On the U-module Structure of the Unipotent Specht Modules of Finite General Linear Groups

Abstract

Let $q$ be a prime power, $G=GL_n(q)$ and let $U\leqslant G$ be the subgroup of (lower) unitriangular matrices in $G$. For a partition $\lambda$ of $n$ denote the corresponding unipotent Specht module over the complex field $\C$ for $G$ by $S^\lambda$. It is conjectured that for $c\in \Z_{\geqslant 0}$ the number of irreducible constituents of dimension $q^c$ of the restriction $\RRes^{G}_U(S^\lambda)$ of $S^\lambda$ to $U$ is a polynomial in $q$ with integer coefficients depending only on $c$ and $\lambda$, not on $q$. In the special case of the partition $\lambda=(1^n)$ this implies a longstanding (still open) conjecture of Higman \cite{higman}, stating that the number of conjugacy classes of $U$ should be a polynomial in $q$ with integer coefficients depending only on $n$ not on $q$. In this paper we prove the conjecture in the case that $\lambda=(n-m,m)$ $(0\leqslant m \leqslant n/2)$ is a 2-part partition. As a consequence, we obtain a new representation theoretic construction of the standard basis of $S^\lambda$ (over fields of characteristic coprime to $q$) defined by M. Brandt, R. Dipper, G. James and S. Lyle in \cite{brandt2}, \cite{dj1} and an explanation of the rank polynomials appearing there.