Regular character tables of symmetric groups.
type: Item
instance of: scholarly article
title: Regular character tables of symmetric groups.
zbMATH Open document ID: 1036.20014
published in: The Electronic Journal of Combinatorics
publication date: 2003-05-07T00:00:00Z
full work available at URL: https://eudml.org/doc/123471
full work available at URL: v10i1n3.html
review text: Let \(X_n\) be the character table of the symmetric group \(S_n\), \(a_n\) the product of all parts of all partitions of \(n\), and \(b_n\) the product of all factorials of multiplicities of parts of partitions of \(n\). It is a well known fact that the absolute value of the determinant of \(X_n\) equals \(a_n\) and also equals \(b_n\); see \textit{F. W. Schmidt} and \textit{R. Simion} [J. Comb. Theory, Ser. A 36, 249-252 (1984; Zbl 0532.05003)]. In this note the author generalizes this result to certain submatrices of \(X_n\), called the regular and singular character tables. For an integer \(l\geq 2\), a partition is called \(l\)-regular if no part is repeated \(l\) or more times, and is called \(l\)-class regular if no part is divisible by \(l\). Partitions which are not (class) regular are called (class) singular. An irreducible character is called regular (singular) if its corresponding partition is regular (singular). We call a conjugacy class regular (singular) if its corresponding partition is class regular (singular). The regular character table \(X_n^{\text{reg}}\) contains the values of regular characters on regular classes and \(X_n^{\text{sing}}\) is defined analogously. The main results are the following: \(|\det(X_n^{\text{reg}})|\) equals the product of all parts of all class regular partitions of \(n\) (when \(l\) is prime, this is also the absolute value of the determinant of the Brauer character table in characteristic \(l\)), and \(|\det(X_n^{\text{sing}})|\) is the product of the factorials of the multiplicities of all parts of class singular partitions of \(n\).
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1907106
zbMATH Keywords: character tables of finite symmetric groups
zbMATH Keywords: singular character tables
zbMATH Keywords: partitions
zbMATH Keywords: irreducible characters
author: Jørn B. Olsson
reviewed by: Cornel Baetica
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1907106