Classification of monoids of Lie type
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 \(M^ +\).
scientific article; zbMATH DE number 554140