Ricci curvature, Bruhat graphs and Coxeter groups

Abstract

We consider the notion of discrete Ricci curvature for graphs defined by Schmuckenschl{ä}ger \cite{shmuck} 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 \cite{Dyer}. 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.

12 pages. arXiv admin note: text overlap with arXiv:2102.10134