The McKay conjecture for exceptional groups and odd primes.
type: Item
instance of: scholarly article
title: The McKay conjecture for exceptional groups and odd primes.
zbMATH Open document ID: 1178.20006
author: Britta Späth
published in: Mathematische Zeitschrift
publication date: 2009-03-26T00:00:00Z
review text: The McKay conjecture is one of the main conjectures in the representation theory of finite groups. In the important paper [Invent. Math. 170, No. 1, 33-101 (2007; Zbl 1138.20010)], \textit{I. M. Isaacs, G. Malle}, and \textit{G. Navarro} have been able to reduce the conjecture to prove that all finite simple non-Abelian groups are ``good'' for every prime \(\ell\) in a certain sense. Now let \(S\) be a finite simple group of Lie type defined in characteristic \(p\neq\ell\) and let \(G\) be the maximal perfect central \(\ell'\)-extension of \(S\). Then it was shown by \textit{G. Malle} [in Represent. Theory 11, 192-220 (2007; Zbl 1139.20008)] that \(S\) is good for the prime \(\ell\), if the normalizers \(N_G(T)\) of Sylow tori \(T\) satisfy a set of three specific conditions. In the same paper, Malle proved that \(N_G(T)\) satisfies the first condition, and also (part of) the second condition, provided that all irreducible characters of \(C_G(T)\) extend to their inertia group in \(N_G(T)\). The main result of this paper establishes this extendibility condition in the case \(G\) is an exceptional group of Lie type. This result also allows the author to verify the McKay conjecture for the exceptional groups of Lie type \(G\) and odd primes \(\ell\neq p\).
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 5533140
zbMATH Keywords: McKay conjecture
zbMATH Keywords: numbers of irreducible characters
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: finite groups of Lie type
describes a project that uses: Magma
reviewed by: Pham Huu Tiep
Wikidata QID: Q122984855
MaRDI profile type: Publication
full work available at URL: s00209-008-0340-7
OpenAlex ID: W2029869183
cites work: Alperin-McKay conjecture for the Chevalley groups \(G_ 2(q)\)
cites work: On a theorem of Shintani
cites work: Discovering mathematics with Magma. Reducing the abstract to the concrete
cites work: Generic Sylow theorems for the reductive groups over finite fields
cites work: Q4303005
cites work: Q5662096
cites work: Q3682663
cites work: Q3997911
cites work: Q4509026
cites work: Q4375273
cites work: Character theory of finite groups
cites work: Q4103728
cites work: A reduction theorem for the McKay conjecture.
cites work: Die unipotenten charaktere von 2F4(q2)
cites work: Height 0 characters of finite groups of Lie type
cites work: The Inductive McKay Condition for Simple Groups Not of Lie Type
cites work: A remark on the Alperin-McKay conjecture.
cites work: Finite Unitary Reflection Groups
cites work: The Character Tables for SL(3, q), SU(3, q2), PSL(3, q), PSU(3, q2)
cites work: Regular elements of finite reflection groups
cites work: Linear algebraic groups.
cites work: Q4930353
cites work: Normalisateurs de tores. I: Groupes de Coxeter etendus
DOI: 10.1007/S00209-008-0340-7
Recommended article: Groups of prime degree and the Bateman-Horn conjecture
Recommended article: The McKay conjecture and Brauer's induction theorem.
Recommended article: The McKay conjecture and Galois automorphisms.
Recommended article: New refinements of the McKay conjecture for arbitrary finite groups
Recommended article: DIRICHLET'S PRIME NUMBER THEOREM FOR MODULAR GROUPS
Recommended article: On Oliver's \(p\)-group conjecture.
Recommended article: Exceptional Groups and Their Modular Forms
Recommended article: Remarks on the McKay conjecture.
Recommended article: Nonsolvable groups satisfying the prime-power hypothesis.
description: scientific article; zbMATH DE number 5533140