论文标题
辫子布置双子体和渗透素体的复曲面品种
Braid arrangement bimonoids and the toric variety of the permutohedron
论文作者
论文摘要
我们表明，置换式体的复曲面品种（=置换界空间）具有物种中的共同交流性双子体的结构，并通过嵌入/投影 - 单位边界分隔剂给出了乘法/集合。就Losev-Manin对置换空间作为模量空间的描述而言，乘法是Riemann Spheres和Comultiplication字符串的串联，这是忘记了明显的点。以这种方式，双子体结构是零标记曲线属的模量空间上的循环作战结构的类似物。通过（CO）乘法推动向前/拉回的数据，可以通过（CO）乘法推动向前/拉回数据来赋予COCOMOTATIVE/Computative Bimonoids的结构。许多众所周知的组合对象索引置换域空间的数据。此外，组合对象通常具有双子体的结构，并通过以某种方式合并/限制对象给出乘法/合并。我们证明，这些索引组合物体所享受的双子型结构与置换式空间的双子体结构引起的组合物体相吻合。因此，渗透界空间可以被视为一种基本的基本对象，该对象几何地解释了许多组合霍普夫代数。 Aguiar-Mahajan表明，经典的组合HOPF理论是基于辫子超平面布置的。本文的目的是类似地建立置换式的空间作为中心对象，提供更加统一的视角。这项工作的主要动机涉及Schwinger参数化中的Feynman振幅，如果一个人在奇异点的分辨率中炸毁了所有内容，则在置换方面成为积分。然后，Feynman图的Hopf代数结构首先出现在Connes-Kreimer的工作中，与由固定式海德空间的双子体结构诱导的相吻合。
We show that the toric variety of the permutohedron (=permutohedral space) has the structure of a cocommutative bimonoid in species, with multiplication/comultiplication given by embedding/projecting-onto boundary divisors. In terms of Losev-Manin's description of permutohedral space as a moduli space, multiplication is concatenation of strings of Riemann spheres and comultiplication is forgetting marked points. In this way, the bimonoid structure is an analog of the cyclic operad structure on the moduli space of genus zero marked curves. Covariant/contravariant data on permutohedral space is endowed with the structure of cocommutative/commutative bimonoids by pushing-forward/pulling-back data along the (co)multiplication. Many well-known combinatorial objects index data on permutohedral space. Moreover, combinatorial objects often have the structure of bimonoids, with multiplication/comultiplication given by merging/restricting objects in some way. We prove that the bimonoid structure enjoyed by these indexing combinatorial objects coincides with that induced by the bimonoid structure of permutohedral space. Thus, permutohedral space may be viewed as a fundamental underlying object which geometrically interprets many combinatorial Hopf algebras. Aguiar-Mahajan have shown that classical combinatorial Hopf theory is based on the braid hyperplane arrangement in a crucial way. This paper aims to similarly establish permutohedral space as a central object, providing an even more unified perspective. The main motivation for this work concerns Feynman amplitudes in the Schwinger parametrization, which become integrals over permutohedral space if one blows-up everything in the resolution of singularities. Then the Hopf algebra structure of Feynman graphs, first appearing in the work of Connes-Kreimer, coincides with that induced by the bimonoid structure of permutohedral space.
