The Hit Problem for the Modular Invariants of Linear Groups

✍ Scribed by Nguyen H.V Hu·ng; Tran Ngoc Nam

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 176 KB
• Volume: 246
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.8974

✦ Synopsis

Let the mod 2 Steenrod algebra, , and the general linear group, GL k = GL k 2 , act on P k = 2 x 1 x k with deg x i = 1 in the usual manner. We prove that, for a family of some rather small subgroups G of GL k , every element of positive degree in the invariant algebra P G k is hit by in P k . In other words, P G k + ⊂ + • P k , where P G k + and + denote respectively the submodules of P G k and 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 G k 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 = GL k , the inclusion P GL k k + ⊂ + • 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.