{"title":"Cuntz algebra automorphisms: Graphs and stability of permutations","authors":"Francesco Brenti, Roberto Conti, Gleb Nenashev","doi":"10.1090/tran/9159","DOIUrl":null,"url":null,"abstract":"<p>We characterize the permutative automorphisms of the Cuntz algebra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O Subscript n\"> <mml:semantics> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"script\">O</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\mathcal {O}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (namely, stable permutations) in terms of two sequences of graphs that we associate to any permutation of a discrete hypercube <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket n right-bracket Superscript t\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mi>n</mml:mi> <mml:msup> <mml:mo stretchy=\"false\">]</mml:mo> <mml:mi>t</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">[n]^t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As applications we show that in the limit of large <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=\"application/x-tex\">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp. <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) 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 <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"4\"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding=\"application/x-tex\">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"5\"> <mml:semantics> <mml:mn>5</mml:mn> <mml:annotation encoding=\"application/x-tex\">5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cycles, as well as a notable class of stable quadratic <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r\"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding=\"application/x-tex\">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-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.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9159","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We characterize the permutative automorphisms of the Cuntz algebra On\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[n]^t. As applications we show that in the limit of large tt (resp. nn) 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 44 and 55-cycles, as well as a notable class of stable quadratic rr-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.
我们用两个图序列来描述 Cuntz 代数 O n \mathcal {O}_n（即稳定的置换）的置换自动形，这两个图序列与离散超立方体 [ n ] t [n]^t 的任何置换相关联。作为应用，我们证明了在大 t t（或 n n ）的极限中，几乎所有的排列都不稳定，从而证明了 Brenti 和 N. N. 的 Conj.12.5 [Adv. Math. 381 (2021), p. 60]，描述（并枚举）了稳定的二次 4 4 循环和五 5 循环，以及一类值得注意的稳定的二次 r r 循环，即那些通过稳定的转置实现兼容循环因式分解的循环。我们的一些结果使用了新的组合概念，可能会引起独立的兴趣。
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