Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Semigroup algebras of the full matrix semigroup over a finite field


Author: L. G. Kovács
Journal: Proc. Amer. Math. Soc. 116 (1992), 911-919
MSC: Primary 16S36; Secondary 20M25
MathSciNet review: 1123658
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ denote the multiplicative semigroup of all $ n$-by-$ n$ matrices over a finite field $ F$ and $ K$ a commutative ring with an identity element in which the characteristic of $ F$ is a unit. It is proved here that the semigroup algebra $ K[M]$ is the direct sum of $ n + 1$ algebras, namely, of one full matrix algebra over each of the group algebras $ K[\operatorname{GL}(r,F)]$ with $ r = 0,1, \ldots ,n$. The degree of the relevant matrix algebra over $ K[\operatorname{GL}(r,F)]$ is the number of $ r$-dimensional subspaces in an $ n$-dimensional vector space over $ F$.

For $ K$ a field of characteristic different from that of $ F$, this result was announced by Faddeev in 1976. He only published an incomplete sketch of his proof, which relied on details from the representation theory of finite general linear groups. The present proof is self-contained.


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


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16S36, 20M25

Retrieve articles in all journals with MSC: 16S36, 20M25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1123658-6
Article copyright: © Copyright 1992 American Mathematical Society