Cellularity and the Jones basic construction

TL;DR

This paper develops a unified framework for establishing cellularity in a class of algebras related to the Jones basic construction, simplifying proofs and providing new insights into their structure.

Contribution

It introduces a general approach to cellularity for algebras connected to the Jones basic construction, using path-labeled bases and compatibility with module operations.

Findings

Cellularity of various algebras proved uniformly

Cellular bases labeled by paths on branching diagrams

Compatibility of cellular structures with restriction and induction

Abstract

We establish a framework for cellularity of algebras related to the Jones basic construction. Our framework allows a uniform proof of cellularity of Brauer algebras, ordinary and cyclotomic BMW algebras, walled Brauer algebras, partition algebras, and others. Our cellular bases are labeled by paths on certain branching diagrams rather than by tangles. Moreover, for the class of algebras that we study, we show that the cellular structures are compatible with restriction and induction of modules. Applied to cyclotomic BMW algebras, our method allows a new a shorter proof of the finite spanning result and isomorphism with cyclotomic Kauffman tangle algebras.