Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

Abstract

We consider a tower of generalized rook monoid algebras over the field 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 k of positive characteristic p, utilizing Jucys-Murphy elements of rook monoid algebras, for 0 Ä i Ä p 1 we define the corresponding i -restriction and i -induction functors along with two extra functors. On the direct sum G C of the Grothendieck groups of module categories over rook monoid algebras over k, these functors induce an action of the tensor product of the universal enveloping algebra U.sl p .C// and the monoid algebra COEB of the bicyclic monoid B. Furthermore, we prove that G C is isomorphic to the tensor product of the basic representation of U.sl p .C// and the unique infinite-dimensional simple module over COEB, and also exhibit that G C is a bialgebra. Under some natural restrictions on the characteristic of k, we outline the corresponding result for generalized rook monoids.