Classical actions of quantum permutation groups

TL;DR

This paper classifies all actions of quantum permutation groups on classical spaces, showing the uniqueness of the ergodic action and extending results to related easy quantum groups.

Contribution

It explicitly characterizes all actions of quantum permutation groups on classical spaces and identifies the unique non-trivial ergodic action, extending to easy quantum groups.

Findings

The defining action is the only non-trivial ergodic action.

Explicit description of all actions of quantum permutation groups.

Extension of results to easy quantum groups associated with non-crossing partitions.

Abstract

We describe explicitly all actions of the quantum permutation groups on classical compact spaces. In particular, we show that the defining action is the only non-trivial ergodic one. We then extend these results to all easy quantum groups associated to non-crossing partitions.