On the partition approach to Schur-Weyl duality and free quantum groups

TL;DR

This paper introduces a unified framework for classical and quantum groups based on partitions, classifies free fusion semirings, and explores their structure and duality properties, advancing understanding of free quantum groups.

Contribution

It provides a general definition for groups determined by partitions, classifies free fusion semirings, and proves their realization within this framework, addressing open questions.

Findings

Classification of free fusion semirings

Realization of all free fusion semirings through the new construction

Decomposition results for free quantum groups

Abstract

We give a general definition of classical and quantum groups whose representation theory is "determined by partitions" and study their structure. This encompasses many examples of classical groups for which Schur-Weyl duality is described with diagram algebras as well as generalizations of P. Deligne's interpolated categories of representations. Our setting is inspired by many previous works on easy quantum groups and appears to be well-suited to the study of free fusion semirings. We classify free fusion semirings and prove that they can always be realized through our construction, thus solving several open questions. This suggests a general decomposition result for free quantum groups which in turn gives information on the compact groups whose Schur-Weyl duality is implemented by partitions. The paper also contains an appendix by A. Chirvasitu proving simplicity results for the reduced C*-algebras of some free quantum groups.