Stanley-Reisner's ring and the occurrence of the Steinberg representation in the hit problem

TL;DR

This paper generalizes a known result relating indecomposable elements in polynomial rings over finite fields to the Steinberg representation, using Stanley-Reisner rings of matroid complexes, and explores their connections to Brown-Gitler modules and spectra.

Contribution

It extends the connection between indecomposable elements and the Steinberg representation from _2 to all finite fields using Stanley-Reisner rings, and links these algebraic structures to topological decompositions.

Findings

Indecomposable elements in degree q^{n-1}-n match the dimension of a complex cuspidal representation.

Decomposition of the Steinberg summand into suspensions of Brown-Gitler modules.

Evidence for a stable topological decomposition related to the Steinberg module.

Abstract

G. Walker and R. Wood proved that in degree $2^n-1-n$, the space of indecomposable elements of $\Bbb F_2[x_1,\ldots,x_n]$, considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of $GL_n(\Bbb F_2)$. We generalize this result to all finite fields by analyzing certain finite quotients of $\Bbb F_q[x_1,\ldots,x_n]$ which come from the Stanley-Reisner rings of some matroid complexes. Our method also shows that the space of indecomposable elements in degree $q^{n-1}-n$ has the dimension equal to that of a complex cuspidal representation of $GL_n(\Bbb F_q)$. As a by product, over the prime field $\Bbb F_2$, we give a decomposition of the Steinberg summand of one of these quotients into a direct sum of suspensions of Brown-Gitler modules. This decomposition suggests the existence of a stable decomposition derived from the Steinberg module of a certain topological space into a wedge of suspensions of Brown-Gitler spectra.