Remark on Hopf images in quantum permutation groups $S_n^+$

TL;DR

This paper investigates the structure and automorphisms of the quantum permutation group on four points, answering a specific faithfulness question and classifying certain subgroups, revealing limitations of existing criteria.

Contribution

It provides a positive answer to an inner faithfulness question for $n=4, k=2$, classifies automorphisms of $S_4^+$, and characterizes embeddings of $O_{-1}(2)$ into $S_4^+$, showing some criteria are not reversible.

Findings

Confirmed inner faithfulness for $n=4, k=2$

Classified automorphisms of $S_4^+$

Described embeddings of $O_{-1}(2)$ into $S_4^+$

Abstract

Motivated by a question of A.~Skalski and P.M.~So{\l}tan about inner faithfulness of the S.~Curran's map, we revisit the results and techniques of T.~Banica and J.~Bichon's Crelle paper and study some group-theoretic properties of the quantum permutation group on $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4, k=2$, but also to classify the automorphisms of $S_4^+$, describe all the embeddings $O_{-1}(2)\subset S_4^+$ and show that all the copies of $O_{-1}(2)$ inside $S_4^+$ are conjugate. We then use these results to show that the criterion we applied to answer the aforementioned question does not admit converse.