Higher orbitals of quizzy quantum group actions

Abstract

The hyperoctahedral group HN is known to have two natural liberations: the “good” one HN+, which is the quantum symmetry group of N segments, and the “bad” one 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 HN,ON. Our results include: (1) an interpretation of the embedding O¯N⊂S2N+, as corresponding to the antisymmetric representation of ON, (2) a study of the liberations of HN, notably with the result <HN+,O¯N>=ON+, and (3) a comparison of the k-orbitals for the inclusions HN⊂HN+ and HN⊂O¯N, for k∈N small.