The Hit Problem for the Modular Invariants of Linear Groups,

Let the mod 2 Steenrod algebra, , and the general linear group, GLk := GL(k, 2), act on Pk := 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 PkG is hit by in Pk. In other words, (PkG)+ + • Pk, where (PkG)+ and + denote respectively the submodules of PkG and 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 PkG 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 (PkGLk)+ + • 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.