Valuations and the Hopf Monoid of Generalized Permutahedra
Valuations and the Hopf Monoid of Generalized Permutahedra
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发表时间：
2020-10
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通讯作者：
Federico Ardila;Mario Sánchez
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作者：
Federico Ardila;Mario Sánchez
我们证明了广义置换体的Hopf结构下降，模的包含-排除关系，到一个指标的广义置换体的Hopf幺半群是同构的加权有序集分区的Hopf幺半群。这个霍普夫理论结构为许多已知的关于广义置换面体的赋值提供了一个简单、统一的解释;这些包括Chern-Schwartz-MacPherson循环、体积多项式、拟阵的Derksen-Fink不变量、偏序集的阶多项式、偏序集Tutte多项式、Poincare多项式、广义置换面体的泛Tutte特征。我们得到几个组合推论，例如，证明了一个nestohedron不能细分成更小的nestohedra。
We prove that the Hopf structure on generalized permutahedra descends, modulo the inclusion-exclusion relations, to an indicator Hopf monoid of generalized permutahedra that is isomorphic to the Hopf monoid of weighted ordered set partitions. This Hopf theoretic construction offers a simple, unified explanation for many of the known valuations on generalized permutahedra; these include the Chern-Schwartz-MacPherson cycle, the volume polynomial, and the Derksen-Fink invariant of a matroid, the order polynomial, poset Tutte polynomial, and Poincare polynomial of a poset, and the universal Tutte character of generalized permutahedra. We obtain several combinatorial corollaries; for example, a proof that a nestohedron cannot be subdivided into smaller nestohedra.