Cuntz algebra automorphisms: transpositions

Permutative automorphisms of the Cuntz algebras ${\mathcal{O}}_n$ are in bijection with the stable permutations of $[n{]}^t$. They are also the elements of the reduced Weyl group of $Aut({\mathcal{O}}_n)$. In this paper, we characterize the stability of transpositions in $S([n{]}^3)$, and thus providing a new family (with $6$ degrees of freedom) of automorphisms of the Cuntz algebras ${\mathcal{O}}_n$ for any $n>1$.