Combinatorial Theory

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, 06B23

Keywords: Coxeter groups, root systems, affine Coxeter groups, lattice theory