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

DOI:
https://doi.org/10.1090/S0002-9939-1992-1123658-6

MathSciNet review:
1123658

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Abstract: Let denote the multiplicative semigroup of all -by- matrices over a finite field and a commutative ring with an identity element in which the characteristic of is a unit. It is proved here that the semigroup algebra is the direct sum of algebras, namely, of one full matrix algebra over each of the group algebras with . The degree of the relevant matrix algebra over is the number of -dimensional subspaces in an -dimensional vector space over .

For a field of characteristic different from that of , 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.

**[1]**Frank W. Anderson and Kent R. Fuller,*Rings and categories of modules*, Graduate Texts in Math., vol. 13, Springer-Verlag, New York, Heidelberg, and Berlin, 1974. MR**0417223 (54:5281)****[2]**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Pure Appl. Math., vol. 11, Interscience-Wiley, New York and London, 1962. MR**0144979 (26:2519)****[3]**D. K. Faddeev,*Representations of the full matrix semigroup over a finite field*, Dokl. Akad. Nauk SSSR**230**(1976), 1290-1293; English transl., Soviet Math. Dokl.**17**(1976), 1483-1486. MR**0422441 (54:10430)****[4]**D. J. Glover,*A study of certain modular representations*, J. Algebra**51**(1978), 425-475. MR**0476841 (57:16392)****[5]**Jan Okniński and Mohan S. Putcha,*Complex representations of matrix semigroups*, Trans. Amer. Math. Soc.**323**(1991), 563-581. MR**1020044 (91e:20047)**

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1123658-6

Article copyright:
© Copyright 1992
American Mathematical Society