Restricting cell modules of partition algebras

TL;DR

This paper investigates how cell modules of partition algebras restrict to symmetric groups and identifies conditions under which these restrictions have Specht filtrations, enhancing understanding of their cellular structure.

Contribution

It provides new criteria for when restricted cell modules of partition algebras admit Specht filtrations, linking cellular structures across algebraic levels.

Findings

Identifies conditions for Specht filtrations in restricted modules

Clarifies the cellular structure relationship between partition and symmetric group algebras

Advances understanding of module restrictions in cellular algebras

Abstract

The restriction of a (dual) Specht module to a smaller symmetric group has a filtration by (dual) Specht modules of this smaller group. In the cellular structure of the group algebra of the symmetric group, the cell modules are exactly the (dual) Specht modules. The partition algebra is a cellular algebra containing the group algebra of the symmetric group. In this article, we study the structure of the restriction of a cell module to the group algebra of a symmetric group (with smaller index) and find conditions for the restriction to possess a (dual) Specht filtration.