Sortable elements in infinite Coxeter groups
Sortable elements in infinite Coxeter groups
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无限 Coxeter 组中的可排序元素
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
David E. Speyer
中科院分区：
文献类型：
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作者：
Nathan Reading;David E. Speyer
在以前的一系列文章中，我们研究了有限Coxeter群中的可排序元，以及与之相关的寒武纪扇子。我们利用可排序元和寒武纪扇子研究了有限型簇代数和与有限型Artin群相关的不相交划分。在这篇文章中，作为将这些应用扩展到有限类型之外的第一步，我们研究了一般Coxeter群W中的可排序元。我们提供了一致论证，将所有以前的有限类型证明转化为一致证明(而不是逐个类型证明)，推广了许多有限类型结果，并证明了新的更精细的结果。在我们的证明中的关键工具包括与箭图理论的欧拉形式有关的斜对称形式(推广)，以及将W的每个元素按弱序映射到其下面的唯一极大c-可排序元素的投影pidown^c。Pidown^c的纤维本质上定义了c-寒武纪扇体。最基本的结果是，首先，精确地描述了可排序元素在(BGP)反射函子下是如何转化的，其次，精确地描述了Pidown^c的纤维。这些基本结果和其他结果进一步导致了关于寒武纪(半)晶格和寒武纪扇形的晶格理论和几何的进一步结果。
In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.