Tensor square of the basic spin representations of Schur covering groups for the symmetric groups
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Tensor square of the basic spin representations of Schur covering groups for the symmetric groups Kazuya Aokage 1 Received: 23 April 2019 / Accepted: 10 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider the tensor square of the basic spin representations of Schur covering groups Sn and Sn for the symmetric group Sn . It is known from work of Stembridge that the irreducible components of the tensor square of the basic spin representations for Sn , for n odd, are multiplicity-free and indexed by hook partitions ([3], pp. 133). In this paper, we derive similar results for Sn when n is even, and for Sn when n is arbitrary. We assume that n ≥ 4, n = 6, when discussing Sn . Keywords Symmetric group · Symmetric function · Projective representation Mathematics Subject Classification 05E05 · 20C25 · 20C30
1 Introduction of a finite group G is introduced by Schur [2], The theory of the covering group G who investigated the projective representations of G. For a finite group G, there exists satisfying the following properties: a group G −→ G −→ 1 such that any (1). There exists a central extension 1 −→ A −→ G projective representation of G over C lifts projectively to a linear representation of G, 2 ∗ (2). |A| = |H (G, C )|, where H 2 (G, C∗ ), called the Schur multiplier of G, denotes the second cohomology is called the Schur group with respect to the trivial action of G on C∗ . The group G covering group of G. Each projective representation of G described by Schur can be linearized by a linear [2]. A theorem of Schur states that for n ≥ 4, the symmetric group representation of G Sn admits exactly two minimal central extensions, also known as covering groups, Sn
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Kazuya Aokage [email protected] Department of Mathematics, National Institute of Technology, Ariake College, 150 Higashihagio-Machi, Omuta Fukuoka, Japan 836-8585
123
Journal of Algebraic Combinatorics
and Sn , both of order 2 · n!, non-isomorphic for n = 6, with corresponding twisted group algebras [1,2]. In this paper, we investigate the irreducible components of the inner tensor square of the basic spin representation for the Schur covering groups Sn and Sn . One of the references for the subject of spin representation of the covering groups for the symmetric group Sn is Hoffman and Humphreys’s book [1]. Representations of the covering groups which do not represent (− 1) faithfully (i.e., (− 1) maps to +1) lift to ordinary representations of the symmetric group. Since tensor squares fall in this class, tensor powers of the covering group characters are characters of Sn . This observation underlies our main result. For the Schur covering group Sn , when n is odd, Stembridge-derived irreducible decomposition for the inner tensor square of the basic spin representations ([3], pp. 133). This decomposition is multiplicity-free and indexed by hook partitions of n. We present similar results for Sn when n is even, and for Sn when n is arbitrary. The plan of the pape
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