论文标题
广义集群综合体：面孔和相关停车位的精致枚举
The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces
论文作者
论文摘要
Fomin和Reading引入了广义集群复合物，是来自有限型群集代数的Fomin-Zelevinsky群集复合物的自然扩展。在这项工作中，与这个综合体的每个面孔相关联，我们将基础有限的Coxeter组的抛物线共轭类别关联。我们表明，根据这些数据，面部的精致枚举（分别为正面面孔）从相应的特征多项式（等效地，就Orlik-Solomon指数而言）提供了明确的公式。这种特征多项式最初来自超平面排列的理论，但可以通过抛物线伯恩赛环方便地定义。这与停车空间的理论有联系：我们的结果最终依赖于在这种情况下获得的非交叉分区链条的一些列举。公式计数面和非交叉分区的一个计数链之间的确切关系是组合互惠，从而概括了Narayana和Kirkman数字之间的互惠。
The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.
