Let rho : G hooked right arrow 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 rho. 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 functions(1) on V. The subalgebra of polynomials invariant under this action is denoted by F[V](G). If U subset of V = F-n is a linear subspace then the pointwise stabilizer of U is denoted by G(U) = {g is an element of G vertical bar g(u) = u for all u is an element of U}. It is known that several properties of F[V](G) are inherited by F[V](GU) (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](GU) 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.
展开▼
