The symmetric group S_n permutes n elements classically. Its quantum analogue — the quantum permutation group S_n⁺ — allows the permutation matrix entries to be projections in a C*-algebra rather than 0-1 scalars. The classical group embeds in the quantum one: every classical permutation is a quantum permutation, but not conversely.
Banica and Bichon conjectured that S_n is maximal in S_n⁺ as a quantum subgroup — there is no quantum group strictly between them. The paper on maximality levels (arXiv: 2603.21759) investigates this conjecture through Tannaka–Krein duality.
By duality, a quantum subgroup between S_n and S_n⁺ corresponds to a partition category strictly between the non-crossing partitions NC and all partitions P. Any such “exotic” category must contain a linear combination of crossing-partition vectors — elements that cross when drawn as string diagrams.
The paper proves that no exotic category can contain a linear combination involving three or fewer crossing partitions. At N = 6, no exotic category exists among linear combinations of 31 specific crossing partitions that are distinguished from NC or P at sixth-order moments. The conjecture holds in every case checked, and each case eliminates more of the space where counterexamples could live.
The through-claim: the gap between classical and quantum is categorical, and the category has no room for intermediates. The non-crossing partitions (classical) and all partitions (quantum) appear to be adjacent in the lattice of partition categories. The proof strategy is elimination: showing that any element you add to NC immediately generates all of P. There is no halfway quantum.
2603.21759. Quantum groups / permutation groups / Tannaka–Krein duality / partition categories / non-crossing partitions.