Invariant algebra and cuspidal representations of finite monoids
type: Item
instance of: scholarly article
title: Invariant algebra and cuspidal representations of finite monoids
zbMATH Open document ID: 0924.20050
published in: Journal of Algebra
publication date: 1999-11-09T00:00:00Z
review text: 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 \(\mathbb{F}_q\).
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20H30
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 14L40
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 1271829
zbMATH Keywords: semigroups of upper triangular matrices
zbMATH Keywords: finite regular monoids
zbMATH Keywords: unit groups
zbMATH Keywords: algebras of invariants
zbMATH Keywords: complex semigroup algebras
zbMATH Keywords: irreducible characters
zbMATH Keywords: induced characters
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: cuspidal characters
zbMATH Keywords: canonical monoids of Lie type
zbMATH Keywords: balanced monoids
zbMATH Keywords: full transformation semigroups
author: Mohan S. Putcha
reviewed by: Jan Okniński
MaRDI profile type: Publication
full work available at URL: jabr.1998.7634
OpenAlex ID: W2065335667
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DOI: 10.1006/JABR.1998.7634
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description: scientific article; zbMATH DE number 1271829