Free arrangements and rhombic tilings
Free arrangements and rhombic tilings
复制标题
自由排列和菱形瓷砖
DOI：
10.1007/bf02711498
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发表时间：
1996
影响因子：
0.8
通讯作者：
V. Reiner
中科院分区：
文献类型：
--
作者：
Paul H. Edelman;V. Reiner
设Z是中心对称的整数边长多边形。我们回答了以下两个问题：（1）在Saito和Terao意义下，什么时候相应的判别超平面是无约束的？（2）当Z的单位镶嵌都是Billera和Sturmfels意义下的斜方相干时，Z的单位镶嵌是什么？
令人惊讶的是，这两个问题的答案非常相似。此外，借助于MacMahon关于平面分拆的一个老结果和Elnitsky关于菱形镶嵌的一些新结果，对第一个问题的回答有助于对第二个问题的回答。这些结果也引起了一些有趣的几何推论。对某些特定八边形的判别式排列的考虑导致了Saito [ER 2]猜想的一个先前宣布的反例，即真实的自由排列的复化补是aK（π，1）空间。
AbstractLet Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:(1)When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?(2)When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels?
Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is aK (π, 1) space.