非交叉划分与碎片交集序
Noncrossing partitions and the shard intersection order
AI总结：
本文定义了有限Coxeter群W元素的新格结构，即碎片交集序，其弱序更弱，并包含非交叉划分格NC(W)。该构造提供了NC(W)为格的新证明，且其格属性与NC(W)相似。
摘要：
我们定义了一个有限Coxeter群W元素的新格结构，称为碎片交集序。该格弱于弱序，并且非交叉划分格NC(W)是其子格。新的NC(W)构造提供了NC(W)为格的新证明。碎片交集序是分级的，其秩生成函数是W-欧拉多项式。碎片交集序的许多格论性质，如Mobius数、最大链数等，与NC(W)的相应性质完全类似。碎片交集序的顺序复杂度中的单纯形与W-排列多面体的某种拉伸三角剖分中的单纯形之间存在自然的维度保持双射。限制该双射到NC(W)的顺序复杂度，得到与W-关联多面体的拉伸三角剖分中的单纯形之间的双射。碎片交集序通过W的反射超平面的多面体几何间接定义。事实上，本文大部分结果是在更一般的简单超平面排列设置中证明的。
英文摘要：
We define a new lattice structure on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new construction of NC(W) yields a new proof that NC(W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of the shard intersection order, like Mobius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC(W). There is a natural dimension-preserving bijection between simplices in the order complex of the shard intersection order (i.e. chains in the shard intersection order) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC(W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. The shard intersection order is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
