Permutations, tensor products, and Cuntz algebra automorphisms

We study the reduced Weyl groups of the Cuntz algebras On from a combinatorial point of view. Their elements correspond bijectively to certain permutations of nr elements, which we call stable. We mostly focus on the case r=2 and general n. A notion of rank is introduced, which is subadditive in a suitable sense. Being of rank 1 corresponds to solving an equation which is reminiscent of the Yang-Baxter equation. Symmetries of stable permutations are also investigated, along with an immersion procedure that allows to obtain stable permutations of (n+1)2 objects starting from stable permutations of n2 objects. A complete description of stable transpositions and of stable 3-cycles of rank 1 is obtained, leading to closed formulas for their number. Other enumerative results are also presented which yield lower and upper bounds for the number of stable permutations.