2025/08/31 by Grant Barkley, Grant T. Barkley, Barkley, Grant +10
Materials Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Quasicrystal Structures and Properties #math.CO #math.GR
paper · pdf · doi:10.48550/arxiv.2509.00871
openalex publication_date 2025/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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.