Extended Weak Order for the Rank 3 Universal Coxeter Group
Combinatorics
2025-09-03 v1 Group Theory
Abstract
The weak order is a classical poset structure on a Coxeter group; it is a lattice when the group is finite but merely a meet-semilattice when the group is infinite. Motivated by problems in Kazhdan--Lusztig theory, Matthew Dyer introduced the extended weak order, a poset that contains a copy of the weak order as an order ideal, and he conjectured that the extended weak order for any Coxeter group is a lattice. We prove Dyer's conjecture for the rank 3 universal Coxeter group. This is the first non-spherical, non-affine Coxeter group for which Dyer's conjecture has been proven.
Keywords
Cite
@article{arxiv.2509.00871,
title = {Extended Weak Order for the Rank 3 Universal Coxeter Group},
author = {Grant Barkley and Colin Defant and Patricia Hersh and Jon McCammond and Thomas McConville and David E Speyer},
journal= {arXiv preprint arXiv:2509.00871},
year = {2025}
}
Comments
28 pages, 15 figures