{"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。
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
$\text{GL}(n,q)$
.
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.