Decomposing Frobenius Heisenberg categories

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 $\mathcal{Heis}{F,k}$.