On the modular representation theory of the partition algebra

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 the quantum groups Uqp x slpq and Uqpsl8q 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 Lpµq is a composition factor of a cell module ∆pλq if and only if λ and µ differ by the action of a transposition.

On the decomposition matrix of the partition algebra in positive characteristic

“…When writing this paper, it was brought to the author's attention that the decomposition numbers of the partition algebra P k n (δ) for n < p were obtained independently, and by different methods, by A. Shalile [Sha14].…”