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Abstract:
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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. |