Polynomial divided differences from pseudoreflections

Let ρ act on a vector space V, and suppose the fixed space of g has codimension one, as it does for a pseudoreflection. This hyperplane is the kernel of a nonzero form ℓ. On polynomial functions Sym(V*), every difference p - g • p is divisible by ℓ, and there is a unique linear operator over the invariant algebra satisfying ℓ * δ p = p - g • p. This operator lowers homogeneous degree by one.

This connects the geometric codimension-one condition with the divided-difference construction and its operator identity in TauCeti.RepresentationTheory.SymmetricAlgebra.DividedDifference. It requires neither finite-dimensionality nor any assumption on the characteristic or the order of the ambient group. Finite order is not needed: the codimension-one condition suffices.

References

• R. Kane, Reflection Groups and Invariant Theory, Springer (2001), §§ 17–18.
• L. Smith, Polynomial Invariants of Finite Groups, A K Peters (1995), § 7.2.

theorem Representation.exists_unique_symmetricAlgebraDividedDifference_of_finrank_quotient_eq_one {K : Type u_1} {G : Type u_2} {V : Type u_3} [Field K] [Group G] [AddCommGroup V] [Module K V] (ρ : Representation K G V) (g : G) (hg : Module.finrank K (V ⧸ (ρ g).fixedSubmodule) = 1) : ∃ (ℓ : Module.Dual K V), ℓ ≠ 0 ∧ LinearMap.ker ℓ = (ρ g).fixedSubmodule ∧ ∃ (δ : SymmetricAlgebra K (Module.Dual K V) →ₗ[↥ρ.dual.symmetricAlgebraInvariants] SymmetricAlgebra K (Module.Dual K V)), (∀ (p : SymmetricAlgebra K (Module.Dual K V)), (SymmetricAlgebra.ι K (Module.Dual K V)) ℓ * δ p = p - (ρ.dual.symmetricAlgebra g) p) ∧ (∀ (n : ℕ), ∀ p ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) (n + 1), δ p ∈ TauCeti.SymmetricAlgebra.homogeneousSubmodule K (Module.Dual K V) n) ∧ ∀ (δ' : SymmetricAlgebra K (Module.Dual K V) →ₗ[↥ρ.dual.symmetricAlgebraInvariants] SymmetricAlgebra K (Module.Dual K V)), (∀ (p : SymmetricAlgebra K (Module.Dual K V)), (SymmetricAlgebra.ι K (Module.Dual K V)) ℓ * δ' p = p - (ρ.dual.symmetricAlgebra g) p) → δ' = δ

If the fixed space of g has codimension one, it has a defining form and a unique divided-difference operator on polynomial functions, linear over the invariant algebra. The operator lowers homogeneous degree by one. In particular this constructs the divided difference of every pseudoreflection, without requiring its order to be two.