三个元环群的群联方案的特尔维利格代数

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2024-04-25 DOI:10.1002/jcd.21941

Jing Yang, Xiaoqian Zhang, Lihua Feng

{"title":"三个元环群的群联方案的特尔维利格代数","authors":"Jing Yang, Xiaoqian Zhang, Lihua Feng","doi":"10.1002/jcd.21941","DOIUrl":null,"url":null,"abstract":"For any finite group <math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math>, the Terwilliger algebra <math>\n \n <mrow>\n <mi>T</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> of the group association scheme satisfies the following inclusions: <math>\n \n <mrow>\n <msub>\n <mi>T</mi>\n \n <mn>0</mn>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>⊆</mo>\n \n <mi>T</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>⊆</mo>\n \n <mover>\n <mi>T</mi>\n \n <mo>˜</mo>\n </mover>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>, where <math>\n \n <mrow>\n <msub>\n <mi>T</mi>\n \n <mn>0</mn>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> is a specific vector space and <math>\n \n <mrow>\n <mover>\n <mi>T</mi>\n \n <mo>˜</mo>\n </mover>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> is the centralizer algebra of the permutation representation of <math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> induced by the action of conjugation. The group <math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> is said to be triply transitive if <math>\n \n <mrow>\n <msub>\n <mi>T</mi>\n \n <mn>0</mn>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mover>\n <mi>T</mi>\n \n <mo>˜</mo>\n </mover>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>. In this paper, we determine the dimensions of <math>\n \n <mrow>\n <msub>\n <mi>T</mi>\n \n <mn>0</mn>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> and <math>\n \n <mrow>\n <mover>\n <mi>T</mi>\n \n <mo>˜</mo>\n </mover>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> for <math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> being <math>\n \n <mrow>\n <msub>\n <mi>T</mi>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n </mrow>\n </msub>\n \n <mo>=</mo>\n \n <mrow>\n <mo>〈</mo>\n \n <mrow>\n <mi>a</mi>\n \n <mo>,</mo>\n \n <mi>b</mi>\n \n <mo>∣</mo>\n \n <msup>\n <mi>a</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>n</mi>\n </mrow>\n </msup>\n \n <mo>=</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <msup>\n <mi>a</mi>\n \n <mi>n</mi>\n </msup>\n \n <mo>=</mo>\n \n <msup>\n <mi>b</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>,</mo>\n \n <mi>b</mi>\n \n <mi>a</mi>\n \n <msup>\n <mi>b</mi>\n \n <mrow>\n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n </msup>\n \n <mo>=</mo>\n \n <msup>\n <mi>a</mi>\n \n <mi>k</mi>\n </msup>\n </mrow>\n \n <mo>〉</mo>\n </mrow>\n </mrow></math>, <math>\n \n <mrow>\n <msub>\n <mi>C</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>⋊</mo>\n \n <msub>\n <mi>C</mi>\n \n <mi>p</mi>\n </msub>\n </mrow></math> and <math>\n \n <mrow>\n <msub>\n <mi>C</mi>\n \n <mi>p</mi>\n </msub>\n \n <mo>⋊</mo>\n \n <msub>\n <mi>C</mi>\n \n <mi>n</mi>\n </msub>\n </mrow></math>, and show that <math>\n \n <mrow>\n <msub>\n <mi>T</mi>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>C</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>⋊</mo>\n \n <msub>\n <mi>C</mi>\n \n <mn>2</mn>\n </msub>\n </mrow></math> and <math>\n \n <mrow>\n <msub>\n <mi>C</mi>\n \n <mn>3</mn>\n </msub>\n \n <mo>⋊</mo>\n \n <msub>\n <mi>C</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>n</mi>\n </mrow>\n </msub>\n </mrow></math> are triply transitive. Additionally, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of <math>\n \n <mrow>\n <msub>\n <mi>T</mi>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n </mrow>\n </msub>\n </mrow></math>, <math>\n \n <mrow>\n <msub>\n <mi>C</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>⋊</mo>\n \n <msub>\n <mi>C</mi>\n \n <mi>p</mi>\n </msub>\n </mrow></math> and <math>\n \n <mrow>\n <msub>\n <mi>C</mi>\n \n <mi>p</mi>\n </msub>\n \n <mo>⋊</mo>\n \n <msub>\n <mi>C</mi>\n \n <mi>n</mi>\n </msub>\n </mrow></math> when they are triply transitive.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"438-463"},"PeriodicalIF":0.5000,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Terwilliger algebras of the group association schemes of three metacyclic groups\",\"authors\":\"Jing Yang, Xiaoqian Zhang, Lihua Feng\",\"doi\":\"10.1002/jcd.21941\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For any finite group <math>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow></math>, the Terwilliger algebra <math>\\n \\n <mrow>\\n <mi>T</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow></math> of the group association scheme satisfies the following inclusions: <math>\\n \\n <mrow>\\n <msub>\\n <mi>T</mi>\\n \\n <mn>0</mn>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>⊆</mo>\\n \\n <mi>T</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>⊆</mo>\\n \\n <mover>\\n <mi>T</mi>\\n \\n <mo>˜</mo>\\n </mover>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow></math>, where <math>\\n \\n <mrow>\\n <msub>\\n <mi>T</mi>\\n \\n <mn>0</mn>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow></math> is a specific vector space and <math>\\n \\n <mrow>\\n <mover>\\n <mi>T</mi>\\n \\n <mo>˜</mo>\\n </mover>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow></math> is the centralizer algebra of the permutation representation of <math>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow></math> induced by the action of conjugation. The group <math>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow></math> is said to be triply transitive if <math>\\n \\n <mrow>\\n <msub>\\n <mi>T</mi>\\n \\n <mn>0</mn>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>=</mo>\\n \\n <mover>\\n <mi>T</mi>\\n \\n <mo>˜</mo>\\n </mover>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow></math>. In this paper, we determine the dimensions of <math>\\n \\n <mrow>\\n <msub>\\n <mi>T</mi>\\n \\n <mn>0</mn>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow></math> and <math>\\n \\n <mrow>\\n <mover>\\n <mi>T</mi>\\n \\n <mo>˜</mo>\\n </mover>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow></math> for <math>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow></math> being <math>\\n \\n <mrow>\\n <msub>\\n <mi>T</mi>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>k</mi>\\n </mrow>\\n </msub>\\n \\n <mo>=</mo>\\n \\n <mrow>\\n <mo>〈</mo>\\n \\n <mrow>\\n <mi>a</mi>\\n \\n <mo>,</mo>\\n \\n <mi>b</mi>\\n \\n <mo>∣</mo>\\n \\n <msup>\\n <mi>a</mi>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>n</mi>\\n </mrow>\\n </msup>\\n \\n <mo>=</mo>\\n \\n <mn>1</mn>\\n \\n <mo>,</mo>\\n \\n <msup>\\n <mi>a</mi>\\n \\n <mi>n</mi>\\n </msup>\\n \\n <mo>=</mo>\\n \\n <msup>\\n <mi>b</mi>\\n \\n <mn>2</mn>\\n </msup>\\n \\n <mo>,</mo>\\n \\n <mi>b</mi>\\n \\n <mi>a</mi>\\n \\n <msup>\\n <mi>b</mi>\\n \\n <mrow>\\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msup>\\n \\n <mo>=</mo>\\n \\n <msup>\\n <mi>a</mi>\\n \\n <mi>k</mi>\\n </msup>\\n </mrow>\\n \\n <mo>〉</mo>\\n </mrow>\\n </mrow></math>, <math>\\n \\n <mrow>\\n <msub>\\n <mi>C</mi>\\n \\n <mi>n</mi>\\n </msub>\\n \\n <mo>⋊</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mi>p</mi>\\n </msub>\\n </mrow></math> and <math>\\n \\n <mrow>\\n <msub>\\n <mi>C</mi>\\n \\n <mi>p</mi>\\n </msub>\\n \\n <mo>⋊</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mi>n</mi>\\n </msub>\\n </mrow></math>, and show that <math>\\n \\n <mrow>\\n <msub>\\n <mi>T</mi>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>k</mi>\\n </mrow>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mi>n</mi>\\n </msub>\\n \\n <mo>⋊</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mn>2</mn>\\n </msub>\\n </mrow></math> and <math>\\n \\n <mrow>\\n <msub>\\n <mi>C</mi>\\n \\n <mn>3</mn>\\n </msub>\\n \\n <mo>⋊</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>n</mi>\\n </mrow>\\n </msub>\\n </mrow></math> are triply transitive. Additionally, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of <math>\\n \\n <mrow>\\n <msub>\\n <mi>T</mi>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>k</mi>\\n </mrow>\\n </msub>\\n </mrow></math>, <math>\\n \\n <mrow>\\n <msub>\\n <mi>C</mi>\\n \\n <mi>n</mi>\\n </msub>\\n \\n <mo>⋊</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mi>p</mi>\\n </msub>\\n </mrow></math> and <math>\\n \\n <mrow>\\n <msub>\\n <mi>C</mi>\\n \\n <mi>p</mi>\\n </msub>\\n \\n <mo>⋊</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mi>n</mi>\\n </msub>\\n </mrow></math> when they are triply transitive.\",\"PeriodicalId\":15389,\"journal\":{\"name\":\"Journal of Combinatorial Designs\",\"volume\":\"32 8\",\"pages\":\"438-463\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-04-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Designs\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21941\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Designs","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21941","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

对于任何有限群，群关联方案的特尔维利格代数满足以下结论：其中，是一个特定的向量空间，是共轭作用诱导的的置换表示的中心化代数。如果 .在本文中，我们确定了、和的维数，并证明和是三传递的。此外，我们还给出了、和的特威里格布尔的韦德伯恩成分的完整特征。

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The Terwilliger algebras of the group association schemes of three metacyclic groups

For any finite group $G$ , the Terwilliger algebra $T (G)$ of the group association scheme satisfies the following inclusions: $T_{0} (G) \subseteq T (G) \subseteq \tilde{T} (G)$ , where $T_{0} (G)$ is a specific vector space and $\tilde{T} (G)$ is the centralizer algebra of the permutation representation of $G$ induced by the action of conjugation. The group $G$ is said to be triply transitive if $T_{0} (G) = \tilde{T} (G)$ . In this paper, we determine the dimensions of $T_{0} (G)$ and $\tilde{T} (G)$ for $G$ being $T_{n, k} = 〈 a, b ∣ a^{2 n} = 1, a^{n} = b^{2}, b a b^{- 1} = a^{k} 〉$ , $C_{n} ⋊ C_{p}$ and $C_{p} ⋊ C_{n}$ , and show that $T_{n, k}, C_{n} ⋊ C_{2}$ and $C_{3} ⋊ C_{2 n}$ are triply transitive. Additionally, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of $T_{n, k}$ , $C_{n} ⋊ C_{p}$ and $C_{p} ⋊ C_{n}$ when they are triply transitive.

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来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.