A note on the scatteredness of reflection orders

TL;DR

This paper characterizes affine Coxeter systems through the structure of their reflection orders, linking affineness to scatteredness and specific order types, and explores properties of non-affine systems.

Contribution

It provides new characterizations of affine and non-affine Coxeter systems based on the structure and order types of their reflection orders.

Findings

Affine systems have reflection orders that are scattered or of type ω+ω*.

Non-affine systems admit reflection orders with order types (ω+ω*)[k] for arbitrarily large k.

The proofs use geometry of roots, isotropic cones, and universal reflection subgroups.

Abstract

In this note, we characterize affine and non-affine Coxeter systems among all Coxeter systems in terms of the structure of their reflection orders. For an infinite irreducible system $(W,S)$, we show that affineness can be characterized in three equivalent ways: by the scatteredness of all reflection orders, by the existence of a reflection order of type $\omega + \omega^*$, and by a finiteness property of intervals determined by dihedral reflection subgroups. We also show that non-affineness can be characterized by the existence of order types $(\omega + \omega^*)[k]$ for arbitrarily large $k$, obtained by restricting any reflection order to a suitable subset. Our proofs exploit the geometry of projective roots, the isotropic cone, and universal reflection subgroups in infinite non-affine Coxeter groups.