论文标题
双斑点集中的组合描述
Combinatorial descriptions of biclosed sets in affine type
论文作者
论文摘要
令$ w $为Coxeter组，让$φ^+$成为其正根。如果我们有$α$，$β$，$β$和$γ$，则称为$φ^+ $的子集$ b $，将其称为双licl，in \ Mathbb {r} _ {> 0}α+ \ \ \ \ \ m mathbb {r} $α$和$β\ not \ in B $，然后在b $中$γ\ not \。有限的双闭合集是$ w $的元素的反转集，有限反转集之间的遏制是$ w $的弱顺序。马修·戴尔（Matthew Dyer）建议研究所有双粘性子集的$φ^+$的poset，该子集按围栏订购，并猜想它是一个完整的晶格。随着朝代猜想的进展，我们对仿射根系中的所有双粘合集进行了分类。我们在经典类型中提供类型统一描述，也提供具体模型，$ \ widetilde {a} $，$ \ widetilde {b} $，$ \ widetilde {c} $，$ \ widetilde {d} $。我们使用模型来证明双重套件形成一个完整的晶格，类型为$ \ widetilde {a} $和$ \ widetilde {c} $。
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 $\widetilde{C}$.
