Combinatorial descriptions of biclosed sets in affine type

Abstract

Let $W$ be a Coxeter group and let $Φ^+$ be its positive roots. A subset $B$ of $Φ^+$ is called biclosed if, whenever we have roots $α$, $β$ and $γ$ with $γ\in \mathbb{R}_{>0} α+ \mathbb{R}_{>0} β$, if $α$ and $β\in B$ then $γ\in B$ and, if $α$ and $β\not\in B$, then $γ\not\in B$. 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 $Φ^+$, 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 $\widetilde{C}$.

Figures (3)

• Figure 1: The Dynkin diagrams associated to affine indecomposable root systems. The extra, unfilled, node corresponds to the extra simple root $\alpha_0$. The remaining filled nodes form the Dynkin diagram for the associated finite crystallographic root system.
• Figure 2: The positive roots in the $\widetilde{A}_1$ root system. A possible reflection order is shown below. An example of a biclosed set is shaded in blue.
• Figure 3: The $A_2$ Coxeter fan. Each face is labeled by an ordered partition of $\{1,2,3\}$. The origin is also a face, with label $(\{1,2,3\})$.

Theorems & Definitions (86)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 3.1
• Remark 3.2
• Remark 3.3
• Example 3.4
• Proposition 3.5: Dyer2019