Representations of Partition Categories
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Date
2023-07-06
Authors
Vargas, Max
Journal Title
Journal ISSN
Volume Title
Publisher
University of Oregon
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 constructsome special projective functors, then apply these functors to give self-contained proofs of results of Comes and
Ostrik on blocks of Deligne’s category $\REP(S_t)$. We then study a restriction functor $\REP(S_t)\to\REP(S_{t-1})$ and prove a conjecture of Comes and Ostrik involving this functor. Finally, we use the restriction functor to verify
a criterion of Benson, Etingof, and Ostrik, thereby identifying the abelian envelope of $\REP(S_t)$ with the Ringel dual of the category of locally finite-dimensional $\Par_t$-modules.
This dissertation includes published co-authored material.