Young tableaux and the Steenrod algebra

TL;DR

This paper establishes a connection between the hit problem for the Steenrod algebra acting on polynomial algebras and semistandard Young tableaux, providing new combinatorial tools to analyze modular representations of GL(n,F_2).

Contribution

It introduces a novel link between the hit problem and Young tableaux, enabling the use of combinatorial formulas to estimate dimensions of certain modules.

Findings

Semistandard Young tableaux index monomials spanning Q^d(n).

Hook formula bounds the dimension of Q^d(n).

Q^d(n) equals the Steinberg module in a specific degree.

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)=F_2[x_1,...,x_n], over the field F_2 of two elements, and semistandard Young tableaux as they apply to the modular representation theory of the general linear group GL(n,F_2). The cohits Q^d(n)=P^d(n)/P^d(n)\cap A^+(P(n)) form a modular representation of GL(n,F_2) 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.