Combinatorics Seminar: Finite reflection groups and graph norms

Seminar | April 3 | 12:10-1 p.m. | 939 Evans Hall

 David Conlon, University of Oxford

 Department of Mathematics

For any given graph $H$, we may define a natural corresponding functional $\|\cdot \|_H$. We then 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. Using results from the theory of finite reflection groups, we identify a much larger class of weakly norming graphs. This result includes all previous examples of weakly norming graphs and adds many more. We also discuss 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. Joint work with Joonkyung Lee.