Decomposing Frobenius Heisenberg categories

TL;DR

This paper provides two new diagrammatic presentations of Frobenius Heisenberg categories when the underlying Frobenius algebra decomposes, and establishes an equivalence with a partial Karoubi envelope of the original category.

Contribution

It introduces alternative planar diagram presentations for Frobenius Heisenberg categories based on algebra decomposition and proves their categorical equivalence with tensor products via a partial Karoubi envelope.

Findings

Two alternate diagrammatic presentations of $ ext{Heis}_{F,k}$.

Equivalence of tensor products with a subcategory of the Karoubi envelope.

Clarification of morphism spaces via colored strands and tokens.

Abstract

We give two alternate presentations of the Frobenius Heisenberg category, $\mathcal{Heis}_{F,k}$, defined by Savage, when the Frobenius algebra $F=F_1\oplus\dotsb\oplus F_n$ decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of $\mathcal{Heis}_{F,k}$ are given in terms of planar diagrams consisting of strands "colored" by integers $i=1,\dotsc,n$, where a strand of color $i$ carries tokens labelled by elements of $F_i.$ In addition, we prove that when $F$ decomposes this way, the tensor product of Frobenius Heisenberg categories, $\mathcal{Heis}_{F_1,k}\otimes\dotsb\otimes\mathcal{Heis}_{F_n,k},$ is equivalent to a certain subcategory of the Karoubi envelope of $\mathcal{Heis}_{F,k}$ that we call the $\textit{partial}$ Karoubi envelope of $\mathcal{Heis}_{F,k}$.