Permutative automorphisms of the Cuntz algebras: quadratic cycles, an involution and a box product
This work can be read as a continuation of \textit{F. Brenti} and \textit{R. Conti} [Adv. Math. 381, Article ID 107590, 61 p. (2021; Zbl 1459.05347)] where the study of stable permutations on \(n^r\) elements is undertaken. These stable permutations are graded by the notion of rank. The ultimate goal is to gain an understanding of the Weyl groups of the Cuntz algebras. The more immediate desire is to characterize and enumerate the stable permutations of a given rank. In this work, the authors achieve this for transpositions of rank one, for 3-cycles of rank at most 2, and for a restricted class of 4-cycles. On this basis, they conjecture that a stable \(r\)-cycle has rank at most \(r - 1\). They also introduce two rank-preserving algebraic operations on the set of stable permutations: (i) an involution that preserves cycle structure and (ii) a binary operation that associates to permutations on \(n^r\) and \(m^r\) objects, respectively, a permutation on \((nm)^r\) objects.
scientific article; zbMATH DE number 7628621
scientific article