Combinatorics and structure of Hecke-Kiselman algebras

Abstract

Hecke-Kiselman monoids ${\text{HK}}_{\Theta }$ and their algebras $K[{\text{HK}}_{\Theta }]$, over a field $K$, associated to finite oriented graphs $\Theta$ are studied. In the case $\Theta$ is a cycle of length $n⩾3$, a hierarchy of certain unexpected structures of matrix type is discovered within the monoid $C_n={\text{HK}}_{\Theta }$ and it is used to describe the structure and the properties of the algebra $K[C_n]$. In particular, it is shown that $K[C_n]$ is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand-Kirillov dimension one. This is used to characterize all Noetherian algebras $K[{\text{HK}}_{\Theta }]$ in terms of the graphs $\Theta$. The strategy of our approach is based on the crucial role played by submonoids of the form $C_n$ in combinatorics and structure of arbitrary Hecke-Kiselman monoids ${\text{HK}}_{\Theta }$.

29 pages