Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem

Abstract

A result of G. Walker and R. Wood states that the space of indecomposable elements in degree 2 n -1-n of the polynomial algebra 𝔽 2 [x 1 ,...,x n ], considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of GL n (𝔽 2 ). We generalize this result to all finite fields by studying a family of finite quotient rings R n,k , k∈ℕ * , of 𝔽 q [x 1 ,...,x n ], where each R n,k is defined as a quotient of the Stanley–Reisner ring of a matroid complex. By considering a variant of R n,k , we also show that the space of indecomposable elements of 𝔽 q [x 1 ,...,x n ] in degree q n-1 -n has dimension equal to that of a complex cuspidal representation of GL n (𝔽 q ), that is (q-1)(q 2 -1)⋯(q n-1 -1).