Classification of charge-conserving loop braid representations

Abstract

Here a loop braid representation is a monoidal functor $F$ from the loop braid category $L$ to a suitable target category, and is N-charge-conserving if the target is the category ${\mathrm{Match}}^N$ of charge-conserving matrices (specifically ${\mathrm{Match}}^N$ is the same rank-N charge-conserving monoidal subcategory of the monoidal category $\mathrm{Mat}$ used to classify braid representations in [27]) with $F$ strict, and surjective on $N$, the object monoid. We classify and construct all such representations. In particular we prove that representations at given N fall into varieties indexed by a set in bijection with the set of pairs of plane partitions of total degree N.