Additional projective representations of the symmetric groups
type: Item
instance of: scholarly article
title: Additional projective representations of the symmetric groups
zbMATH Open document ID: 0998.20014
DOI: 10.1007/BF03042012
author: Yanyan Li
published in: Advances in Applied Clifford Algebras
publication date: 2002-08-25T00:00:00Z
review text: Let \(G\) be a finite group and let \(F\) be an algebraically closed field of characteristic zero with multiplicative group \(F^*\). A mapping \(T\colon G\to\text{GL}_n(F)\) from \(G\) into the general linear group \(\text{GL}_n(F)\) is called a projective representation of \(G\) of degree \(n\) over \(F\) if there exists a function \(\alpha\colon G\times G\to F^*\) such that \(T(g)T(h)=\alpha(g,h)T(gh)\) for all \(g,h\in G\). Then, \(T\) is called irreducible if the vector space \(V=F^n\) has no nontrivial proper subspace invariant under all \(T(g)\), \(g\in G\). The projective representation theory of the symmetric groups from the point of view of Clifford algebras was considered for the first time in 1962 by \textit{A. O. Morris} [Proc. Lond. Math. Soc., III. Ser. 12, 55-76 (1962; Zbl 0104.25202)]. It was the reason to call them spin representations. In the paper under review, the authors attempt to follow \textit{V. F. Molchanov}'s approach [Vestn. Mosk. Univ., Ser. I 21, No. 1, 52-57 (1966; Zbl 0207.33404)], for ordinary representations of the symmetric group to construct the projective (spin) representations of the symmetric group. They investigate an alternative strategy which leads to the proof, more along the lines used by Molchanov. They determine all recurrence relations which should be useful in providing such a proof.
reviewed by: A. Iranmanesh
Mathematics Subject Classification ID: 20C25
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 1568597
zbMATH Keywords: projective representations
zbMATH Keywords: symmetric groups
zbMATH Keywords: spin representations
zbMATH Keywords: recurrence relations
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03042012
OpenAlex ID: W2081087603
description: scientific article; zbMATH DE number 1568597