On the maximality conjecture for quantum permutation groups

Paper (68 pages, PDF and LaTeX source) and the complete code behind its computational statements. The quantum permutation group S_N^+ contains S_N, and the maximality conjecture states that there is no quantum group strictly in between. We revisit the conjecture with three approaches: quantum subgroups as seeds, the theory of a hypothetical exotic intermediate quantum group, and the level-by-level analysis of the intermediate tensor categories, which is the main contribution: the conjecture holds up to level 7 for all N ≥ 6, level 8 is closed computationally for 8 ≤ N ≤ 37, and level 9 numerically at eight values of N. Orthogonal and projective versions are discussed as well.