有限一般线性群的交点定理

Alena Ernst, K. Schmidt
{"title":"有限一般线性群的交点定理","authors":"Alena Ernst, K. Schmidt","doi":"10.1017/S0305004123000075","DOIUrl":null,"url":null,"abstract":"Abstract A subset Y of the general linear group \n$\\text{GL}(n,q)$\n is called t-intersecting if \n$\\text{rk}(x-y)\\le n-t$\n for all \n$x,y\\in Y$\n , or equivalently x and y agree pointwise on a t-dimensional subspace of \n$\\mathbb{F}_q^n$\n for all \n$x,y\\in Y$\n . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of \n$\\mathbb{F}_q^n$\n . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of \n$\\text{GL}(n,q)$\n that intersect not necessarily pointwise in t-dimensional subspaces of \n$\\mathbb{F}_q^n$\n and for cross-intersecting subsets of \n$\\text{GL}(n,q)$\n . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of \n$\\text{GL}(n,q)$\n .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Intersection theorems for finite general linear groups\",\"authors\":\"Alena Ernst, K. Schmidt\",\"doi\":\"10.1017/S0305004123000075\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract A subset Y of the general linear group \\n$\\\\text{GL}(n,q)$\\n is called t-intersecting if \\n$\\\\text{rk}(x-y)\\\\le n-t$\\n for all \\n$x,y\\\\in Y$\\n , or equivalently x and y agree pointwise on a t-dimensional subspace of \\n$\\\\mathbb{F}_q^n$\\n for all \\n$x,y\\\\in Y$\\n . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of \\n$\\\\mathbb{F}_q^n$\\n . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of \\n$\\\\text{GL}(n,q)$\\n that intersect not necessarily pointwise in t-dimensional subspaces of \\n$\\\\mathbb{F}_q^n$\\n and for cross-intersecting subsets of \\n$\\\\text{GL}(n,q)$\\n . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of \\n$\\\\text{GL}(n,q)$\\n .\",\"PeriodicalId\":18320,\"journal\":{\"name\":\"Mathematical Proceedings of the Cambridge Philosophical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-05-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Proceedings of the Cambridge Philosophical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/S0305004123000075\",\"RegionNum\":3,\"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":"Mathematical Proceedings of the Cambridge Philosophical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/S0305004123000075","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3

摘要

摘要一般线性群$\text{GL}(n,q)$的子集Y称为t相交,如果$\text{rk}(x- Y)\ n-t$对于所有$x, Y \in Y$,或者等价地,x和Y在$\mathbb{F}_q^n$的t维子空间上对所有$x, Y \in Y$点方向一致。我们证明,如果n相对于t足够大,则每一个这样的t相交集的大小不超过$\mathbb{F}_q^n$的t维子空间的基的稳定子的大小。在相等的情况下,Y的特征向量是这些稳定器的协集的特征向量的线性组合。对于$\text{GL}(n,q)$在$\mathbb{F}_q^n$的t维子空间中不一定点向相交的子集,以及$\text{GL}(n,q)$的交叉子集,我们也给出了类似的结果。这些结果可以看作是极端集合理论中经典Erdős-Ko-Rado定理的变体,并且是对称群中已知的相应结果的q-类似。我们的方法基于特征值技术来估计图中最大独立集的大小,并且关键地涉及到$\text{GL}(n,q)$的表示理论。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Intersection theorems for finite general linear groups
Abstract A subset Y of the general linear group $\text{GL}(n,q)$ is called t-intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$ . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of $\mathbb{F}_q^n$ . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t-dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$ . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$ .
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
1.70
自引率
0.00%
发文量
39
审稿时长
6-12 weeks
期刊介绍: Papers which advance knowledge of mathematics, either pure or applied, will be considered by the Editorial Committee. The work must be original and not submitted to another journal.
期刊最新文献
The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves Generalised knotoids Multiplicative dependence of rational values modulo approximate finitely generated groups Tropical curves in abelian surfaces I: enumeration of curves passing through points Domination inequalities and dominating graphs
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1