This paper continues our study of the action of the mod 2 Steenrod algebra A 2 on the polynomial algebra P ( n ) = F 2 [ t 1 , … , t n ] . 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 A 2 -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 , F 2 ) , i.e. as a module over the finite group algebra F 2 G ( 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 F ¯ 2 , the algebraic closure of F 2 , we obtain commuting actions of A 2 and G ( n ) on P ( n ) = F ¯ 2 [ t 1 , … , t n ] . 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 G ¯ ( n ) = GL ( n , F ¯ 2 ) . In particular, we make use of tilting modules for G ¯ ( n ) to show that for every irreducible representation L ( λ ) of G ( n ) , a minimal set of A 2 -generators of P ( n ) must contain a copy of the corresponding dual Weyl module ∇ ( λ ) .