Classification of monoids of Lie type
type: Item
instance of: scholarly article
title: Classification of monoids of Lie type
zbMATH Open document ID: 0801.20051
published in: Journal of Algebra
publication date: 1994-10-16T00:00:00Z
review text: Let \(G\) be a finite group of Lie type. The paper is concerned with an intriguing class of finite regular monoids \(M\) with unit group \(G\), called monoids of Lie type, whose structure strongly reflects the structure of \(G\). The simplest example being \(M = M_ n(\mathbb{F}_ q)\) and \(G = GL_ n(\mathbb{F}_ q)\). These monoids were first classified by the author, [J. Algebra 120, No. 1, 139-169 (1989; Zbl 0683.20051)], into discrete types with the type map \(\lambda : {\mathcal U} \to 2^{\Lambda}\), where \(\mathcal U\) is the set of \(\mathcal J\)-classes of \(M\) and \(\Gamma\) is the Coxeter graph of \(G\). The type map determines the system of idempotents of \(M\). For any given type \(\lambda\) there exist universal monoids \(M^ +(\lambda)\), \(M^ -(\lambda)\) such that all monoids of type \(\lambda\) are obtained as \(M^ +(\lambda) \twoheadrightarrow M \twoheadrightarrow M^ -(\lambda)\). These monoids \(M\) are then classified by certain families of normal subgroups of Levi subgroups. Moreover, a description of \(M^ -(\lambda)\) is known. The main result of the present paper establishes a description of the monoid \(M^ +(\lambda)\). Some of these universal monoids are then considered in connection with group representation theory. In particular, it is shown that, if \(M^ +\) is the universal monoid of the same type as \(M_ n(\mathbb{F}_ q)\), then every irreducible representation of \(GL_ n(\mathbb{F}_ q)\) of degree \(>1\) has a unique extension to \(M^ +\).
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20M30
zbMATH DE Number: 554140
zbMATH Keywords: general linear groups
zbMATH Keywords: finite fields
zbMATH Keywords: finite group of Lie type
zbMATH Keywords: finite regular monoids
zbMATH Keywords: unit group
zbMATH Keywords: monoids of Lie type
zbMATH Keywords: Coxeter graph
zbMATH Keywords: type map
zbMATH Keywords: system of idempotents
zbMATH Keywords: universal monoids
zbMATH Keywords: Levi subgroups
zbMATH Keywords: irreducible representation
Wikidata QID: Q115396271
author: Mohan S. Putcha
reviewed by: Jan Okniński
MaRDI profile type: Publication
full work available at URL: jabr.1994.1035
OpenAlex ID: W1988327846
DOI: 10.1006/JABR.1994.1035
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description: scientific article; zbMATH DE number 554140