Combinatorial Gelfand models for some semigroups and q-rook monoid algebras

TL;DR

This paper develops combinatorial Gelfand models for various finite semigroup algebras, including the symmetric inverse semigroup and the q-rook monoid algebra, extending previous models and providing new algebraic insights.

Contribution

It introduces new combinatorial Gelfand models for several semigroup algebras and extends the Gelfand model to the q-rook monoid algebra, broadening the scope of algebraic representations.

Findings

Constructed Gelfand models for symmetric inverse semigroup and dual structures

Extended Gelfand model to q-rook monoid algebra

Provided algebraic and combinatorial insights into semigroup representations

Abstract

Inspired by the results of [R. Adin, A. Postnikov, Y. Roichman, Combinatorial Gelfand model, preprint math.RT arXiv:0709.3962], we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup, and the factor power of the symmetric group. Furthermore we extend the Gelfand model for the semigroup algebras of the symmetric inverse semigroup to a Gelfand model for the $q$-rook monoid algebra.