On the representation theory of partition (easy) quantum groups

TL;DR

This paper demonstrates that the representation theory of easy quantum groups, including fusion rules, can be fully understood through their combinatorial partition data, unifying several key quantum groups.

Contribution

It shows that the fusion rules of easy quantum groups are determined by their combinatorial partition structures, providing a unified approach for multiple quantum groups.

Findings

Fusion rules are fully determined by partition combinatorics.

Unified framework for quantum permutation, free orthogonal, and hyperoctahedral quantum groups.

Representation theory can be derived from combinatorial data.

Abstract

Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S.L. Woronowicz. In the case of easy quantum groups, the intertwiner spaces are given by the combinatorics of partitions, see the inital work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group $S_N^+$, the free orthogonal quantum group $O_N^+$ as well as the hyperoctahedral quantum group $H_N^+$.