On the modular representation theory of the partition algebra

TL;DR

This paper determines the decomposition numbers of the partition algebra over fields of characteristic zero or large characteristic, characterizes semisimplicity, and connects block structures to quantum group actions and Weyl group combinatorics.

Contribution

It provides explicit decomposition numbers for the partition algebra and links its block structure to quantum group actions and Weyl group combinatorics.

Findings

Decomposition numbers are determined for characteristic zero or large characteristic fields.

Partition algebra blocks categorify weight spaces of type A quantum group actions.

Semisimplicity criteria are established based on the parameter values.

Abstract

We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine for which exact values of the parameter the partition algebras are semisimple over an arbitrary field. Furthermore, we show that the blocks of the partition algebra over an arbitrary field categorify weight spaces of an action of type A quantum groups on an analogue of the Fock space. In particular, we recover the block structure which was recently determined by Bowman et al. In order to do so, we use induction and restriction functors as well as analogues of Jucys-Murphy elements. The description of decomposition numbers will be in terms of combinatorics of partitions but can also be given a Lie theoretic interpretation in terms of a Weyl group of type A: A simple module $L(\mu)$ is a composition factor of a cell module $\Delta(\lambda)$ if and only if $\lambda$ and $\mu$ differ by the action of a transposition.