2021/06/03 by Hai, Nguyen Dang Ho
#Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2106.01537
G. Walker and R. Wood proved that in degree 2n-1-n, the space of indecomposable elements of \Bbb F2[x1,…,xn], considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of GLn(\Bbb F2). We generalize this result to all finite fields by analyzing certain finite quotients of \Bbb Fq[x1,…,xn] which come from the Stanley-Reisner rings of some matroid complexes. Our method also shows that the space of indecomposable elements in degree qn-1-n has the dimension equal to that of a complex cuspidal representation of GLn(\Bbb Fq). As a by product, over the prime field \Bbb F2, 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.