Enriched order polytopes and enriched Hibi rings
Enriched order polytopes and enriched Hibi rings
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DOI：
10.1007/s40879-020-00403-2
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发表时间：
2020-03
影响因子：
0.6
通讯作者：
Hidefumi Ohsugi;Akiyoshi Tsuchiya
中科院分区：
文献类型：
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作者：
Hidefumi Ohsugi;Akiyoshi Tsuchiya
Stanley引入了两类与偏序集相关的格多面体，称为偏序集P的序多面体O _P OP和链多面体C _P CP.已知给定偏序集P，O _P OP和C _P CP的Ehrhart多项式等于P的计数P-划分的序多项式.本文引入偏序集P的扩充序多胞形，并利用Gröbner基理论证明了它是一个自反多胞形，其Ehrhart多项式等于P的扩充链多胞形的Ehrhart多项式和P的左扩充序多胞形的Ehrhart多项式，从而计数左扩充P-划分.富序多面体的环面环称为富Hibi环。事实证明，丰富的Hibi环是正常的，Gorenstein和Koszul。上述结果意味着在和的伸缩格点之间存在一个双射。对于这样的双射，我们给出了丰富序和链多面体的刻面表示。
Stanley introduced two classes of lattice polytopes associated to posets, which are called the order polytope O _P OP and the chain polytope C _P CP of a poset P. It is known that, given a poset P, the Ehrhart polynomials of O _P OP and C _P CP are equal to the order polynomial of P that counts the P-partitions. In this paper, we introduce the enriched order polytope of a poset P and show that it is a reflexive polytope whose Ehrhart polynomial is equal to that of the enriched chain polytope of P and the left enriched order polynomial of P that counts the left enriched P-partitions, by using the theory of Gröbner bases. The toric rings of enriched order polytopes are called enriched Hibi rings. It turns out that enriched Hibi rings are normal, Gorenstein, and Koszul. The above result implies the existence of a bijection between the lattice points in the dilations of and. Towards such a bijection, we give the facet representations of enriched order and chain polytopes.