Skew-symmetric elements in the group algebra of a symmetric group
type: Item
instance of: scholarly article
title: Skew-symmetric elements in the group algebra of a symmetric group
zbMATH Open document ID: 0581.20008
DOI: 10.1007/BF01979702
author: A. N. Zyrichev
author: Yuri P. Razmyslov
published in: Algebra and Logic
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187259
review text: Let \(S_ n\) be the symmetric group on \(n\) letters and let \(\rho_ s=\sum_{t\in S_ n}(\text{sign}\;t)t^{-1}st\) for \(s\in S_ n\). The element \(\rho_ s\) depends only (up to a sign) on the conjugacy class of \(s\) and is non-zero if and only if the conjugacy class of \(s\) is determined by a partition with odd pairwise distinct parts. Let \(\lambda =(\lambda_ 1,\ldots,\lambda_ k)\), \(\lambda_ 1>\lambda_ 2>\ldots>\lambda_ k>0\), be such a partition of \(n\), and let \(K\) be any field of characteristic zero. The main result of the paper says that if \(s\) is an element of cycle-type \(\lambda\) in \(S_ n\), then \(\rho_ s\) belongs to the minimal two-sided ideal of the group ring \(K[S_ n]\) corresponding to the Young diagram whose hooks of boxes on the main diagonal are symmetric of lengths \(\lambda_ 1,\lambda_ 2,\ldots,\lambda_ k\), respectively. The proof given in the paper is in terms of the theory of matrix identities with trace developed by one of the authors in [Izv. Akad. Nauk SSSR, Ser. Mat. 38, 723--756 (1974; Zbl 0311.16016)]. However, there is a remark in a footnote that an alternative proof of the main theorem follows from the Frobenius results concerning representations of the alternating group.
reviewed by: Tadeusz Józefiak
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 3929252
zbMATH Keywords: symmetric group
zbMATH Keywords: conjugacy class
zbMATH Keywords: partition
zbMATH Keywords: group ring
zbMATH Keywords: Young diagram
zbMATH Keywords: hooks
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3929252