Lannes' T-Functor, Pointwise Stabilizers, and Degree Bounds

Abstract

Let ρ: G↪GL(n, 𝔽) be a representation of a finite group G over the field 𝔽, and denote by V the vector space 𝔽 n on which G acts via ρ. By means of the dual (contragredient) representation G also acts on the symmetric algebra S(V*) of the vector space V* dual to V. Following [10] we denote S(V*) by 𝔽[V] and regard it as the algebra of polynomial functions1 on V. The subalgebra of polynomials invariant under this action is denoted by 𝔽[V] G . If U ⊆ V = 𝔽 n is a linear subspace then the pointwise stabilizer of U is denoted by G U = {g ∈ G | g(u) = u ∀ u ∈ U}. It is known that several properties of 𝔽[V] G are inherited by 𝔽[V] G U (see, e.g., [6] Section 10.6 and the references there). For finite fields, following the pioneering work of Dwyer and Wilkerson [1], many such properties have been demonstrated using the T -functor introduced by Lannes [4] (see also [9]) in his study of unstable modules over the Steenrod algebra. In this note we show that given a degree bound for the generators of 𝔽[V] G as an algebra, this bound is inherited by 𝔽[V] G U when 𝔽 = 𝔽 q is a Galois field with q elements. To do so we examine some finiteness properties of unstable algebras over the Steenrod algebra and show that the T -functor preserves them, extending results in [6] Section 10.2. 1Polynomial functions is meant in the sense of algebraic geometry: See e.g., [10] §1.2 for a discussion of how elements of S(V*) can be regarded as polynomial functions from to the algebraic closure of 𝔽 that are defined over 𝔽. So 𝔽[−] is a contravariant functor of its argument.