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

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