{"title":"最大类群族的表示zeta函数：非特殊素数","authors":"Shannon Ezzat","doi":"10.1515/jgth-2022-0213","DOIUrl":null,"url":null,"abstract":"We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>M</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2022-0213_ineq_0001.png\" /> <jats:tex-math>M_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with maximal nilpotency class. For representation dimensions coprime to all primes <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2022-0213_ineq_0002.png\" /> <jats:tex-math>p<n</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we construct all irreducible representations of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>M</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2022-0213_ineq_0001.png\" /> <jats:tex-math>M_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> by defining a standard form for the matrices of these representations.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Representation zeta function of a family of maximal class groups: Non-exceptional primes\",\"authors\":\"Shannon Ezzat\",\"doi\":\"10.1515/jgth-2022-0213\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>M</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2022-0213_ineq_0001.png\\\" /> <jats:tex-math>M_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with maximal nilpotency class. For representation dimensions coprime to all primes <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2022-0213_ineq_0002.png\\\" /> <jats:tex-math>p<n</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we construct all irreducible representations of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>M</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2022-0213_ineq_0001.png\\\" /> <jats:tex-math>M_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> by defining a standard form for the matrices of these representations.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-03-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/jgth-2022-0213\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2022-0213","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们用一种构造方法来获得具有最大无幂级数的有限生成无幂群 M n M_{n} 族的所有𝑝局部表示 zeta 函数。对于与所有素数 p < n p<n 共价的表示维数，我们通过定义这些表示的矩阵的标准形式来构造 M n M_{n} 的所有不可还原表示。
Representation zeta function of a family of maximal class groups: Non-exceptional primes
We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups MnM_{n} with maximal nilpotency class. For representation dimensions coprime to all primes p<np<n, we construct all irreducible representations of MnM_{n} by defining a standard form for the matrices of these representations.
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