A new approach to the representation theory of the partition 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 ReppS t q., . . . of Jucys-Murphy elements in these partition algebras, which were studied further by Enyang [E1, E2]. Enyang worked out a recursive definition for the Jucys-Murphy elements and used them to construct an analog of Young's orthogonal form for the irreducible P n ptq-modules. His definition involves a complicated five term recurrence relation, making the Jucys-Murphy elements for partition algebras considerably harder to work with than the classical Jucys-Murphy elements of the symmetric groups. Recently, Creedon [Cr] has revisited Enyang's work,

Defining an Affine Partition Algebra

“…In this last section we relate our affine partition algebra to the work of J. Brundan and M. Vargas in [2] and prove a new result on their category. We start by recalling the definition of their affine partition category APar as a subcategory of Heis generated by certain objects and morphisms, and of their affine partition algebra AP k , which is an endomorphism algebra within APar.…”

“…J. Brundan and M. Vargas recently defined in [2] an affine partition category APar as a monoidal subcategory of the Heisenberg category introduced by Khovanov in [12] generated by certain objects and morphisms. This was based on the observation made by S. Likeng and A.…”

“…For α 1 , as was done in the previous case we may pull one of the crossings outside of the curl, and thus decrease the number of intersections in R by one, and hence α 1 = 0 by induction. For α 2 the curve containing h 1 and the original left curl have been turned into the two new curves h (1) 1 and h (2) 1 . Note the original left curl is no longer present, but regardless of how the original curve containing h 1 intersected the curl, at least one of the new curves h (1) 1 and h (2) 1 must form a new, smaller, left curl.…”

“…For α 2 the curve containing h 1 and the original left curl have been turned into the two new curves h (1) 1 and h (2) 1 . Note the original left curl is no longer present, but regardless of how the original curve containing h 1 intersected the curl, at least one of the new curves h (1) 1 and h (2) 1 must form a new, smaller, left curl. The region bounded by this new curl is a subregion of R containing strictly less number of intersections.…”

“…While writing this paper, J. Brundan and M. Vargas produced a preprint [2] defining an affine partition category APar as a monoidal subcategory of the Heisenberg category generated by some objects and morphisms. Taking an endomorphism algebra in their category gives an alternative definition of an affine partition algebra, which they denote by AP k .…”