A new approach to the representation theory of the partition category

TL;DR

This paper introduces an innovative approach to the representation theory of the partition category using a new graphical monoidal category and Jucys-Murphy elements, providing simplified proofs of existing results.

Contribution

It develops the affine partition category as a subcategory of Khovanov's Heisenberg category and utilizes Jucys-Murphy elements to construct projective functors, advancing the understanding of Deligne's category.

Findings

Defined the affine partition category within Khovanov's Heisenberg category

Constructed projective functors using Jucys-Murphy elements

Provided new proofs of results on blocks of Deligne's category

Abstract

We explain a new approach to the representation theory of the partition category based on a reformulation of the definition of the Jucys-Murphy elements introduced originally by Halverson and Ram and developed further by Enyang. Our reformulation involves a new graphical monoidal category, the affine partition category, which is defined here as a certain monoidal subcategory of Khovanov's Heisenberg category. We use the Jucys-Murphy elements to construct some special projective functors, then apply these functors to give self-contained proofs of results of Comes and Ostrik on blocks of Deligne's category Rep(S_t).