Tracing the orbitals of the quantum permutation group

TL;DR

This paper demonstrates that the quantum permutation group possesses free orbitals using a noncommutative matrix model, and explores the structure of intermediate quantum subgroups with explicit Haar state formulas.

Contribution

It introduces a noncommutative flat matrix model to establish the existence of free orbitals in quantum permutation groups and characterizes intermediate subgroups.

Findings

Quantum permutation group has free orbitals.

Intermediate quantum subgroups must have free three-orbitals.

Explicit Haar state formulas for degree four monomials.

Abstract

Using a suitably noncommutative flat matrix model, it is shown that the quantum permutation group has free orbitals: that is, a monomial in the generators of the algebra of functions can be zero for trivial reasons only. It is shown that any strictly intermediate quantum subgroup between the classical and quantum permutation groups must have free three-orbitals. This is used to give explicit formulae for the Haar state on degree four monomials that hold for such intermediate quantum subgroups as well as the quantum permutation group itself.