Finite reflection groups and graph norms
Finite reflection groups and graph norms
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DOI：
10.1016/j.aim.2017.05.009
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发表时间：
2017-07-31
影响因子：
1.7
通讯作者：
Lee, Joonkyung
中科院分区：
文献类型：
--
作者：
Conlon, David;Lee, Joonkyung
给定顶点集{1，2，中心点中心点中心点，n}和一个函数f：[0,1]（2）（2） - > r，定义与（h）f平行于（h）：=垂直栏积分pi（ij是e（h））f（x（i），x（j））d mu（垂直杆V（h）垂直条）垂直条（1/垂直杆）的元素e（h）垂直条），其中mu是[0,1]上的lebesgue度量。我们说，如果平行于（H）平行于中心点，则H是规范的。平行于（r（h））平行于平行于（r（h））的中心点的类似概念是通过平行于（r（h））平行的f定义的：=平行于平行于（H）的垂直杆f垂直杆，h据说h弱如果平行于平行于（r（h））的中心点，则规范是标准。经典结果表明，弱规范的图一定是二分。在另一个方向上，哈塔米（Hatami）表明，即使是循环，完整的二分图和超振管都弱规范性。我们证明，在某种自然自然家族的作用下，其边缘以适当方式渗透的任何图形都是薄弱的。该结果包括所有以前已知的弱规范图的示例，但也使我们能够确定有限反射组引起的更广泛的类别。我们包括我们结果的几种应用。特别是，我们定义并比较了Gowers八面体规范的许多概括，并证明了Sidorenko的猜想的一些新实例。 （c）2017 Elsevier Inc.保留所有权利。
Given a graph H on vertex set {1, 2, center dot center dot center dot , n} and a function f : [0,1](2) -> R, defineparallel to f parallel to(H) :=vertical bar integral Pi(ij is an element of E(H)) f(x(i),x(j))d mu(vertical bar V(H)vertical bar)vertical bar(1/vertical bar E(H)vertical bar) , where mu is the Lebesgue measure on [0,1]. We say that H is norming if parallel to center dot parallel to(H) is a semi-norm. A similar notion parallel to center dot parallel to(r(H)) is defined by parallel to f parallel to(r(H)) := parallel to vertical bar f vertical bar parallel to(H) and H is said to be weakly norming if parallel to center dot parallel to(r(H)) is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. We demonstrate that any graph whose edges percolate in an appropriate way under the action of a certain natural family of automorpbisms is weakly forming. This result includes all previously known examples of weakly norming graphs, but also allows us to identify a much broader class arising from finite reflection groups. We include several applications of our results. In particular, we define and compare a number of generalisations of Gowers' octahedral norms and we prove some new instances of Sidorenko's conjecture. (C) 2017 Elsevier Inc. All rights reserved.