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• W2080436064 abstract "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 [Citation10] we denote S(V*) by 𝔽[V] and regard it as the algebra of polynomial functionsFootnote 1 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., [Citation6] Section 10.6 and the references there). For finite fields, following the pioneering work of Dwyer and Wilkerson [Citation1], many such properties have been demonstrated using the T -functor introduced by Lannes [Citation4] (see also [Citation9]) 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 [Citation6] Section 10.2. 1Polynomial functions is meant in the sense of algebraic geometry: See e.g., [Citation10] §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. Key Words: Degree boundsLannes T-factorPointwise stabilizers2000 Mathematics Subject Classification: Primary 55S10Secondary 13A50 Notes 1Polynomial functions is meant in the sense of algebraic geometry: See e.g., [Citation10] §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. We write |θ| for the degree of an element θ of this basis. Communicated by M. Vigue." @default.
• W2080436064 created "2016-06-24" @default.
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• W2080436064 date "2009-12-31" @default.
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• W2080436064 title "Lannes'<i>T</i>-Functor, Pointwise Stabilizers, and Degree Bounds" @default.
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• W2080436064 doi "https://doi.org/10.1080/00927870903448732" @default.
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