Cambrian fans
Cambrian fans
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寒武纪扇
DOI：
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发表时间：
2006
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
.对于有限Coxeter群W和W的Coxeter元素c，c -寒武纪扇是由W的反射超平面定义的扇的粗化。它的最大锥自然由W的可排序元素索引。本文的主要结果是，c -可排序元素和c -簇之间的已知的双射cl c诱导了fans的组合同构。特别地，c -寒武纪扇与W的广义协面体的正规扇组合同构。c -寒武纪扇的射线是由基本权的W -轨道中的某些向量产生的，而c -星团扇的射线是由某些根产生的。对于特定的（“二分”）选择的c，我们表明，c -寒武纪风扇是线性同构的c -集群风扇。我们的特点，在集团的组合学方面，诱导的偏序，通过地图cl c，在c -集团的c -寒武纪格。我们给出了一个简单的双射从c -集群的c -非交叉分区，尊重refined（Narayana）枚举。我们涉及到寒武纪风扇的集群代数理论中众所周知的对象，提供了一个几何背景下的g -向量和准嘉当同伴。
. For a finite Coxeter group W and a Coxeter element c of W, the c -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W . Its maximal cones are naturally indexed by the c -sortable elements of W . The main result of this paper is that the known bijection cl c between c -sortable elements and c -clusters induces a combinatorial isomorphism of fans. In particular, the c -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W . The rays of the c -Cambrian fan are generated by certain vectors in the W -orbit of the fundamental weights, while the rays of the c -cluster fan are generated by certain roots. For particular (“bi-partite”) choices of c , we show that the c -Cambrian fan is linearly isomorphic to the c -cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map cl c , on c -clusters by the c -Cambrian lattice. We give a simple bijection from c -clusters to c -noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric context for g -vectors and quasi-Cartan companions.