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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.09275 |
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Table of Contents:
- 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 $ω+ ω^*$, 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 $(ω+ ω^*)[k]$ 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.