Irreducible representations of Hecke-Kiselman monoids

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.