Partial inner products.
type: Item
instance of: scholarly article
title: Partial inner products.
zbMATH Open document ID: 1123.20005
author: Reinhard Knörr
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: \(G\) is a finite group, \(S\) is a subset of \(\text{Irr}(G)\). For any class functions \(\alpha\) and \(\beta\) of \(G\), \(\sum_{\sigma\in S}[\alpha,\sigma][\sigma,\beta]\) is called the \(S\)-partial inner product of \(\alpha\) and \(\beta\). Assume characters and class functions are over \(\mathbb{C}\) and \(\chi\mapsto\overline\chi\) is the complex conjugation. For any \(g\) and \(h\) in different conjugacy classes of \(G\), the orthogonal property is well known: \(\sum_{\chi\in\text{Irr}(G)}\chi(g)\overline{\chi(h)}=0\). A weak second orthogonal property is introduced in a Clifford setting. That is, for any normal subgroup \(N\) of \(G\), and \(\varphi\in\text{Irr}(N)\), and \(\text{Irr}(G|\varphi)\) is the set of irreducible constituents of \(\varphi^G\), then \(\sum_{\chi\in\text{Irr}(G)}\chi(g)\overline{\chi(h)}=0\), with \(\overline g\) and \(\overline h\) not conjugate in \(\overline G=G/N\). The paper also looks into the Burnside-Brauer theorem for two nonfaithful characters \(\eta,\delta\) and the exponents \(\delta\eta^i\), and proves that those which are constituents of the exponents are exactly those with nonzero \([\chi|_K,\delta|_K]\), where \(K\) is the kernel of \(\eta\). That is, for any two characters \(\gamma\) and \(\delta\) of \(G\) with class function \(\eta\) such that the kernel \(K\) of \(\eta\) is \(G(\eta,\eta(1))\), the following are equivalent: (i) \([\gamma|_K,\delta|_K]\neq 0\); (ii) \([\gamma,\delta\eta^i]\neq 0\) for some \(i<m\); (iii) \([\gamma,\delta\eta^i]\neq 0\) for some \(i\), where \(m\) is the number of different values of \(\eta\).
reviewed by: Chuang Peng
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 5078826
zbMATH Keywords: finite group representations
zbMATH Keywords: ordinary characters
zbMATH Keywords: Clifford theory
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: inner products of class functions
zbMATH Keywords: partial inner products
zbMATH Keywords: orthogonality relations
zbMATH Keywords: constituents
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.05.027
OpenAlex ID: W2000884845
cites work: Q4103728
DOI: 10.1016/J.JALGEBRA.2006.05.027
description: scientific article; zbMATH DE number 5078826