Real representations of the finite orthogonal and symplectic groups of odd characteristic
type: Item
instance of: scholarly article
title: Real representations of the finite orthogonal and symplectic groups of odd characteristic
zbMATH Open document ID: 0576.20026
DOI: 10.1016/0021-8693(85)90049-3
author: B. George
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: The author addresses the general question: which complex irreducible representations of a finite classical group can be realized over the real field? In an earlier paper [Proc. Lond. Math. Soc., III. Ser. 47, 493--506 (1983; Zbl 0551.20023)], he dealt with the general linear groups. Now he turns to the orthogonal and symplectic groups of odd characteristic. As before, the general method is to investigate in detail the \(\mathbb R\)-elementary subgroups. The main results are as follows. Theorem 1. Each complex irreducible character of an orthogonal group \(O(n,q)\) (\(q\) odd) is the character of a real representation. A real-valued complex irreducible character of \(\mathrm{Sp}(2n,q)\) (\(q\) odd) has Schur index 2 or 1 over \(\mathbb R\) according as it is faithful or not. Theorem 2. Every real-valued complex irreducible character of a special orthogonal group \(\mathrm{SO}(n,q)\) (\(q\) odd) is the character of a real representation. The author also proves that, when \(q\equiv 1\pmod 4\), the sum of the degrees of the irreducible complex characters of \(\mathrm{Sp}(2n,q)\) is \(q^{n(n+1)}\prod^{n}_{i=1}(q^ i+1)\); and he conjectures that this remains true when \(q\equiv 3\pmod 4\).
reviewed by: G. E. Wall
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 3920786
zbMATH Keywords: complex irreducible representations
zbMATH Keywords: finite classical group
zbMATH Keywords: general linear groups
zbMATH Keywords: symplectic groups
zbMATH Keywords: orthogonal group
zbMATH Keywords: real representation
zbMATH Keywords: Schur index
zbMATH Keywords: special orthogonal group
zbMATH Keywords: degrees
zbMATH Keywords: irreducible complex characters
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90049-3
OpenAlex ID: W2078996099
cites work: The Sylow 2-subgroups of the finite classical groups
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cites work: Schur indices of sympletic groups
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description: scientific article; zbMATH DE number 3920786