论文标题
估值和概括的普遍定义
Valuations and the Hopf Monoid of Generalized Permutahedra
论文作者
论文摘要
本文的目的是表明估值理论和HOPF理论在广义定居者的类别中兼容。我们证明了这些多面体降低的HOPF结构$ \ MATHBF {GP}^+$ modulo the Canne-Oxclusion与指标hopf hopf hopf hopf hopf hopf hopf hopf noid $ \ mathbb {i}（\ mathbf {gp}^+）的概括性定位均为订单，均为订购均值。此商Hopf Monoid $ \ Mathbb {i}（\ Mathbf {gp}^+）$是cof的。它是带有多项式字符的Hopf Monoid类别中的终端对象。这部分解释了霍普夫·莫尼德（Hopf Monoids）理论中广义定位的无处不在。 这个HOPF理论框架为许多有关广义定居者及其亚家族的新估值提供了简单，统一的解释。示例包括用于矩形：Chern-Schwartz-Macpherson Cycles，Eur的体积多项式，Kazhdan-Lusztig多项式，动机Zeta函数和Derksen-Fink不变性；对于POSET：多项式，庞加莱多项式和Poset Tutte多项式；对于广义定居者：通用的tutte特征和固定型圆环的相应类。 我们获得了几个代数和组合推论。例如：$ \ mathbf {gp}^+$的估值角色组的存在，以及巢增载体的不可分性性化为较小的巢穴。
The goal of this paper is to show that valuation theory and Hopf theory are compatible on the class of generalized permutahedra. We prove that the Hopf structure $\mathbf{GP}^+$ on these polyhedra descends, modulo the inclusion-exclusion relations, to an indicator Hopf monoid $\mathbb{I}(\mathbf{GP}^+)$ of generalized permutahedra that is isomorphic to the Hopf monoid of weighted ordered set partitions. This quotient Hopf monoid $\mathbb{I}(\mathbf{GP}^+)$ is cofree. It is the terminal object in the category of Hopf monoids with polynomial characters; this partially explains the ubiquity of generalized permutahedra in the theory of Hopf monoids. This Hopf theoretic framework offers a simple, unified explanation for many new and old valuations on generalized permutahedra and their subfamilies. Examples include, for matroids: the Chern-Schwartz-MacPherson cycles, Eur's volume polynomial, the Kazhdan-Lusztig polynomial, the motivic zeta function, and the Derksen-Fink invariant; for posets: the order polynomial, Poincaré polynomial, and poset Tutte polynomial; for generalized permutahedra: the universal Tutte character and the corresponding class in the Chow ring of the permutahedral variety. We obtain several algebraic and combinatorial corollaries; for example: the existence of the valuative character group of $\mathbf{GP}^+$, and the indecomposability of a nestohedron into smaller nestohedra.
