Cell structures on the blob algebra

TL;DR

This paper investigates the cellular structures of the blob algebra derived from conjectures on the Hecke algebra of type B, focusing on the case r=0, and establishes a natural cell structure via restriction to bipartitions.

Contribution

It demonstrates that the r=0 case of certain conjectures induces a natural cellular structure on the blob algebra through restriction to bipartitions.

Findings

Established the natural cell structure on the blob algebra from the conjectures.

Connected the cellular structures of the Hecke algebra of type B to the blob algebra.

Provided a specific case (r=0) where the conjectures apply.

Abstract

We consider the $ r = 0 $ case of the conjectures by Bonnaf\'e, Geck, Iancu and Lam on cellular structures on the Hecke algebra of type $ B $. We show that this case induces the natural cell structure on the blob algebra $ b_n $ by restriction to one-line bipartitions.