Irreducible representations of Hecke-Kiselman monoids

TL;DR

This paper characterizes all irreducible representations of Hecke-Kiselman monoid algebras over algebraically closed fields, revealing a structure similar to finite semigroup representations and providing a complete description for the case of oriented cycles.

Contribution

It provides a complete classification of irreducible representations of Hecke-Kiselman monoid algebras satisfying a polynomial identity, linking graph properties to algebraic representations.

Findings

Irreducible representations are either 1-dimensional from idempotents or from matrix-type semigroups.

The polynomial identity condition relates to a simple graph-theoretic criterion.

The prime spectrum is fully characterized for oriented cycle graphs.

Abstract

Let $K[HK_{\Theta}]$ denote the Hecke-Kiselman algebra of a finite oriented graph $\Theta$ over an algebraically closed field $K$. All irreducible representations, and the corresponding maximal ideals of $K[HK_{\Theta}]$, are characterized in case this algebra satisfies a polynomial identity. The latter condition corresponds to a simple condition that can be expressed in terms of the graph $\Theta$. The result shows a surprising similarity to the classical results on representations of finite semigroups; namely every representation either comes form an idempotent in the Hecke-Kiselman monoid $HK_{\Theta}$ (and hence it is $1$-dimensional), or it comes from certain semigroup of matrix type (which is an order in a completely $0$-simple semigroup over an infinite cyclic group). The case when $\Theta$ is an oriented cycle plays a crucial role; the prime spectrum of $K[HK_{\Theta}]$ is completely characterized in this case.