Invariant algebra and cuspidal representations of finite monoids
For a finite regular monoid \(M\) with unit group \(G\) the algebra of invariants \(\mathbb{C}[M]^G\) of the complex semigroup algebra \(\mathbb{C}[M]\) is studied. So, \(\mathbb{C}[M]^G\) is the centralizer of \(G\) in \(\mathbb{C}[M]\). For every irreducible character \(\theta\) of a maximal subgroup \(H\) of \(M\), certain induced characters \(\theta^+\), \(\theta^-\) and \(\widetilde\theta\) of \(G\) were defined by the author [in Proc. Lond. Math. Soc., III. Ser. 73, No. 3, 623-641 (1996; Zbl 0860.20051)]. Then \(M\) is called balanced, if for every \(H\), \(\theta\), every component of \(\theta^+\cap\theta^-\) is a component of \(\widetilde\theta\). For such monoids it is shown that \(\mathbb{C}[M]^G\) is a quasi-hereditary algebra and the blocks of \(\mathbb{C}[M]^G\) are determined. Moreover, a theory of cuspidal characters is developed, extending the case of universal canonical monoids of Lie type, considered by the author and the reviewer [Int. J. Algebra Comput. 1, No. 1, 33-47 (1991; Zbl 0752.20034)]. So the irreducible characters of \(G\) are classified into series depending on which \(\mathcal J\)-class of \(M\) they come from. The examples of balanced monoids include the full transformation semigroup \({\mathcal T}^n\) of all self-maps on \(\{1,\dots,n\}\) and the semigroup \(T_n(\mathbb{F}_q)\) of upper triangular matrices over the field \(\mathbb{F}_q\).
scientific article; zbMATH DE number 1271829
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