A bound on norms of generalized characters with applications
type: Item
instance of: scholarly article
title: A bound on norms of generalized characters with applications
zbMATH Open document ID: 0940.20015
author: Geoffrey R. Robinson
published in: Journal of Algebra
publication date: 2000-07-17T00:00:00Z
review text: In the paper under review the author investigates lower bounds for \(\sum_{g\in G^\#}|\Theta(g)|^2\) where \(\Theta\) is a generalized character of \(G\). He shows that this number can not get too small unless \(\Theta\) is a multiple of the regular character. More precisely, the sum is (lower) bounded by the number of conjugacy classes of \(G\) minus 1 provided \(\Theta\) does not vanish identically on \(G^\#\). The proof depends on some kind of the Cauchy-Schwarz lemma which the author proves in a cyclotomic number field. As consequences he obtains \(k_H(G)\leq|H|\cdot|C_G(x):C_H(x)|\) for some \(x\in H^\#\) where \(H\) is a subgroup of \(G\) and \(k_H(G)\) denotes the number of irreducible characters of \(G\) not vanishing identically on \(H^\#\). If \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(k_+(G)\leq|P|\cdot|C_G(x):C_P(x)|\) for some \(x\in P^\#\) where \(k_+(G)\) denotes the number of irreducible characters of \(G\) not lying in a \(p\)-block of defect zero. Furthermore he analyzes non-trivial situations in which the sum \(\sum_{g\in G^\#}|\Theta(g)|^2\) takes minimum value.
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1271825
zbMATH Keywords: \(p\)-blocks
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: generalized characters
zbMATH Keywords: numbers of conjugacy classes
zbMATH Keywords: Cauchy-Schwarz lemma
zbMATH Keywords: cyclotomic number fields
zbMATH Keywords: numbers of irreducible characters
MaRDI profile type: Publication
full work available at URL: jabr.1998.7589
OpenAlex ID: W1974252761
cites work: ON THE NUMBER OF IRREDUCIBLE CHARACTERS OF FINITE GROUPS IN A GIVEN BLOCK
cites work: Finite Groups Having Only One Irreducible Representation of Degree Greater than One
DOI: 10.1006/JABR.1998.7589
description: scientific article; zbMATH DE number 1271825