Cellularity of a Larger Class of Diagram Algebras

TL;DR

This paper demonstrates the cellularity of certain diagram algebras, including $bZ_2$-relation, signed partition, and partition algebras, and explores their modular representations.

Contribution

It introduces a new realization of these algebras as tabular algebras and establishes their cellularity, extending existing algebraic frameworks.

Findings

Proves cellularity of the algebra of $bZ_2$-relations, signed partition, and partition algebras.

Provides modular representation theory for these algebras.

Realizes these algebras as tabular algebras.

Abstract

In this paper, we realize the algebra of $\mathbb{Z}_2$-relations, signed partition algebras and partition algebras as tabular algebras and prove the cellularity of these algebras using the method of \cite{GM1}. Using the results of Graham and Lehrer in \cite{GL}, we give the modular representations of the algebra of $\mathbb{Z}_2$-relations, signed partition algebras and partition algebras.