Ricci curvature, Bruhat graphs and Coxeter groups

Abstract

We consider the notion of discrete Ricci curvature for graphs defined by Schmuckenschläger [12] 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 [6]. 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.

The intermediate orders of a Coxeter group

“…We also introduce the k-Bruhat graph of a Coxeter system, which turns out to be always locally finite and in some sense approximates the Bruhat graph when the group is infinite. Recent results on the Ricci curvature of Bruhat and Cayley graphs of Coxeter groups ( [13], [12]) lead to consider the Ricci curvature of a k-Bruhat graph; this problem closes the paper.…”

“…We end this section with a problem on the Ricci curvature of k-Bruhat graphs. The Ricci curvature of the graph G 0 (the Cayley graph of (W, S)) is studied (and in many cases explicitly computed) in [13]; the Ricci curvature of the Bruhat graph of a finite Coxeter group is proved to be 2 in [12]. We refer to these articles for definitions and preliminary results.…”

Ricci curvature of Bruhat orders

“…In this work we consider the Ricci curvature of the Hasse graphs of the Bruhat order of finite irreducible Coxeter systems. This is a considerably harder problem that those studied by the same author on [25] and [26] because not all vertices are 'isomorphic' in these graphs.…”

“…In [24], [25] and [26] the Ricci curvature of Bruhat graphs of Coxeter groups and of the Hasse graphs of the weak orders of Coxeter and affine Weyl groups is studied.…”