Combinatorics Seminar

When: Sunday, March 20, 10am

Where: Schreiber 309

Speaker: David Conlon, Oxford

Title: Finite reflection groups and graph norms


Given a graph H and a symmetric function f:[0,1]2→ℝ, define the absolute H-norm of f by

‖f‖H := (∫ ∏ij∈E(H) |f(xi,xj)| dμ|V(H)|)1/|E(H)|,

where μ is the Lebesgue measure on [0,1]. We say that H is weakly norming if the absolute H-norm is a norm on the vector space of symmetric functions. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, it is known that even cycles, complete bipartite graphs and hypercubes are all weakly norming. Using results from the theory of finite reflection groups, we demonstrate that any graph which is edge-transitive under the action of a certain natural family of automorphisms is weakly norming. This result includes all previous examples of weakly norming graphs and adds many more. Joint work with Joonkyung Lee.

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