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A note on the scatteredness of reflection orders

Weijia Wang, Rui Wang

TL;DR

This work characterizes affine Coxeter systems among all Coxeter groups through the structure of reflection orders. It leverages normalized and lexicographic root representations, limit roots, and the isotropic cone, together with a rank-3 universal Coxeter subgroup, to analyze order types and density properties of reflection orders. The main results show that an irreducible infinite Coxeter group is affine iff all reflection orders are scattered, and provide two new characterizations: existence of a reflection order with order type $ω+ω^*$ and a dihedral-interval finiteness condition. These findings illuminate the combinatorial geometry of reflection orders and offer practical, order-theoretic criteria to distinguish affine types, with implications for related combinatorics and representation theory.

Abstract

In this note, we characterize 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 is equivalently characterized by the scatteredness of all reflection orders, by the existence of a reflection order of type $ω+ω^*$, and by a finiteness property of intervals determined by dihedral reflection subgroups. Our proof exploits the geometry of projective roots, the isotropic cone and the universal reflection subgroups in an infinite non-affine Coxeter group.

A note on the scatteredness of reflection orders

TL;DR

This work characterizes affine Coxeter systems among all Coxeter groups through the structure of reflection orders. It leverages normalized and lexicographic root representations, limit roots, and the isotropic cone, together with a rank-3 universal Coxeter subgroup, to analyze order types and density properties of reflection orders. The main results show that an irreducible infinite Coxeter group is affine iff all reflection orders are scattered, and provide two new characterizations: existence of a reflection order with order type and a dihedral-interval finiteness condition. These findings illuminate the combinatorial geometry of reflection orders and offer practical, order-theoretic criteria to distinguish affine types, with implications for related combinatorics and representation theory.

Abstract

In this note, we characterize affine Coxeter systems among all Coxeter systems in terms of the structure of their reflection orders. For an infinite irreducible system , we show that affineness is equivalently characterized by the scatteredness of all reflection orders, by the existence of a reflection order of type , and by a finiteness property of intervals determined by dihedral reflection subgroups. Our proof exploits the geometry of projective roots, the isotropic cone and the universal reflection subgroups in an infinite non-affine Coxeter group.
Paper Structure (11 sections, 12 theorems, 38 equations, 2 figures)

This paper contains 11 sections, 12 theorems, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coxeter system
  4. The root system of a Coxeter system
  5. The order type of a total order
  6. Reflection order
  7. Normalized roots
  8. Lexicographic reflection order
  9. Limit root and isotropic cone
  10. Dense Subset
  11. non-scatteredness of reflection orders of non-affine, infinite Coxeter groups

Key Result

Lemma 2.7

$\prec_{\mathrm{reflex}}$ is a reflection order.

Figures (2)

  • Figure 1: The normalized roots and the normalized isotropic cone of rank 3 universal Coxeter system
  • Figure 2: Density of the first barycentric coordinates of the positive roots of a rank 3 universal Coxeter group

Theorems & Definitions (23)

  • Lemma 2.7
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.5
  • proof
  • ...and 13 more