Semigroup algebras of the full matrix semigroup over a finite field
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- by L. G. Kovács PDF
- Proc. Amer. Math. Soc. 116 (1992), 911-919 Request permission
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
- Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR 0417223, DOI 10.1007/978-1-4684-9913-1
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
- D. K. Faddeev, Representations of the full semigroup of matrices over a finite field, Dokl. Akad. Nauk SSSR 230 (1976), no. 6, 1290–1293 (Russian). MR 0422441
- D. J. Glover, A study of certain modular representations, J. Algebra 51 (1978), no. 2, 425–475. MR 476841, DOI 10.1016/0021-8693(78)90116-3
- Jan Okniński and Mohan S. Putcha, Complex representations of matrix semigroups, Trans. Amer. Math. Soc. 323 (1991), no. 2, 563–581. MR 1020044, DOI 10.1090/S0002-9947-1991-1020044-8
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 911-919
- MSC: Primary 16S36; Secondary 20M25
- DOI: https://doi.org/10.1090/S0002-9939-1992-1123658-6
- MathSciNet review: 1123658