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Young tableaux and the Steenrod algebra

Grant Walker and R M W Wood

Geometry & Topology Monographs 11 (2007) 379–397

DOI: 10.2140/gtm.2007.11.379

arXiv: 0903.5003

Abstract

The purpose of this paper is to forge a direct link between the hit problem for the action of the Steenrod algebra A on the polynomial algebra P(n)=F2[x1,…,xn], over the field F2 of two elements, and semistandard Young tableaux as they apply to the modular representation theory of the general linear group GL(n,F2). The cohits Qd(n)=Pd(n)/Pd(n)∩A+(P(n)) form a modular representation of GL(n,F2) and the hit problem is to analyze this module. In certain generic degrees d we show how the semistandard Young tableaux can be used to index a set of monomials which span Qd(n). The hook formula, which calculates the number of semistandard Young tableaux, then gives an upper bound for the dimension of Qd(n). In the particular degree d where the Steinberg module appears for the first time in P(n) the upper bound is exact and Qd(n) can then be identified with the Steinberg module.

Keywords

Steenrod algebra, hit problem, Young tableaux, Steinberg module

Mathematical Subject Classification

Primary: 55S10

Secondary: 20C20

References
Publication

Received: 28 December 2004
Revised: 17 July 2005
Accepted: 18 December 2005
Published: 14 November 2007

Authors
Grant Walker
School of Mathematics
University of Manchester
Manchester
M13 9PL
United Kingdom
R M W Wood
School of Mathematics
University of Manchester
Manchester
M13 9PL
United Kingdom