Algorithmic Problems in Categories of Partitions
Nicolas Faroß, Sebastian Volz
TL;DR
This work studies categories of partitions, objects arising in easy quantum groups and diagram algebras, by developing efficient data-structures and quasi-linear-time algorithms for partitions and their category operations, with an actionable OSCAR implementation. It extends partitions to colored and spatial generalizations, and demonstrates that category problems can be undecidable in general by embedding varieties of groups into hyperoctahedral categories of partitions. The paper provides rigorous algorithmic foundations (normalize, tensor, involution, composition via union-find, and spatial/color extensions) and a concrete software pipeline, enabling computational exploration of representation-theoretic structures behind diagram algebras. The undecidability results pose fundamental limits on automatic reasoning in this landscape while leaving open questions about finite generation and the full scope of both colored and spatial categorical frameworks.
Abstract
Categories of partitions are combinatorial structures arising from the representation theory of certain compact quantum groups and are linked to classical diagram algebras such as the Temperley-Lieb algebra. In this paper, we present efficient algorithms and data-structures for partitions of sets and their corresponding category operations, including a concrete implementation in the computer algebra system OSCAR. Moreover, we show that there exists a category of partitions for which the natural computational problems of deciding membership of a given partition as well as counting partitions of a given size are algorithmically undecidable.