Complex group algebras of the double covers of the symmetric and alternating groups

Bessenrodt, Christine; Nguyen, Hung; Olsson, Jørn; Tong-Viet, Hung

doi:10.2140/ant.2015.9.601

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Complex group algebras of the double covers of the symmetric and alternating groups

Christine Bessenrodt, Hung Ngoc Nguyen, Jørn B. Olsson and Hung P. Tong-Viet

Vol. 9 (2015), No. 3, 601–628

DOI: 10.2140/ant.2015.9.601

Abstract

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let $n \geq 5$ be an integer, $G$ a finite group, and let ${\hat{A}}_{n}$ and ${\hat{S}}_{n}^{\pm}$ denote the double covers of $A_{n}$ and $S_{n}$ , respectively. We prove that $ℂ G ≅ ℂ {\hat{A}}_{n}$ if and only if $G ≅ {\hat{A}}_{n}$ , and $ℂ G ≅ ℂ {\hat{S}}_{n}^{+} ≅ ℂ {\hat{S}}_{n}^{-}$ if and only if $G ≅ {\hat{S}}_{n}^{+}$ or ${\hat{S}}_{n}^{-}$ . This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasisimple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.

Keywords

symmetric groups, alternating groups, complex group algebras, Schur covers, double covers, irreducible representations, character degrees

Mathematical Subject Classification 2010

Primary: 20C30

Secondary: 20C15, 20C33

Milestones

Received: 19 May 2014

Revised: 13 January 2015

Accepted: 23 February 2015

Published: 17 April 2015

Authors

Christine Bessenrodt
Institut für Algebra, Zahlentheorie und Diskrete Mathematik Fakultät für Mathematik und Physik Leibniz Universität Hannover Welfengarten 1 D-30167 Hannover Germany

Hung Ngoc Nguyen
Department of Mathematics The University of Akron Akron, OH 44325 United States

Jørn B. Olsson
Department of Mathematical Sciences University of Copenhagen DK-2100 Copenhagen Ø Denmark

Hung P. Tong-Viet
Department of Mathematics and Applied Mathematics University of Pretoria Private Bag X20 Hatfield, Pretoria 0002 South Africa

Vol. 9, No. 3, 2015