The Gelfand-Kirillov dimension of Hecke-Kiselman algebras

Hecke-Kiselman algebras $A_{\mathrm{\Theta }}$, over a field $k$, associated to finite oriented graphs $\mathrm{\Theta }$ are considered. It has been known that every such algebra is an automaton algebra in the sense of Ufranovskii. In particular, its Gelfand-Kirillov dimension is an integer if it is finite. In this paper, a numerical invariant of the graph $\mathrm{\Theta }$ that determines the dimension of $A_{\mathrm{\Theta }}$ is found. Namely, we prove that the Gelfand-Kirillov dimension of $A_{\mathrm{\Theta }}$ is the sum of the number of cyclic subgraphs of $\mathrm{\Theta }$ and the number of oriented paths of a special type in the graph, each counted certain specific number of times.