Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

Abstract

We consider a tower of generalized rook monoid algebras over the field $\mathbb{C}$ of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys–Murphy elements for generalized rook monoid algebras.

Over an algebraically closed field $\mathbb{k}$ of positive characteristic $p$, utilizing Jucys–Murphy elements of rook monoid algebras, for $0\le i\le p-1$ we define the corresponding $i$-restriction and $i$-induction functors along with two extra functors. On the direct sum ${\mathcal{G}}_{\mathbb{C}}$ of the Grothendieck groups of module categories over rook monoid algebras over $\mathbb{k}$, these functors induce an action of the tensor product of the universal enveloping algebra $U({\widehat{\mathfrak{sl}}}_p(\mathbb{C}))$ and the monoid algebra $\mathbb{C}[\mathcal{B}]$ of the bicyclic monoid $\mathcal{B}$. Furthermore, we prove that ${\mathcal{G}}_{\mathbb{C}}$ is isomorphic to the tensor product of the basic representation of $U({\widehat{\mathfrak{sl}}}_p(\mathbb{C}))$ and the unique infinite-dimensional simple module over $\mathbb{C}[\mathcal{B}]$, and also exhibit that ${\mathcal{G}}_{\mathbb{C}}$ is a bialgebra. Under some natural restrictions on the characteristic of $\mathbb{k}$, we outline the corresponding result for generalized rook monoids.