On the irreducible representations of a finite semigroup

Abstract

Abstract. Work of Clifford, Munn and Ponizovskiȋ 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-Ponizovskiȋ 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.

On simple modules over twisted finite category algebras

“…For inverse semigroup algebras and inverse category algebras this follows also from [21] and [14]. Our proof of Theorem 1.1 follows closely the lines of that given in [6] for semigroups, which has motivated the present work. As in [6], this is based on Green's work on Schur functors in [10, 6.2].…”

Matrix Mortality and the Černý-Pin Conjecture

“…We take a minimalist approach, stating exactly what we need in order to prove our main result. Details can be found in [9,15,24,30,33]. To fix notation, if S is a monoid, we use Irr(S) to denote the set of equivalence classes of irreducible representations of S. The reader should verify that every irreducible representation of a group is by invertible maps.…”

“…Let C : B ×A → G 0 J be the sandwich matrix for J and denote by ρ⊗C the d|B|×d|A| matrix obtained by applying ρ entrywise to C (where we take ρ(0) to be the d×d zero matrix). The following result can be extracted from [33] and [9,Chapter 5]; see also [30] and [31,Chapter 15] for a summary without proofs or [15,23] for module-theoretic statements and proofs. Now we are ready to prove that f (n) = n(n+1)/2 is a superadditive mortality function for monoids in EDS.…”

“…Set V = Q n . Then tr e = dim V e. But the theory of Munn and Ponizovsky implies that the restriction of the action of G to V e gives an irreducible representation of G [9,15,30,33]. Since G is the trivial group, this implies dim V e = 1 and so tr e = 1.…”

Combinatorial Gelfand models for some semigroups and q-rook monoid algebras

“…We are left to show that V S is a Gelfand model for S. We will use the fact that simple modules over the complex semigroup algebra of a finite semigroup S are in bijective correspondence with simple modules of G e i , 1 ≤ i ≤ k (see [CP,Chapter 5] or [GMS,Theorem 7] for a more modern approach). In view of this, it is enough to show that for a maximal subgroup G of S and a simple CG-module V the corresponding CS-module V ′ is isomorphic to a submodule of V S , and then to make sure that the sum of the dimensions of all V ′ -s equals the dimension of V S .…”