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Extended weak order in all rank three Coxeter groups

  • Grant Barkley
  • Colin Defant
  • Patricia Hersh
  • Jon McCammond
  • Thomas McConville
  • David E Speyer
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Abstract

We prove that extended weak order is a lattice for all rank 3 Coxeter groups. This was previously known in finite and affine types, and in the universal Coxeter group of rank 3. All other cases are new to this paper. As in earlier work, the proof proceeds by showing that all rank 3 Coxeter arrangements are clean arrangements, meaning that every biclosed set is weakly separated from its complement by a hyperplane. As a consequence, we also deduce Dyer's Conjecture A for all rank 3 Coxeter groups, which asserts that cover relations in extended weak order are given by toggling a single root.

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Cite

BibTeXRIS
BibTeX
@misc{barkley2026extended,
  title = {{Extended weak order in all rank three Coxeter groups}},
  author = {Grant Barkley and Colin Defant and Patricia Hersh and Jon McCammond and Thomas McConville and David E Speyer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38849},
  eprint = {2610.38849},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38849},
}
RIS
TY  - UNPB
TI  - Extended weak order in all rank three Coxeter groups
AU  - Grant Barkley
AU  - Colin Defant
AU  - Patricia Hersh
AU  - Jon McCammond
AU  - Thomas McConville
AU  - David E Speyer
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38849
UR  - https://arxiv.org/abs/2610.38849
AB  - We prove that extended weak order is a lattice for all rank 3 Coxeter groups. This was previously known in finite and affine types, and in the universal Coxeter group of rank 3. All other cases are new to this paper. As in earlier work, the proof proceeds by showing that all rank 3 Coxeter arrangements are clean arrangements, meaning that every biclosed set is weakly separated from its complement by a hyperplane. As a consequence, we also deduce Dyer's Conjecture A for all rank 3 Coxeter groups, which asserts that cover relations in extended weak order are given by toggling a single root.
ER  -