# Permutation modules for cellularly stratified algebras

## TL;DR

This paper generalizes the concept of permutation modules from symmetric and Brauer algebras to a broader class called cellularly stratified algebras, including partition algebras, under certain conditions.

## Contribution

It introduces a new construction of permutation modules for cellularly stratified algebras, extending previous definitions to partition algebras in suitable characteristics.

## Key findings

- Permutation modules are defined for cellularly stratified algebras.
- Partition algebras satisfy the conditions for these modules when the field characteristic is large.
- The new framework broadens the applicability of permutation modules in algebra representation theory.

## Abstract

Permutation modules play an important role in the representation theory of the symmetric group. Hartmann and Paget defined permutation modules for non-degenerate Brauer algebras. We generalise their construction to a wider class of algebras, namely cellularly stratified algebras, satisfying certain conditions. Partition algebras are shown to satisfy these conditions, provided the characteristic of the underlying field is large enough. Thus we obtain a definition of permutation modules for partition algebras.