Flag modules and the hit problem for the Steenrod algebra

Abstract

AbstractThe ‘hit problem’ of F. P. Peterson in algebraic topology asks for a minimal generating set for the polynomial algebraP(n) =2[x1,. . .,xn] as a module over the Steenrod algebra2. An equivalent problem is to find an2-basis for the subringK(n) of elementsfin the dual Hopf algebraD(n), a divided power algebra, such thatSqk(f)=0 for allk> 0. The Steenrod kernelK(n) is a2GL(n,2)-module dual to the quotientQ(n) ofP(n) by the hit elements+2P(n). A submoduleS(n) ofK(n) is obtained as the image of a family of maps from the permutation moduleFl(n) ofGL(n,2) on complete flags in ann-dimensional vector spaceVover2. We use the Schubert cell decomposition of the flags to calculateS(n) in degrees$d =\sum_{i=1}^n (2^{\lambda_i}-1)$, where λ1> λ2> ⋅⋅⋅ > λn≥ 0. When λn= 0, we define a2GL(n,2)-module map δ:Qd(n) →Q2d+n−1(n) analogous to the well-known isomorphismQd(n) →Q2d+n(n) of M. Kameko. When λn−1≥ 2, we show that δ is surjective and δ*:S2d+n−1(n)→Sd(n) is an isomorphism.