Algorithmic Problems in Categories of Partitions

TL;DR

This paper explores the computational aspects of categories of partitions, presenting efficient algorithms and data structures, and demonstrating undecidability in certain cases related to their membership and counting problems.

Contribution

It introduces new algorithms and data structures for partitions, implements them in OSCAR, and proves undecidability results for specific category problems.

Findings

Efficient algorithms for partition operations

Implementation in the OSCAR system

Existence of undecidable membership and counting problems

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.