Isometries parfaites entre blocs de groupes linéaires ou unitaires. (Perfect isometries between blocks of linear or unitary groups)
type: Item
instance of: scholarly article
title: Isometries parfaites entre blocs de groupes linéaires ou unitaires. (Perfect isometries between blocks of linear or unitary groups)
zbMATH Open document ID: 0694.20025
DOI: 10.1007/BF02571321
author: Michel Enguehard
published in: Mathematische Zeitschrift
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174203
review text: The purpose of this paper is to prove the following theorem ``Let G and \(G'\) be general linear groups, or unitary groups, on the same finite field. Let \(\ell\) be a prime different from the characteristic of the basic field, and b and \(b'\) be \(\ell\)-blocks of G and \(G'\) respectively. If the weights of b and \(b'\) are equal, there exists a perfect isometry between b and \(b'.''\) The weight of an \(\ell\)-block of a general linear or unitary finite group is a family of integers and was introduced by \textit{J. B. Olsson} and \textit{G. O. Michler} [Math. Z. 184, 203-233 (1983; Zbl 0505.20009)]. A perfect isometry between two blocks - in fact between their Grothendieck groups on the complex field - consists of a bijection between the sets of irreducible ordinary characters and a sign-valued function with some good arithmetical properties; this notion was introduced by M. Broué in 1988. The existence of a perfect isometry between b and \(b'\) implies that corresponding ordinary characters have equal heights, that the blocks have the same number of irreducible modular characters, and that the algebras of b and \(b'\) on a suitable complete valued ring have isomorphic centers. The method of proof is similar to the one already used by the same author for the blocks of equal weights of symmetric groups. So it confirms ``the particularly close fit of the Deligne-Luztig theory, the Brauer theory, and the combinatorial theory underlying the character theory of the symmetric group'' [\textit{P. Fong} and \textit{B. Srinivasan}, Invent. Math. 69, 109-153 (1982; Zbl 0507.20007)]. By a theorem of \textit{M. Broué} and \textit{J. Michel} [J. Reine Angew. Math. 395, 56-67 (1989; Zbl 0654.20048)] the proof reduces immediately to the case where each of the blocks contains a unipotent character. Then the weight is an integer and the block is defined by a partition. The sets of conjugacy classes of Brauer's elements are in natural bijection. Elementary facts on partitions, ho…
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 4136220
zbMATH Keywords: general linear groups
zbMATH Keywords: unitary groups
zbMATH Keywords: l-blocks
zbMATH Keywords: weight
zbMATH Keywords: perfect isometry
zbMATH Keywords: Grothendieck groups
zbMATH Keywords: irreducible ordinary characters
zbMATH Keywords: height
zbMATH Keywords: irreducible modular characters
zbMATH Keywords: unipotent character
zbMATH Keywords: generalized characters
MaRDI profile type: Publication
cites work: Q3800275
cites work: Q3784301
cites work: Foncteurs de Lusztig et caractères des groupes linéaires et unitaires sur un corps fini. (Lusztig functors and characters of linear and unitary groups over a finite field)
cites work: On the Representation of the Symmetric Group
cites work: The blocks of finite general linear and unitary groups
cites work: The Characters of the Finite General Linear Groups
cites work: Representations of reductive groups over finite fields
cites work: The characters of the finite unitary groups
cites work: Representations of finite Chevalley groups. A survey
description: scientific article; zbMATH DE number 4136220