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

Abstract

Let q be a prime power, G=GLn(q) and let U⩽G be the subgroup of (lower) unitriangular matrices in G. For a partition λ of n denote the corresponding unipotent Specht module over the complex field C for G by Sλ. It is conjectured that for c∈Z⩾0 the number of irreducible constituents of dimension qc of the restriction ResUG(Sλ) of Sλ to U is a polynomial in q with integer coefficients depending only on c and λ, not on q. In the special case of the partition λ=(1n) 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 λ=(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 Sλ (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.