论文标题
仿射Coxeter组的SHI布置和低元素
Shi arrangements and low elements in affine Coxeter groups
论文作者
论文摘要
鉴于一个仿射Coxeter $ W $，相应的SHI布置是Shi引入的相应Coxeter超平面布置的改进，该布置用于研究Kazhdan-Lusztig细胞以$ W $。特别是，什（Shi）表明，什（Shi）布置的每个区域都包含$ w $的最小长度的一个元素。引入了$ W $中的低元素来研究相应的Artin-tits（编织）组的单词问题，并原来生产自动机以研究$ W $中简化单词的组合。 在本文中，我们在一个仿射Coxeter组的情况下表明，SHI布置中区域的最小长度要素的集合恰好是低元素的集合，在这种情况下解决了Dyer的猜想，在这种情况下是第二作者。作为我们证明的副产品，我们表明，SHI布置中任何区域的下降壁 - 将区域与基本壁co区分开的壁完全是其相应低元素的壁co的下降壁。
Given an affine Coxeter group $W$, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan-Lusztig cells for $W$. In particular, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in $W$. Low elements in $W$ were introduced to study the word problem of the corresponding Artin-Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in $W$. In this article, we show in the case of an affine Coxeter group that the set of minimal length elements of the regions in the Shi arrangement is precisely the set of low elements, settling a conjecture of Dyer and the second author in this case. As a byproduct of our proof, we show that the descent-walls -- the walls that separate a region from the fundamental alcove -- of any region in the Shi arrangement are precisely the descent walls of the alcove of its corresponding low element.
