Low elements and small inversion sets are in bijection in rank 3 Coxeter groups

TL;DR

This paper proves a bijection between low elements and small inversion sets in rank 3 Coxeter groups, supporting a conjecture and utilizing bipodality of small roots to analyze inversion polytopes.

Contribution

It establishes a bijection in rank 3 Coxeter groups and confirms part of a previous conjecture, advancing understanding of inversion sets and root structures.

Findings

Low elements correspond bijectively to small inversion sets in rank 3 Coxeter groups

Supports Conjecture 2 from Dyer and Hohlweg (2016)

Uses bipodality of small roots to analyze inversion polytopes

Abstract

In this extended abstract we announce a proof that, in a Coxeter group of rank 3, low elements are in bijection with small inversion sets. This gives a partial confirmation of Conjecture 2 in [Dyer, Hohlweg '16]. That same article provides the main ingredient: the bipodality of the set of small roots is used to propagate information on the vertices of inversion polytopes.