A note on the Cuntz algebra automorphisms

Permutative automorphisms of the Cuntz algebras ${\mathcal{O}}_n$ are in bijection with the stable permutations of $[n{]}^k$. They are also the elements of the restricted Weyl group of $Aut({\mathcal{O}}_n)$. In this note, we characterize a class of stable involutions of $[n{]}^2$. More precisely, we prove Conjecture 12.2 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], and thus providing a new family (with $6$ degrees of freedom) of automorphisms of the Cuntz algebras ${\mathcal{O}}_n$ for any $n>1$.