The hyperoctahedral quantum group

TL;DR

This paper identifies the quantum symmetry group of the hypercube as a specific q-deformation of the orthogonal group at q=-1, and introduces a new quantum group for a graph of segments that extends existing quantum symmetry groups.

Contribution

It establishes the quantum symmetry group of the hypercube as a q-deformation of O_n at q=-1 and introduces a new quantum group H_n^+ for a graph of segments, expanding the series of known quantum symmetry groups.

Findings

Quantum symmetry group of the hypercube is a q-deformation of O_n at q=-1.

The quantum symmetry group of a graph of n segments is a new free quantum group H_n^+.

H_n^+ extends Wang's series of quantum symmetry groups.

Abstract

We consider the hypercube in $\mathbb R^n$, and show that its quantum symmetry group is a $q$-deformation of $O_n$ at $q=-1$. Then we consider the graph formed by $n$ segments, and show that its quantum symmetry group is free in some natural sense. This latter quantum group, denoted $H_n^+$, enlarges Wang's series $S_n^+,O_n^+,U_n^+$.