Degrees of homogeneous algebraically independent elements

Let A = ⨁ₙ 𝒜 n be an ℕ-graded algebra over a commutative ring R, and let g j ∈ 𝒜 (e j) be homogeneous elements. A homogeneous element of degree n of the subalgebra generated by the g j is a polynomial in those g j of degree at most n alone (TauCeti.GradedAlgebra.mem_adjoin_image_of_mem_adjoin_range): it is the evaluation of the weight-n component of any polynomial expressing it, and a monomial of weight n only involves variables of weight at most n.

Consequently, over a domain, if algebraically independent homogeneous elements f i ∈ 𝒜 (d i) lie in the subalgebra generated by the g j, then for every n there are at most as many f i of degree at most n as there are g j of degree at most n (TauCeti.GradedAlgebra.card_filter_le_le_card_filter_le): the f i of degree at most n are algebraically independent elements of a quotient of the polynomial ring on the g j of degree at most n, whose transcendence degree is the number of those g j.

The main application compares two polynomial algebras A ⊆ B with homogeneous generators: after sorting, each generator of B has degree at most that of the corresponding generator of A. For the invariants Sym(V*)^G ⊆ Sym(V*)^H of a finite group and of its subgroup generated by pseudoreflections, this is the key step in the converse of the Chevalley–Shephard–Todd theorem.

Main results

• TauCeti.GradedAlgebra.mem_adjoin_image_of_mem_adjoin_range: a homogeneous element of degree n of the subalgebra generated by homogeneous elements g j is generated by the g j of degree at most n.
• TauCeti.GradedAlgebra.card_filter_le_le_card_filter_le: algebraically independent homogeneous elements in the subalgebra generated by homogeneous elements g j include, in each degree range ≤ n, at most as many elements as there are g j in that range.

References

• 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 TauCeti.GradedAlgebra.mem_adjoin_image_of_mem_adjoin_range {R : Type u_1} {A : Type u_2} {κ : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {g : κ → A} {e : κ → ℕ} (hg : ∀ (j : κ), g j ∈ 𝒜 (e j)) {x : A} (hx : x ∈ Algebra.adjoin R (Set.range g)) {n : ℕ} (hxn : x ∈ 𝒜 n) : x ∈ Algebra.adjoin R (g '' {j : κ | e j ≤ n})

Homogeneous elements of a subalgebra generated by homogeneous elements. If g j ∈ 𝒜 (e j), then every element of degree n of the subalgebra generated by the g j is generated by the g j of degree at most n.

theorem TauCeti.GradedAlgebra.card_filter_le_le_card_filter_le {R : Type u_1} {A : Type u_2} {κ : Type u_3} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {g : κ → A} {e : κ → ℕ} [IsDomain R] {ι : Type u_4} [Fintype ι] [Fintype κ] {f : ι → A} {d : ι → ℕ} (hf : ∀ (i : ι), f i ∈ 𝒜 (d i)) (hind : AlgebraicIndependent R f) (hg : ∀ (j : κ), g j ∈ 𝒜 (e j)) (hfg : ∀ (i : ι), f i ∈ Algebra.adjoin R (Set.range g)) (n : ℕ) : {i : ι | d i ≤ n}.card ≤ {j : κ | e j ≤ n}.card

Counting algebraically independent homogeneous elements by degree. Over a domain, let f i ∈ 𝒜 (d i) be algebraically independent homogeneous elements of the subalgebra generated by finitely many homogeneous elements g j ∈ 𝒜 (e j). Then for every n there are at most as many f i of degree at most n as there are g j of degree at most n.