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,…,x_n], 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 Q^d(n)=P^d(n)/P^d(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 Q^d(n). The hook formula, which calculates the number of semistandard Young tableaux, then gives an upper bound for the dimension of Q^d(n). In the particular degree d where the Steinberg module appears for the first time in P(n) the upper bound is exact and Q^d(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