Ricci curvature, Bruhat graphs and Coxeter groups

Abstract

We consider the discrete Ricci curvature for graphs as defined by Schmuckenschläger [Convex geometric analysis, MSRI Publications, 1998] and compute its value for Bruhat graphs associated to finite Coxeter groups. To do so we work with the geometric realization of a finite Coxeter group and a classical result obtained by Dyer in [Compositio Math. 78 (1991), pp. 185–191]. As an application we obtain a bound for the spectral gap of the Bruhat graph of any finite Coxeter group and an isoperimetric inequality for them. Our proofs are case-free.