On the irreducible representations of a finite semigroup

By Olexandr Ganyushkin, Volodymyr Mazorchuk, Ben Steinberg. 2007

82 citations recorded by the source corpus.

Work of Clifford, Munn and Ponizovskii 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-Ponizovskii 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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/444c369f085dff0036237d8df7073d40bce16304

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In Computational Theory and Mathematics

1. Ben Steinberg
2. Stuart W. Margolis
3. Volodymyr Mazorchuk
4. Olexandr Ganyushkin
5. Franco V. Saliola
6. Mark V. Lawson
7. L. O. Clark
8. Aidan Sims

In Discrete Mathematics and Combinatorics

1. Stuart W. Margolis
2. Ben Steinberg
3. Volodymyr Mazorchuk
4. Olexandr Ganyushkin
5. Franco V. Saliola

In Geometry and Topology

1. Stuart W. Margolis
2. Ben Steinberg
3. Volodymyr Mazorchuk
4. Olexandr Ganyushkin
5. Franco V. Saliola
6. Mark V. Lawson
7. C. Stroppel
8. L. O. Clark

Citations in the corpus

• On Kiselman quotients of 0-Hecke monoids (2010, background): “[11] or [10, Chapter 11]) applies in a straightforward way.”
• A Groupoid Approach to Discrete Inverse Semigroup Algebras (2009, background): “The final section of the paper applies the results to inverse semigroups via their universal groupoids.”
• Combinatorial Topology and the Global Dimension of Algebras Arising in Combinatorics (2012, background): “In general, the simple kB-modules are the modules of the form (k/m)X where m is a maximal ideal of k, as follows from the results of [43].”
• Quivers of monoids with basic algebras (2011, background): “The reader interested in more details about the representation theory of finite semigroups in general is referred to [36] for a modern approach or to [22, Chapter 5] and [57,58,72,79] for the classical approach; see also [55,73] for connections with the theory of quasi-hereditary algebras and [3,4,2”