Representations of the quasi-partition algebras

TL;DR

This paper generalizes the concept of quasi-partition algebras, constructs their simple modules, introduces new related algebras, and explores their representation theory through combinatorial and diagrammatic methods.

Contribution

It provides a broader definition of quasi-partition algebras, constructs simple modules, and introduces two new algebras with isomorphisms to centralizer algebras for certain parameters.

Findings

New generalized definitions of quasi-partition algebras

Construction of simple modules for these algebras

Identification of isomorphic relations between new algebras and centralizer algebras

Abstract

The quasi-partition algebras were introduced by Daugherty and the first author as centralizers of the symmetric group. In this article, we give a more general definition of these algebras and give a construction of their simple modules. In addition, we introduce two new algebras, we give linear bases and show that for specializations of their parameters, these new algebras are isomorphic to centralizer algebras. We provide a generalized Bratteli diagram that illustrates how the representation theory of the three algebras discussed in this paper are related. Moreover, we give combinatorial formulas for the dimensions of the simple modules of these algebras.