Finite reflection groups and graph norms

By David Conlon, Joonkyung Lee. 2016

70 citations recorded by the source corpus.

Given a graph $H$ on vertex set $\{1,2,\cdots, n\}$ and a function $f:[0,1]^2 \rightarrow \mathbb{R}$, define \begin{align*} \|f\|_{H}:=\left\vert\int \prod_{ij\in E(H)}f(x_i,x_j)d\mu^{|V(H)|}\right\vert^{1/|E(H)|}, \end{align*} where $\mu$ is the Lebesgue measure on $[0,1]$. We say that $H$ is norming if $\|\cdot\|_H$ is a semi-norm. A similar notion $\|\cdot\|_{r(H)}$ is defined by $\|f\|_{r(H)}:=\||f|\|_{H}$ and $H$ is said to be weakly norming if $\|\cdot\|_{r(H)}$ 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 automorphisms is weakly norming. 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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8bbcbc03e7dcb5b94b48fe8a52ba31974199e998

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In Computational Theory and Mathematics

  1. Joonkyung Lee
  2. David Conlon
  3. A. Sah
  4. Peter Keevash

In Discrete Mathematics and Combinatorics

  1. Joonkyung Lee
  2. David Conlon
  3. A. Sah
  4. Peter Keevash

In Geometry and Topology

  1. Joonkyung Lee
  2. David Conlon
  3. A. Sah
  4. Peter Keevash

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