On the irreducible representations of a finite semigroup

TL;DR

This paper offers a modern proof of the classification of irreducible representations of finite semigroups, extending previous results by avoiding the use of Rees's theorem and working over any base ring.

Contribution

It provides a simplified, ring-agnostic proof of the Clifford-Munn-Ponizovski{} result using Green's lemma, bypassing the need for 0-simple semigroup theory.

Findings

Proof works over any base ring.

Circumvents Rees's theorem in classification.

Simplifies the understanding of irreducible representations.

Abstract

Work of Clifford, Munn and Ponizovski{\u\i} parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovski{\u\i} result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.