Higher orbitals of quizzy quantum group actions

TL;DR

This paper explores the higher orbitals of quantum group actions related to hyperoctahedral groups, revealing new embeddings, liberation results, and orbital comparisons within the framework of quizzy quantum groups.

Contribution

It introduces a unified framework for studying liberations and twists of hyperoctahedral and orthogonal quantum groups, providing new insights into their embeddings and orbitals.

Findings

Embedding of ar{O}_N into S_{2^N}^+ interpreted via antisymmetric representation

Liberation of hyperoctahedral groups with ig<H_N^+, ar{O}_Nig>=O_N^+

Comparison of k-orbitals for H_Nig H_N^+ and H_Nig ar{O}_N for small k

Abstract

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 $\bar{O}_N\subset S_{2^N}^+$, as corresponding to the antisymmetric representation of $O_N$, (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.