Cuntz algebra automorphisms: Graphs and stability of permutations

Brenti, Francesco; Conti, Roberto; Nenashev, Gleb

doi:10.1090/tran/9159

Cuntz algebra automorphisms: Graphs and stability of permutations
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by Francesco Brenti, Roberto Conti and Gleb Nenashev;

Trans. Amer. Math. Soc. 377 (2024), 8433-8476

DOI: https://doi.org/10.1090/tran/9159

Published electronically: September 17, 2024

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Abstract:

We characterize the permutative automorphisms of the Cuntz algebra $\mathcal {O}_n$ (namely, stable permutations) in terms of two sequences of graphs that we associate to any permutation of a discrete hypercube $[n]^t$. As applications we show that in the limit of large $t$ (resp. $n$) almost all permutations are not stable, thus proving Conj. 12.5 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], characterize (and enumerate) stable quadratic $4$ and $5$-cycles, as well as a notable class of stable quadratic $r$-cycles, i.e. those admitting a compatible cyclic factorization by stable transpositions. Some of our results use new combinatorial concepts that may be of independent interest.

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