On a Conjecture of Dyer on the Join in the Weak Order of a Coxeter group

TL;DR

This work addresses Dyer's conjectures on the extended weak order of a Coxeter group, focusing on a finite-type reformulation of the join description and proving it for types $A$ (symmetric group) and $I$ (dihedral groups $I_2(m)$). The authors develop a reformulation using root-system data, inversion sets, and $(u,v)$-Bruhat paths, and establish the key equivalence between the original conjecture and its reformulation in finite type. For type $A$, they prove that the left-reflection set of the join satisfies $T_L(u\vee_R v)=T\cap V_W(u,v)$ by leveraging inversions and a transitive-closure property of $T_L$; for type $I_2(m)$ they perform a case analysis to verify the join description. They also report computational confirmations for types $H_3$ and $F_4$ via Sage and outline ongoing work toward a uniform proof across classical types, including potential geometric approaches. The results deepen the understanding of join operations in the extended weak order and connect combinatorial Bruhat-path descriptions with reflection-sets in finite Coxeter groups.

Abstract

In one of his papers on the weak order of Coxeter groups, Dyer formulates several conjectures. Among these, one affirms that the extended weak order forms a lattice, while another offers an algebraic-geometric description of the join of two elements in this poset. The former was recently proven for affine types by Barkley and Speyer. In this paper, we establish the latter for Coxeter groups of types $A$ and $I$. Moreover, we verified the validity of this conjecture for types $H_3$ and $F_4$ through the use of Sage.

Figures (7)

• Figure 1: Coxeter graphs of types $A_n$ and $I_2(m)$.
• Figure 2: From left to right, the Bruhat graph (where the label $t$ denotes the reflection $s_1s_2s_1$), the Hasse diagrams of $(W,\leq_R)$ and of $(\mathscr{B}(\Phi^+),\subseteq)$ for $W$ of type $A_2$.
• Figure 3: Coxeter graphs of types $H_3$ and $F_4$.
• Figure 4: All the possible $(s_1s_2,s_1s_3)$-Bruhat paths in a Coxeter group of type $A$, where the label $t$ denotes the reflection $s_1s_2s_1$. We highlighted the elements of $T\cap V_W(s_1s_2,s_1s_3)$.
• Figure 5: The Bruhat graph $B(I_2(m))$ for $m$ even and odd: the edge labels are omitted.

Theorems & Definitions (20)

• Definition 1.1
• Definition 2.1
• Lemma 2.2
• Example 2.3
• Definition 2.4: $(u,v)$-Bruhat path
• Example 2.5
• Theorem 2.6
• proof
• Theorem 3.1
• proof