Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20M30, 20G05, 20G40, 20M25

Retrieve articles in all journals with MSC: 20M30, 20G05, 20G40, 20M25

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society