Classification of charge-conserving loop braid representations

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 MatchN of charge-conserving matrices (specifically MatchN is the same rank-N charge-conserving monoidal subcategory of the monoidal category 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.