Matrix representations of semigroups

Abstract

In a previous paper, the author gave necessary and sufficient conditions for the algebra of a finite semigroup S over a field of suitable characteristic to be semisimple. When this algebra is semisimple, the matrix representations of S over are completely reducible.

Representation theory of finite semigroups, semigroup radicals and formal language theory

“…By way of contrast, the theory of semigroup representations, which was intensively developed during the 50s and 60s in classic work such as Clifford [15], Munn [29,30] and Ponizovsky (see [16,Chapter 5] for an account of this work, as well as [26,56] for nicer treatments restricting to the case of finite semigroups), has found almost no applications in the theory of finite semigroups. It was pointed out by McAlister in his survey of 1971 [28] that the only paper applying representation theoretic results to finite semigroups was the paper [51] of Rhodes.…”

Regularity of Semigroup Rings

“…Q is 0-simple by definition and inverse by [6,Corollary 3]. Munn [5,Lemma 4.2] has shown that in this case Q is isomorphic to a nXn Rees matrix semigroup over a group with zero G° (G a subgroup of Q, and hence of D) with nXn identity matrix as sandwich matrix. Using this fact, it is easy to verify (see [8,Lemma 3.1], or [2, Lemma 5.17]) that (RQ)o = (RG)n the ring of nXn matrices over the group ring RG.…”

On the irreducible representations of a finite semigroup

“…Clearly Ann S (M ) is an ideal of S. The following definition, due to Munn [16], is crucial to semigroup representation theory.…”

“…Work of Clifford [5,4], Munn [16,15] and Ponizovskiȋ [17] parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. (See [6,Chapter 5] for a full account of this work.)…”