Weyl modules and the mod 2 Steenrod algebra

This paper continues our study of the action of the mod 2 Steenrod algebra A2 on the polynomial algebra P(n)=F2[t1,…,tn]. We obtain further partial results on the ‘hit problem’ of F.P. Peterson, which asks for a minimal generating set for P(n) as an A2-module. We also study the structure of the quotient by the ‘hit elements’ as a graded representation of the finite general linear group G(n)=GL(n,F2), i.e. as a module over the finite group algebra F2G(n). These results were obtained in previous work of the authors for the special case of the Steinberg module for G(n). By extending the scalars to , the algebraic closure of F2, we obtain commuting actions of A2 and G(n) on . While this makes no essential difference to the representation theory of G(n) or to the hit problem, it allows us to treat the action of G(n) on P(n) as the restriction of that of the algebraic group . In particular, we make use of tilting modules for to show that for every irreducible representation L(λ) of G(n), a minimal set of A2-generators of P(n) must contain a copy of the corresponding dual Weyl module ∇(λ).