Permutative automorphisms of the Cuntz algebras: Quadratic cycles, an involution and a box product

Permutative automorphisms of the Cuntz algebra On are in bijection with a class of permutations of nk elements, that are called stable, and are further partitioned by rank. In this work we mainly focus on stable cycles in the quadratic case (i.e., k=2). More precisely, in such a quadratic case we provide a characterization of the stable cycles of rank one (so proving Conjecture 12.1 in [3]), exhibit a closed formula for the number of stable r-cycles of rank one (valid for all n and r), and characterize and enumerate the stable 3-cycles of any given rank. We also show that the set of stable permutations is equipped with a natural involution that preserves the cycle-type and the rank, and that there is a map that associates to two stable permutations of nk and mk elements, respectively, a stable permutation of (nm)k elements.