发表机构
Beijing International Center for Mathematical Research, Peking University; Qiuzhen College, Tsinghua University; School of Mathematical Sciences, Peking University（北京大学北京国际数学研究中心; 清华大学邱耀学院; 北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文对所有有限Coxeter群证明了Dyer关于正根有界闭集并的猜想，并推广到2-闭集，通过闭包准则和根选择论证刻画了反转集与可达元素。
AI 中文摘要
Dyer猜想，两个正根的有界闭集的并可以通过递增Bruhat路径来描述，这些路径的反射标签属于它们的并集。我们给出了对所有有限Coxeter群这一猜想的类型一致证明。更一般地，对于包含在反转集中的正根的$2$-闭集$C$，我们证明其$2$-闭包是反转集$I(w)$，并且从单位元出发使用标签为$C$中根的反射可达的元素恰好构成集合$[e,w]_B w^{-1}$。证明结合了Dyer的闭包准则与根选择论证。
英文摘要
Dyer conjectured that the join of two biclosed sets of positive roots can be described by increasing Bruhat paths whose reflection labels belong to their union. We give a type-uniform proof of this conjecture for all finite Coxeter groups. More generally, for a $2$-coclosed set $C$ of positive roots contained in an inversion set, we show that its $2$-closure is an inversion set $I(w)$ and that the elements reachable from the identity using reflections labeled by roots in $C$ form exactly the set $[e,w]_B w^{-1}$. The proof combines Dyer's closure criteria with a root-selection argument.
