The hyperoctahedral quantum group

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 $S_n^+,O_n^+,U_n^+$.