Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

  • Volodymyr Mazorchuk

    Uppsala University, Sweden
  • Shraddha Srivastava

    Uppsala University, Sweden
Jucys–Murphy elements and Grothendieck groups for generalized rook monoids cover
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Abstract

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

Cite this article

Volodymyr Mazorchuk, Shraddha Srivastava, Jucys–Murphy elements and Grothendieck groups for generalized rook monoids. J. Comb. Algebra 6 (2022), no. 1/2, pp. 185–222

DOI 10.4171/JCA/65