The Gelfand-Kirillov dimension of Hecke-Kiselman algebras

TL;DR

This paper establishes a formula for the Gelfand-Kirillov dimension of Hecke-Kiselman algebras based on graph invariants, linking algebraic properties to combinatorial features of the underlying graph.

Contribution

It introduces a new numerical invariant of the graph that precisely determines the Gelfand-Kirillov dimension of the associated algebra.

Findings

Gelfand-Kirillov dimension equals the sum of cyclic subgraphs and specific oriented paths.

The dimension is an integer when finite, linked to graph structure.

Provides a combinatorial method to compute algebraic complexity.

Abstract

Hecke-Kiselman algebras $A_{\Theta}$, over a field $k$, associated to finite oriented graphs $\Theta$ 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 $\Theta$ that determines the dimension of $A_{\Theta}$ is found. Namely, we prove that the Gelfand-Kirillov dimension of $A_{\Theta}$ is the sum of the number of cyclic subgraphs of $\Theta$ and the number of oriented paths of a special type in the graph, each counted certain specific number of times.