A note on the scatteredness of reflection orders
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908837115592704 |
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| author | Wang, Weijia Wang, Rui |
| author_facet | Wang, Weijia Wang, Rui |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09275 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on the scatteredness of reflection orders Wang, Weijia Wang, Rui Group Theory 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. |
| title | A note on the scatteredness of reflection orders |
| topic | Group Theory |
| url | https://arxiv.org/abs/2601.09275 |