寒武纪扇形
Cambrian fans
AI总结：
本文研究了Coxeter群W的c-Cambrian扇形，证明其与广义associahedron的正常扇形在组合上同构，并探讨了c-Cambrian扇形与c-簇的线性同构关系。
摘要：
对于有限Coxeter群W和其Coxeter元素c，c-Cambrian扇形是反射超平面定义的扇形的粗化。其最大锥自然由W的c-sortable元素索引。本文的主要结果是已知的bijection cl_c在c-sortable元素与c-clusters之间诱导了扇形的组合同构。特别地，c-Cambrian扇形与W的广义associahedron的正常扇形在组合上同构。c-Cambrian扇形的射线由某些在W轨道中的基本权重向量生成，而c-簇扇形的射线由某些根生成。对于特定的(
英文摘要：
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 ("bipartite") 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.
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