Jellyfish partition categories

TL;DR

This paper introduces a new monoidal category based on partition diagrams and demonstrates its equivalence to a subcategory of the representation category of the alternating group, linking diagrammatic and algebraic structures.

Contribution

It constructs the monoidal category (JP)(n) and establishes its equivalence to a subcategory of (Rep(A_n)) for certain characteristics, connecting diagrammatic and group representation theories.

Findings

(JP)(n) is monoidally equivalent to a subcategory of (Rep(A_n))

The equivalence holds when the characteristic of the ground field is 0 or at least n

The category (JP)(n) generalizes partition diagrams to a new algebraic context

Abstract

For each positive integer $n$, we introduce a monoidal category $\mathcal{JP}(n)$ using a generalization of partition diagrams. When the characteristic of the ground field is either 0 or at least $n$, we show $\mathcal{JP}(n)$ is monoidally equivalent to the full subcategory of $\operatorname{Rep}(A_n)$ whose objects are tensor powers of the natural $n$-dimensional permutation representation of the alternating group $A_n$.