The hit problem for the modular invariants of linear groups

Abstract

Let the mod 2 Steenrod algebra, A, and the general linear group, GLk := GL(k, F2), act on Pk := F 2[x1,...,xk] with deg (xi) = 1 in the usual manner. We prove that, for a family of some rather small subgroups G of GLk, every element of positive degree in the invariant algebra Pk G is hit by A in Pk. In other words, (Pk G)+ ?? A+ ? Pk, where (Pk G)+ and A+ denote respectively the submodules of Pk G and ?a consisting of all elements of positive degree. This family contains most of the parabolic subgroups of GLk. It should be noted that the smaller the group G is, the harder the problem turns out to be. Remarkably, when G is the smallest group of the family, the invariant algebra Pk G is a polynomial algebra in k variables, whose degrees are ?? 8 and fixed while k increases. It has been shown by Hu'ng [Trans. Amer. Math. Soc. 349 (1997), 3893-3910] that, for G = GLk, the inclusion (Pk GL k)+ ?? A+ ? Pk is equivalent to a weak algebraic version of the long-standing conjecture stating that the only spherical classes in Q0S0 are the elements of Hopf invariant 1 and those of Kervaire invariant 1. ?? 2001 Elsevier Science.