The hit problem for the modular invariants of linear groups

[摘要] Let the mod 2 Steenrod algebra, A, and the general linear group, GL(k) := GL(k, F-2), act on P-k := F-2[x(1,)...,x(k)] with deg (x(1)) == 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 P-k(G) is hit by A in P-k. In other words, (P-k(G))(+) subset of A(+) . P-k, where (P-k(G))(+) and A(+) denote respectively the submodules of P-k(G) and A consisting of all elements of positive degree. This family contains most of the parabolic subgroups of GL(k). 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 P-k(G) is a polynomial algebra in k variables, whose degrees are less than or equal to 8 and fixed while k increases. It has been shown by Hu'ng [Trans. Amen Math. Soc. 349 (1997), 3893-3910] that, for G = GL(k), the inclusion (P-k(GLk))(+) subset of A(+) . P-k is equivalent to a weak algebraic version of the long-standing conjecture stating that the only spherical classes in Q(0)S(0) are the elements of Hopf invariant 1 and those of Kervaire invariant 1. (C) 2001 Elsevier Science.

[发布日期] 2001-12-01

[关键词] Steenrod algebra;invariant theory;Dickson invariant;Mui invariant