On Specht modules for general linear groups.
type: Item
instance of: scholarly article
title: On Specht modules for general linear groups.
zbMATH Open document ID: 1071.20041
author: Richard Dipper
author: Gordon James
published in: Journal of Algebra
publication date: 2004-10-04T00:00:00Z
review text: Let \(q\) be a power of the prime \(p\). Let \(F\) be a field of characteristic different from \(p\) and containing \(p\)-th roots of unity. Fix \(n\geq 2\). For each partition \(\lambda\) of \(n\) one has the Specht module \(S^\lambda\), which is a representation over \(F\) of the finite group \(\text{GL}_n(q)\). As the authors explain, it is important to understand these Specht modules, even if one cares only about irreducible representations of symmetric groups in arbitrary characteristic, or irreducible rational representations of \(\text{GL}_n(\overline{\mathbb{F}_q})\) in the defining characteristic \(p\). They conjecture that there is a natural basis of \(S^\lambda\) with the following properties. To each basis element is associated a standard \(\lambda\)-tableau \(t\) and the number of basis elements that is associated with the same \(t\) is given by a polynomial \(g_t(\lambda)\) in \(q\). The aim of the paper is to provide evidence for this phenomenon, in particular for two part partitions \(\lambda=(n,n-m)\) with \(m\leq 11\).
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 2105790
zbMATH Keywords: Specht modules
zbMATH Keywords: standard tableaux
reviewed by: Wilberd van der Kallen
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.11.022
OpenAlex ID: W1993603758
cites work: Q5662096
cites work: Identification of the irreducible modular representations of \(GL_n(q)\).
cites work: The q ‐Schur Algebra
cites work: The representation theory of the symmetric groups
cites work: Q3328735
cites work: On Specht modules of general linear groups.
cites work: Q4138735
DOI: 10.1016/J.JALGEBRA.2003.11.022
description: scientific article; zbMATH DE number 2105790