On the U-module structure of the unipotent Specht modules of finite general linear groups

摘要

Let q be a prime power, G = GL(n) (q) and let U <= 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 is an element of Z(>= 0) the number of irreducible constituents of dimension q(c) of the restriction Res(U)(G) (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, 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 for the case that lambda = (n - m, m) (0 <= m <= 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 and an explanation of the rank polynomials appearing there. (C) 2016 Elsevier Inc. All rights reserved.