张量拟随机群

Proceedings of the American Mathematical Society, Series B Pub Date : 2021-03-19 DOI:10.1090/bproc/86

Mark Sellke

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Comput. 17 (2008), pp. 363–387] elegantly characterized the finite groups <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in which <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A 1 upper A 2 upper A 3 equals upper G\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo>=</mml:mo>\n <mml:mi>G</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">A_1A_2A_3=G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for any positive density subsets <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A 1 comma upper A 2 comma upper A 3\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">A_1,A_2,A_3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. This property, <italic>quasi-randomness</italic>, holds if and only if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> does not admit a nontrivial irreducible representation of constant dimension. We present a dual characterization of <italic>tensor quasi-random</italic> groups in which multiplication of subsets is replaced by tensor product of representations.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"35 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Tensor quasi-random groups\",\"authors\":\"Mark Sellke\",\"doi\":\"10.1090/bproc/86\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Gowers [Combin. Probab. Comput. 17 (2008), pp. 363–387] elegantly characterized the finite groups <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G\\\">\\n <mml:semantics>\\n <mml:mi>G</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in which <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A 1 upper A 2 upper A 3 equals upper G\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mn>1</mml:mn>\\n </mml:msub>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mn>3</mml:mn>\\n </mml:msub>\\n <mml:mo>=</mml:mo>\\n <mml:mi>G</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A_1A_2A_3=G</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> for any positive density subsets <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A 1 comma upper A 2 comma upper A 3\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mn>1</mml:mn>\\n </mml:msub>\\n <mml:mo>,</mml:mo>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:mo>,</mml:mo>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mn>3</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A_1,A_2,A_3</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. 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引用次数: 1

摘要

(Combin高尔。Probab。计算，17 (2008)，pp. 363-387]对任意正密度子集a1, a2, a3a_1,A_2,A_3的A 1,A 2,A 3=G的有限群G G进行了优雅的刻画。当且仅当G G不承认常维的非平凡不可约表示时，拟随机性这一性质成立。我们给出了张量拟随机群的对偶刻画，其中子集的乘法用表示的张量积代替。

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Tensor quasi-random groups

Gowers [Combin. Probab. Comput. 17 (2008), pp. 363–387] elegantly characterized the finite groups G G in which A 1 A 2 A 3 = G A_1A_2A_3=G for any positive density subsets A 1 , A 2 , A 3 A_1,A_2,A_3 . This property, quasi-randomness, holds if and only if G G does not admit a nontrivial irreducible representation of constant dimension. We present a dual characterization of tensor quasi-random groups in which multiplication of subsets is replaced by tensor product of representations.

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Proceedings of the American Mathematical Society, Series B

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1.60

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0.00%

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