on a finite-dimensional vector space V and is generated by pseudoreflections, then Sym(V*) is a free module over the invariants Sym(V*)^G. Its rank is |G| when the action is faithful (Representation.finrank_symmetricAlgebraInvariants), and homogeneous lifts of any basis of the coinvariant algebra form a basis (Representation.linearIndependent_of_linearIndependent_mk).

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theorem Representation.finrank_quotient_symmetricAlgebraHilbertIdeal_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 = ⊤) (hρ : Function.Injective ⇑ρ) :
Module.finrank K (SymmetricAlgebra K (Module.Dual K V) ⧸ ρ.dual.symmetricAlgebraHilbertIdeal) = Nat.card G

The coinvariant algebra of a pseudoreflection group has dimension |G|. If a finite group G, whose order is invertible in the field K, acts faithfully and linearly on a finite-dimensional vector space V and is generated by pseudoreflections, then the coinvariant algebra Sym(V*) ⧸ A₊ · Sym(V*), for A = Sym(V*)^G, has dimension |G|.