Jucys-Murphy elements and Grothendieck groups for generalized rook monoids

TL;DR

This paper explores the algebraic structure of generalized rook monoid algebras, constructs modules and Jucys-Murphy elements, and links their Grothendieck groups to representations of affine Lie algebras and monoid algebras.

Contribution

It introduces new module constructions and functors for generalized rook monoid algebras, connecting their Grothendieck groups to affine Lie algebra representations.

Findings

Construction of simple modules and Jucys-Murphy elements.

Identification of Grothendieck groups with tensor products of representations.

Establishment of a bialgebra structure on Grothendieck groups.

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 $\Bbbk$ of positive characteristic $p$, utilizing Jucys-Murphy elements of rook monoid algebras, for $0\leq i\leq p-1$ 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 $\Bbbk$, these functors induce an action of the tensor product of the universal enveloping algebra $U(\hat{\mathfrak{sl}}_p(\mathbb{C}))$ 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 $U(\hat{\mathfrak{sl}}_{p}(\mathbb{C}))$ 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 $\Bbbk$, we outline the corresponding result for generalized rook monoids.