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

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.