Semigroup algebras of the full matrix semigroup over a finite field

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.