# Branching rules for cell modules on a tower of the partition algebras

## TL;DR

This paper develops a diagrammatic method to analyze the branching rules of cell modules in a tower of partition algebras, enabling calculation of structure constants in the Grothendieck ring.

## Contribution

It introduces a novel diagrammatic approach to study branching rules and structure constants in partition algebras with non-zero parameters.

## Key findings

- Derived explicit branching rules for cell modules
- Calculated structure constants of the Grothendieck ring
- Enhanced understanding of the algebraic structure of partition algebras

## Abstract

Partition algebras with non-zero parameters are cellularly stratified and thus have the features of both cellular algebras and stratified algebras. Also, partition algebras form a tower of algebras. In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a tower of the partition algebras. This also allows us to calculate the structure constants of the Grothendieck ring of the tower.