PreprintarXiv licencearXiv
On inversion sets of joins in weak Bruhat order
Abstract
It is well-known that any Coxeter group W is a meet-semilattice with respect to weak order. Furthermore, when W is finite, it is a lattice. In the first part of this paper, we give a recursive algorithm that computes joins in weak order for arbitrary finite W. In the second part, we complete the proof of a conjecture of Dyer expressing the inversion set of a join in terms of the union of inversion sets of each factor, under the assumption that W is finite. Some proofs were developed in collaboration with Claude Opus 5.
Related
- Weak Order on the MacNeille Completion of Bruhat Order
- Bounded joins of biclosed sets
- Weak Bruhat interval modules of the 0-Hecke algebras for stable Grothendieck polynomials
- Finite bases and joins for semirings defined by the divisibility order
- Quantum roots for Kac-Moody root systems and finiteness properties of the Kac-Moody affine Bruhat order
BibTeX
@misc{lim2026inversion,
title = {{On inversion sets of joins in weak Bruhat order}},
author = {Ming Yean Lim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25395},
eprint = {2609.25395},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25395},
}
RIS
TY - UNPB
TI - On inversion sets of joins in weak Bruhat order
AU - Ming Yean Lim
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25395
UR - https://arxiv.org/abs/2609.25395
AB - It is well-known that any Coxeter group W is a meet-semilattice with respect to weak order. Furthermore, when W is finite, it is a lattice. In the first part of this paper, we give a recursive algorithm that computes joins in weak order for arbitrary finite W. In the second part, we complete the proof of a conjecture of Dyer expressing the inversion set of a join in terms of the union of inversion sets of each factor, under the assumption that W is finite. Some proofs were developed in collaboration with Claude Opus 5.
ER -