Year: 2024
Type: article
Abstract: Let \(W\) be a Coxeter group and let \(\Phi^+\) be the positive roots. A subset \(B\) of \(\Phi^+\) is called "biclosed" if, whenever we have roots \(\alpha\), \(\beta\) and \(\gamma\) with \(\gamma \in \mathbb{R}_{›0} \alpha + \mathbb{R}_{›0} \beta\), if \(\alpha\) and \(\beta \in B\) then \(\gamma \in B\) and, if \(\alpha\) and \(\beta \not\in B\), then \(\gamma \not\in B\). The finite biclosed sets are the inversion sets of the elements of \(W\), and the containment between finite inversion sets is the weak order on \(W\). Dyer suggested studying the poset of all biclosed subsets of \(\Phi^+\), ordered by containment, and conjectured that it is a complete lattice. As progress towards Dyer's conjecture, we classify all biclosed sets in the affine root systems. We provide both a type uniform description, and concrete models in the classical types \(\widetilde{A}\), \(\widetilde{B}\), \(\widetilde{C}\), \(\widetilde{D}\). We use our models to prove that biclosed sets form a complete lattice in types \(\widetilde{A}\) and \(\widetilde{C}\), and we classify which biclosed sets are separable and which are weakly separable.Mathematics Subject Classifications: 20F55, 17B22, 06B23Keywords: Coxeter groups, root systems, affine Coxeter groups, lattice theory
Source: Combinatorial Theory
DOI: 10.5070/c64264235
Authors: Grant T. Barkley, David E Speyer
Institutions: Harvard University, University of Michigan
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funder: National Science Foundation
OpenAlex ID: W4402927491