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

2025/10/13 by Biagioli, Riccardo, Perrone, Lorenzo · 1 citation
#05E16 (Secondary) #20F55 (Primary) 06F15 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2510.11446

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 H₃ and F₄ through the use of Sage.

Cited by

• Bruhat Preclosure

Related

• Extended Weak Order for the Rank 3 Universal Coxeter Group
• Affine extended weak order is a lattice
• Extended weak order for the affine symmetric group
• Bruhat Preclosure
• Combinatorics of cone types in Coxeter groups