{"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":50188,"journal":{"name":"Journal of Group Theory","volume":"67 1","pages":""},"PeriodicalIF":0.4000,"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\":50188,\"journal\":{\"name\":\"Journal of Group Theory\",\"volume\":\"67 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-03-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Group Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/jgth-2022-0213\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Group Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2022-0213","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","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.
期刊介绍：
The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.
Topics:
Group Theory-
Representation Theory of Groups-
Computational Aspects of Group Theory-
Combinatorics and Graph Theory-
Algebra and Number Theory
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