Maximality Levels of the classical permutation group in the quantum permutation group

TL;DR

This paper investigates the maximality of the classical permutation group within the quantum permutation group, focusing on categories of partitions and their linear combinations to understand potential counterexamples.

Contribution

It analyzes categories generated by non-crossing partitions and crossing-partition vectors, establishing limitations on exotic categories that could challenge maximality.

Findings

No exotic category contains a linear combination of three crossing-partition vectors.

At N=6, no exotic category contains a linear combination of 31 crossing-partition vectors distinguished from known categories.

Results support the conjecture that the classical permutation group is maximal in the quantum permutation group.

Abstract

Progress on the conjecture of Banica and Bichon that the classical permutation group is a maximal quantum subgroup of the quantum permutation group remains limited to a handful of small-parameter results. By Tannaka--Krein duality, any counterexample to this Maximality Conjecture must arise from a category strictly intermediate between the category $\mathcal{NC}$ of non-crossing partitions and the category $\mathcal{P}$ of all partitions. Any such exotic category must therefore contain a linear combination of crossing-partition vectors. The categories generated by $\mathcal{NC}$ together with some such vectors are studied, with a number of generation results. It is shown that no exotic category can contain a linear combination of three crossing-partition vectors, and, at $N=6$, there is no exotic category containing a linear combination of 31 crossing-partition vectors that is distinguished from $\mathcal{NC}$ or $\mathcal{P}$ at moments of order six.