Decomposition matrices for spin characters of symmetric groups at characteristic \(3\)
type: Item
instance of: scholarly article
title: Decomposition matrices for spin characters of symmetric groups at characteristic \(3\)
zbMATH Open document ID: 0835.20018
author: Christine Bessenrodt
author: Jørn B. Olsson
author: Alun O. Morris
published in: Journal of Algebra
publication date: 1994-06-09T00:00:00Z
review text: Let \(\widehat{S}_n\) denote either of the double covers of the symmetric group \(S_n\) with \(\langle z\rangle\) a central subgroup of order 2 in \(\widehat{S}_n\). The irreducible characters of \(\widehat{S}_n\) over an arbitrary field \(F\) fall into two categories. Those with \(\langle z \rangle\) in their kernels and those which do not have \(\langle z\rangle\) in their kernels. The latter characters are called ordinary or modular spin characters of \(S_n\) according to whether \(F\) has characteristic 0 or non-zero respectively. In a series of papers G. D. James developed methods for finding the decomposition matrices of \(S_n\). He also generalized the classical construction of Specht modules over a field of characteristic 0 to obtain irreducible modules for \(S_n\) over a field of arbitrary characteristic. In the present paper the authors attempt to obtain the shape of the decomposition matrix of a spin block of \(S_n\) in the case \(p=3\). In section 3 of the paper the authors describe the partitions of \(n\) which label the modular spin characters. These partitions are called Schur regular partitions and Schur singular partitions. The section 4 of the paper contains a result called by the authors an ``approximation matrix'' whose consequence is the decomposition matrix of a spin block of \(S_n\). Finally in section 5 they pose a number of questions and conjectures.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 563134
zbMATH Keywords: double covers
zbMATH Keywords: symmetric groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: modular spin characters
zbMATH Keywords: decomposition matrices
zbMATH Keywords: Specht modules
zbMATH Keywords: irreducible modules
zbMATH Keywords: spin blocks
zbMATH Keywords: Schur regular partitions
zbMATH Keywords: Schur singular partitions
reviewed by: M. R. Darafsheh
MaRDI profile type: Publication
full work available at URL: jabr.1994.1058
OpenAlex ID: W1986086951
DOI: 10.1006/JABR.1994.1058
description: scientific article; zbMATH DE number 563134