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 $\beta \notin B$, then $\gamma \notin B$. 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 $\tilde{A}$, $\tilde{B}$, $\tilde{C}$, $\tilde{D}$. We use our models to prove that biclosed sets form a complete lattice in types $\tilde{A}$ and $\tilde{C}$.

24 pages, 3 figures