Ricci curvature, Bruhat graphs and Coxeter groups

TL;DR

This paper computes discrete Ricci curvature for Bruhat graphs of finite Coxeter groups, deriving bounds on spectral gaps and isoperimetric inequalities, using geometric realizations and classical Coxeter group results.

Contribution

It provides a case-free computation of Ricci curvature for Bruhat graphs of finite Coxeter groups and derives spectral gap bounds and isoperimetric inequalities.

Findings

Computed Ricci curvature for Bruhat graphs of finite Coxeter groups

Established bounds for the spectral gap of these graphs

Proved isoperimetric inequalities for Bruhat graphs

Abstract

We consider the notion of discrete Ricci curvature for graphs defined by Schmuckenschl{\"a}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.