On the irreducible representations of a finite semigroup

Ganyushkin, Olexandr; Mazorchuk, Volodymyr; Steinberg, Benjamin

doi:10.1090/S0002-9939-09-09857-8

On the irreducible representations of a finite semigroup
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by Olexandr Ganyushkin, Volodymyr Mazorchuk and Benjamin Steinberg PDF

Proc. Amer. Math. Soc. 137 (2009), 3585-3592 Request permission

Abstract:

Work of Clifford, Munn and Ponizovskiĭ 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-Ponizovskiĭ 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.

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