Primitive representations of finite semigroups I

Abstract

Primitive representations of finite groups as well as primitive finite groups were classified in the O'Nan-Scott Theorem. In this paper we classify faithful finite primitive semigroup representations. To each finite primitive representation, we associate an invariant, a finite dimensional matirix with entries in a primitive finite groups representation. To a large extent, this matrix determines the representation. In later papers in this series the invariant is further explored. Our primitivity resoullts rely on two main ideas, one of which is a small but important part of the theory of tame algebras developed by R. McKenzie, while the other is our adaptation of Rees matrix theory to the representation theory of finite semigroups. Using the latter idea we are able to provide a description of all representations of a regular T class of a finite semigroup.