Combinatorial descriptions of biclosed sets in affine type

Abstract

Let $W$ be a Coxeter group and let $\Phi^+$ be its 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 $W$. Matthew 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 $\widetilde{C}$.