Jucys-Murphy elements and Grothendieck groups for generalized rook monoids (Q2154783)

Summary: 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 \(U(\widehat{\mathfrak{sl}}_p(\mathbb{C}))\) 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 \(\Bbbk \), we outline the corresponding result for generalized rook monoids.