Commutators of signed $n$-cycles

Abstract

We show that for $n \geq 6$ each element of the commutator subgroup in the symmetric group $\mathfrak{S}_n$ resp. in the signed symmetric group $(\mathbb{Z}/2\mathbb{Z})^n\rtimes\mathfrak{S}_n$ is the commutator of two $n$-cycles resp. the commutator of two $n$-cycles with a negative sign product; with one exception. If $n \equiv 2$ $\mathrm{mod}~4$, the element $-\mathrm{id}$ of $(\mathbb{Z}/2\mathbb{Z})^n\rtimes\mathfrak{S}_n$ is not such a commutator. In the language of Coxeter groups, this yields a description of commutators of Coxeter elements in types $A$ and $B$.