Gelfand Models for Diagram Algebras

TL;DR

This paper introduces a unified method to construct Gelfand models for a broad class of semisimple diagram algebras, generalizing known models and explicitly producing irreducible representations.

Contribution

It provides a uniform construction of Gelfand models for various diagram algebras using the Jones basic construction, extending the Saxl model to new algebra classes.

Findings

Constructed Gelfand models for multiple diagram algebras.

Model representations are given by diagrams acting via signed conjugation.

In the planar case, the models produce all irreducible representations.

Abstract

A Gelfand model for a semisimple algebra A over C is a complex linear representation that contains each irreducible representation of A with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of semisimple, combinatorial diagram algebras including: the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, and planar rook monoid algebras. In each case, the model representation is given by diagrams acting via "signed conjugation" on the linear span of their vertically symmetric diagrams. This representation is a generalization of the Saxl model for the symmetric group, and, in fact, our method is to use the Jones basic construction to lift the Saxl model from the symmetric group to each diagram algebra. In the case of the planar diagram algebras, our construction exactly produces the irreducible representations of the algebra.