The hull metric on Coxeter groups

TL;DR

This paper explores a new metric on Coxeter groups based on convex hull inequalities, extending known results from symmetric groups and hyperoctahedral groups, with potential applications in geometric group theory.

Contribution

It generalizes Sidorenko's inequality to arbitrary Coxeter groups, proves it for specific classes, and introduces a novel combinatorial insertion map related to promotion operators.

Findings

Proved inequalities for hyperoctahedral groups $B_n$

Extended inequalities to all right-angled Coxeter groups

Introduced a new invariant metric based on convex hulls

Abstract

We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for $B_n$ (and new proof for $\mathfrak{S}_n$) use a combinatorial insertion map closely related to the well-studied promotion operator on linear extensions; this map may be of independent interest. We also note that the inequalities in question can be interpreted as a triangle inequalities, so that convex hulls can be used to define a new invariant metric on $W$ whenever our conjecture holds. Geometric properties of this metric are an interesting direction for future research.