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Flag modules and the hit problem for the Steenrod algebra

Published online by Cambridge University Press:  01 July 2009

G. WALKER  and
R. M. W. WOOD
G. WALKER
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL. e-mail: [email protected], [email protected]
R. M. W. WOOD
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL. e-mail: [email protected], [email protected]
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Abstract

The ‘hit problem’ of F. P. Peterson in algebraic topology asks for a minimal generating set for the polynomial algebra P(n) = 2[x1,. . ., xn] as a module over the Steenrod algebra 2. An equivalent problem is to find an 2-basis for the subring K(n) of elements f in the dual Hopf algebra D(n), a divided power algebra, such that Sqk(f)=0 for all k > 0. The Steenrod kernel K(n) is a 2GL(n,2)-module dual to the quotient Q(n) of P(n) by the hit elements +2P(n). A submodule S(n) of K(n) is obtained as the image of a family of maps from the permutation module Fl(n) of GL(n,2) on complete flags in an n-dimensional vector space V over 2. We use the Schubert cell decomposition of the flags to calculate S(n) in degrees , where λ1 > λ2 > ⋅⋅⋅ > λn ≥ 0. When λn = 0, we define a 2GL(n,2)-module map δ: Qd(n) → Q2d+n−1(n) analogous to the well-known isomorphism Qd(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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 143 - 171
Copyright
Copyright © Cambridge Philosophical Society 2009

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