论文标题
Hopf有序的简单复合物
Hopf monoids of ordered simplicial complexes
论文作者
论文摘要
我们从HOPF理论的角度研究了纯有序的简单络合物（即，在地面组合的线性秩序上具有线性顺序的简单络合物）。我们将A \ TextIt {Hopf Class}定义为一个纯有序的简单复合物的家族，从而在连接和删除/收缩下引起Hopf Monoid。典型的HOPF类是有序的矩阵家族。 HOPF类的想法使我们能够对与矩形相关的简单复合物进行系统的研究，包括转移的复合物，破碎电路络合物和\ textit {未结合的矩阵}（由无绑定的概括性化的定位螺旋体引起，具有0/1坐标）。 我们在两种情况下计算对抗植物：\ textit {facet-initial complectes}（比移位的复合物大得多）和无界有序的矩形。在后一种情况下，我们将有序的矩阵的Hopf嵌入到有序的广义置换螺旋体的HOPF中，从而使我们能够使用Aguiar和Ardila的拓扑方法来计算Antipode。我们称之为\ textit {scrope complextes}的某些辅助简单复合物的出现使计算变得复杂，它们的欧拉特征控制着对反座的某些系数。所得的反模型公式无数次，无多重性。
We study pure ordered simplicial complexes (i.e., simplicial complexes with a linear order on their ground sets) from the Hopf-theoretic point of view. We define a \textit{Hopf class} to be a family of pure ordered simplicial complexes that give rise to a Hopf monoid under join and deletion/contraction. The prototypical Hopf class is the family of ordered matroids. The idea of a Hopf class allows us to give a systematic study of simplicial complexes related to matroids, including shifted complexes, broken-circuit complexes, and \textit{unbounded matroids} (which arise from unbounded generalized permutohedra with 0/1 coordinates). We compute the antipodes in two cases: \textit{facet-initial complexes} (a much larger class than shifted complexes) and unbounded ordered matroids. In the latter case, we embed the Hopf monoid of ordered matroids into the Hopf monoid of ordered generalized permutohedra, enabling us to compute the antipode using the topological method of Aguiar and Ardila. The calculation is complicated by the appearance of certain auxiliary simplicial complexes that we call \textit{Scrope complexes}, whose Euler characteristics control certain coefficients of the antipode. The resulting antipode formula is multiplicity-free and cancellation-free.
