Combinatorial Gelfand models for semisimple diagram algebras

TL;DR

This paper develops combinatorial Gelfand models for various semisimple diagram algebras, providing explicit constructions for algebras like Brauer, Temperley-Lieb, and partition algebras.

Contribution

It introduces explicit combinatorial Gelfand models for multiple classes of semisimple diagram algebras, expanding the understanding of their representation theory.

Findings

Constructed Gelfand models for Brauer and partition algebras.

Extended models to partial and walled variants of these algebras.

Provided combinatorial descriptions facilitating representation analysis.

Abstract

We construct combinatorial (involutory) Gelfand models for the following diagram algebras in the case when they are semi-simple: Brauer algebra, its partial analogue, walled Brauer algebra, its partial analogue, Temperley-Lieb algebra, its partial analogue, walled Temperley-Lieb algebra, its partial analogue, partition algebra and its Temperley-Lieb analogue.