Abstract

Generalized permutohedra are deformations of regular permutohedra and arise in many different fields of mathematics. One important characterization of generalized permutohedra is the Submodularity Theorem, which is related to the deformation cone of the Braid fan. We lay out general techniques for determining deformation cones of a fixed polytope and apply it to the Braid fan to obtain a natural combinatorial proof for the Submodularity Theorem. We also consider a refinement of the Braid fan, called the nested Braid fan, and construct usual (respectively, generalized) nested permutohedra that have the nested Braid fan as (respectively, a coarsening of) their normal fan. We extend many results on generalized permutohedra to this new family of polytopes, including a one-to-one correspondence between faces of nested permutohedra and chains in ordered partition posets, and a theorem analogous to the Submodularity Theorem. Finally, we show that the nested Braid fan is the barycentric subdivision of the Braid fan, which gives another way to construct this new combinatorial object.

中文翻译：

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嵌套编织扇的变形锥

摘要

广义 permutohedra 是规则 permutohedra 的变形，出现在许多不同的数学领域。广义 permutohedra 的一个重要表征是 Submodularity Theorem，它与 Braid fan 的变形锥有关。我们列出了确定固定多面体变形锥的一般技术，并将其应用于编织扇，以获得子模定理的自然组合证明。我们还考虑了 Braid fan 的改进，称为嵌套 Braid fan，并构建通常的（分别为广义的）嵌套 permutohedra，其中嵌套的 Braid 风扇（分别为粗化）它们的正常风扇。我们将广义 permutohedra 的许多结果扩展到这个新的多面体家族，包括嵌套 permutohedra 的面和有序分区中的链之间的一一对应，以及类似于 Submodularity Theorem 的定理。最后，我们展示了嵌套的 Braid fan 是 Braid fan 的重心细分，这为构建这个新的组合对象提供了另一种方法。