2013/04/16 by Qiong Guo, Guo, Qiong
Computer Science · Mathematics · #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1304.4370
openalex publication_date 2013/04/16 · arxiv created 2013/04/17 · arxiv updated 2013/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let q be a prime power, G=GLn(q) and let U\leqslant 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\geqslant 0 the number of irreducible constituents of dimension qc of the restriction \RResGU(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 \citehigman, 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 λ=(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 Sλ (over fields of characteristic coprime to q) defined by M. Brandt, R. Dipper, G. James and S. Lyle in \citebrandt2, \citedj1 and an explanation of the rank polynomials appearing there.