The Schur indices of the irreducible characters of the special linear groups
type: Item
instance of: scholarly article
title: The Schur indices of the irreducible characters of the special linear groups
zbMATH Open document ID: 0977.20036
author: Alexandre Turull
published in: Journal of Algebra
publication date: 2001-10-10T00:00:00Z
review text: The author provides the definitive solution to the problem of computing the Schur index over the field of rational numbers of any irreducible complex character of the finite special linear group \(\text{SL}_n(q)\), where \(q\) is a power of the prime \(p\). The reviewer has worked on this problem and obtained partial information, with complete solutions in some cases. Turull's solution shows that the outstanding cases not considered by the reviewer are substantially more complicated than might have been anticipated. A brief history of the problem is as follows. Let \(G\) denote the general linear group \(\text{GL}_n(q)\) and let \(U\) denote a Sylow \(p\)-subgroup of \(G\). Let \(H\) denote \(\text{SL}_n(q)\). \(H\) is a normal subgroup of \(G\) and the quotient group \(G/H\) is cyclic of order \(q-1\). Let \(\chi\) be an irreducible complex character of \(G\). \textit{A.~V.~Zelevinsky} showed [in Representations of finite classical groups, Lect. Notes Math. 869, Springer, Berlin (1981; Zbl 0465.20009)] that the restriction of \(\chi\) to \(U\) contains some complex linear character of \(U\) (depending on \(\chi\)) with multiplicity 1. Given the structure of the normalizer of \(U\) in \(G\), this is sufficient to prove that \(\chi\) has Schur index 1 over the rationals. While Zelevinsky's theorem also applies to the irreducible characters of \(H\) (note that \(H\) contains the subgroup \(U\)), this does not imply that all characters of \(H\) have Schur index 1, since the normalizer of \(U\) in \(H\) does not necessarily have the required structure. Nevertheless, one can prove without too much extra work that the Schur index of an irreducible complex character of \(H\) is 1 if either \(q\) is a power of 2, or \(n\) is odd, or if the power of 2 dividing \(n\) is greater than the power of 2 dividing \(p-1\). On the other hand, while the Schur index is always at most 2, examples exist where it is exactly 2. In previous papers [in J. Algebra 227, No. 1, 124-132, 133-158 …
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1563183
zbMATH Keywords: Schur index
zbMATH Keywords: special linear groups
zbMATH Keywords: irreducible complex characters
MaRDI profile type: Publication
full work available at URL: jabr.2000.8485
OpenAlex ID: W2071260254
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DOI: 10.1006/JABR.2000.8485
description: scientific article; zbMATH DE number 1563183