The h-Vector of a Gorenstein Toric Ring of a Compressed Polytope
The h-Vector of a Gorenstein Toric Ring of a Compressed Polytope
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DOI：
10.37236/1891
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发表时间：
2005-10
期刊：
影响因子：
--
通讯作者：
Hidefumi Ohsugi;T. Hibi
中科院分区：
文献类型：
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作者：
Hidefumi Ohsugi;T. Hibi
压缩多面体是一个整凸多面体，它的所有拉三角剖分都是幺模的。一个（q1）-单形的每个顶点都是凸多面体P的一个顶点，如果P的每个面恰好包含q1个顶点，则称它是P中的特殊单形。证明了压缩多胞形P中存在特殊单形当且仅当其复曲面环K[P]是Gorenstein环。由此推论，若P是压缩的，则Gorenstein环面环K[P]的h-向量是单峰的。一个压缩多面体[10，p.337]是一个整凸多面体，它的所有“拉三角剖分”都是幺模的。（回想一下，整数凸多面体是凸多面体，其每个顶点都有整数坐标。压缩多面体的一个典型例子是Birkho多面体[10，例2.4（B）]。后来，在[6]中，提出了包括Birkho多面体在内的一大类压缩多面体。最近，Seth Sullivant [12]证明了一个令人惊讶的结果，即[6]中给出的类本质上包含所有压缩多面体。
A compressed polytope is an integral convex polytope all of whose pulling triangulations are unimodular. A (q 1)-simplex each of whose vertices is a vertex of a convex polytope P is said to be a special simplex in P if each facet of P contains exactly q 1 of the vertices of . It will be proved that there is a special simplex in a compressed polytope P if (and only if) its toric ring K[P] is Gorenstein. In consequence it follows that the h-vector of a Gorenstein toric ring K[P] is unimodal if P is compressed. A compressed polytope [10, p. 337] is an integral convex polytope all of whose \pulling triangulations" are unimodular. (Recall that an integral convex polytope is an convex polytope each of whose vertices has integer coordinates.) A typical example of compressed polytopes is the Birkho polytopes [10, Example 2.4 (b)]. Later, in [6], a large class of compressed polytopes including the Birkho polytopes is presented. Recently, Seth Sullivant [12] proved a surprising result that the class given in [6] does essentially contain all compressed polytopes.