McKay natural correspondences on characters.
type: Item
instance of: scholarly article
title: McKay natural correspondences on characters.
zbMATH Open document ID: 1320.20010
author: Gabriel Navarro
author: Pham Huu Tiep
author: Carolina Vallejo Rodríguez
published in: Algebra \& Number Theory
publication date: 2014-12-22T00:00:00Z
review text: Let \(G\) be a finite group, \(p\) a prime, \(P\in\text{Syl}_p(G)\) and \(N=N_G(P)\). The well-known McKay conjecture states that if \(\text{Irr}_{p'}(G)\) is the set of irreducible (complex) characters of \(p'\)-degree of \(G\), then \(|\text{Irr}_{p'}(G)|=|\text{Irr}_{p'}(N)|\). It is well known that, in general, no choice-free correspondence can exist between \(\text{Irr}_{p'}(G)\) and \(\text{Irr}_{p'}(N)\). In the paper under review the authors study situations, where a natural correspondence exists. The main result is the following Theorem. Let \(p\) be odd. Suppose that \(N=PC_G(P)\). If \(\chi\in\text{Irr}_{p'}(G)\) lies in the principal block \(B_0(G)\) of \(G\), then \(\chi_N=\chi^*+\Delta\), where \(\chi^*\in\text{Irr}(N)\) is linear in the principal block and \(\Delta\) is either zero or a character whose irreducible constituents all have degree divisible by \(p\). Furthermore, the map \(\chi\mapsto\chi^*\) is a bijection \(\text{Irr}_{p'}(B_0(G))\to\text{Irr}_{p'}(B_0(N))\) where \(\text{Irr}_{p'}(B_0(G))\) is the set of irreducible characters in \(B_0(G)\) of degree not divisible by \(p\). The authors point out that the Theorem is not true for \(p=2\) (as \(S_5\) shows); moreover, it is not true for \(p\)-blocks of maximal defect. The authors discuss several consequences of the theorem and also show that for \(p\)-solvable groups, there even exists a natural correspondence between \(\text{Irr}_{p'}(G)\) and \(\text{Irr}_{p'}(N)\) which extends both the correspondence of the Theorem above and the Glauberman correspondence.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 6382482
zbMATH Keywords: finite groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: McKay conjecture
zbMATH Keywords: character correspondences
zbMATH Keywords: self-normalizing Sylow subgroups
zbMATH Keywords: \(p\)-decomposable Sylow normalizers
zbMATH Keywords: principal block
zbMATH Keywords: irreducible constituents
zbMATH Keywords: \(p\)-solvable groups
MaRDI profile type: Publication
full work available at URL: ant.2014.8.1839
OpenAlex ID: W2080424555
DOI: 10.2140/ANT.2014.8.1839
Recommended article: Character Theory and the McKay Conjecture
Recommended article: Q3371025
Recommended article: McKay correspondence and Hilbert schemes
Recommended article: The Projective McKay Correspondence
Recommended article: Correspondences between constituents of projective characters.
Recommended article: Q3158973
Recommended article: A note on derived McKay correspondence
Recommended article: Q3385817
Recommended article: Q4818188
Recommended article: Extensions of unipotent characters and the inductive McKay condition.
description: scientific article; zbMATH DE number 6382482