Higher orbitals of quizzy quantum group actions

The hyperoctahedral group $H_N$ is known to have two natural liberations: the “good” one $H_N^{+}$, which is the quantum symmetry group of $N$ segments, and the “bad” one ${\bar{O}}_N$, which is the quantum symmetry group of the $N$-hypercube. We study here this phenomenon, in the general “quizzy” framework, which covers the various liberations and twists of $H_N,O_N$. Our results include: (1) an interpretation of the embedding $\bar{O}N\subset S{2^N}^+$,ascorrespondingtotheantisymmetricrepresentationof$O_N$, (2) a study of the liberations of $H_N$, notably with the result $<H_N^{+},{\bar{O}}_N>=O_N^{+}$, and (3) a comparison of the $k$-orbitals for the inclusions $H_N\subset H_N^{+}$ and $H_N\subset {\bar{O}}_N$, for $k\in \mathbb{N}$ small.