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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Complex representations of matrix semigroups


Authors: Jan Okniński and Mohan S. Putcha
Journal: Trans. Amer. Math. Soc. 323 (1991), 563-581
MSC: Primary 20M30; Secondary 20G05, 20G40, 20M25
MathSciNet review: 1020044
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Abstract: Let $ M$ be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units $ G$. The multiplicative semigroup $ {\mathcal{M}_n}(F)$, where $ F$ is a finite field, is a particular example. Using Harish-Chandra's theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of $ M$ is completely reducible. Using this we characterize the representations of $ G$ extending to irreducible representations of $ M$ as being those induced from the irreducible representations of certain parabolic subgroups of $ G$. We go on to show that if $ F$ is any field and $ S$ any multiplicative subsemigroup of $ {\mathcal{M}_n}(F)$, then the semigroup algebra of $ S$ over any field of characteristic zero has nilpotent Jacobson radical. If $ S = {\mathcal{M}_n}(F)$, then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of $ {\mathcal{M}_n}(F)$ over a field of characteristic zero is regular if and only if $ \operatorname{ch} (F) = p > 0$ and $ F$ is algebraic over its prime field.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1020044-8
PII: S 0002-9947(1991)1020044-8
Article copyright: © Copyright 1991 American Mathematical Society