Semigroup algebras of the full matrix semigroup over a finite field

Abstract

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[GL(r, F)] with r = 0, 1, ... , ». The degree of the relevant matrix algebra over A" [GL(r, F)] is the number of /■-dimensional subspaces in an «-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.

Effective dimension of finite semigroups

“…The non-trivial maximal subgroup is isomorphic to F × q . A result of Kovács in [23] implies that the structure matrix is invertible whenever the characteristic of k does not divide q. The case of the complex field was first proved by Okniński and Putcha in [28].…”

“…The distinguished ideal of Mat n×n (F 2 ) has rank 2 n − 1 by a result of Kovács [23] (see the discussion below where general q is considered). The unique minimal effective representation is then the linearization of the action by partial maps on F n 2 \ {0}.…”

Generic representation theory of finite fields in nondescribing characteristic

“…Kovács [12]. This followed a related announcement by Faddeev [5], and related results by Okniński and Putcha [18].…”

Décompositions à la Steinberg sur une catégorie additive

“…Si k est un corps fini, le foncteur K[Gr n ] est encore simple si K est de caractéristique différente de celle de k, mais la démonstration précédente ne s'applique plus. La simplicité se déduit des résultats de Kuhn [42] (reposant sur le travail de Kovács [36] sur les représentations des monoïdes multiplicatifs M n (k)) déjà invoqués au corollaire 8.32. En revanche, si k est fini et de même caractéristique que K, alors K[Gr n ] n'est jamais simple si n > 0 -on peut même montrer qu'il est de dimension de Krull au moins n (et conjecturalement, de dimension de Krull exactement n -cf.…”