论文标题
热带完全积极的集群品种
Tropical totally positive cluster varieties
论文作者
论文摘要
我们研究集群品种的整数热带点（满足完整的goncharov猜想）与理想的热带化的完全积极的部分，呈现相应的群集代数。假设我们通过khovanskii的基础给出了群集代数的介绍，以收集$ {\ bf g} $ - 与突变相关的几种种子相关的矢量估值。在完全秩的完全扩展的交换矩阵的情况下，我们构建了理想的完全积极部分的子粉丝的光线，该射线与与种子集合相对应的集群代数的交换图的子图相结合。此外，通过在热带化中从最大锥体获得的gröbner曲折变性，可以鉴定出有关与种子相关的毛keel-keel-kontsevich的复曲面变性的几何信息。作为应用，我们证明了关于Rietsch-Williams对Grassmannians的关系的猜想，该估值是由Plabic Graphs \ Cite \ cite {RW17}到Kaveh-Manon对估值的工作，从理想\ Cite {KM16}的热带化中获得的估值。在第二个应用程序中，我们对问题的部分答案是，如果$ \ mathtt类型中的全旗品种的Feigin-fourier-littelmann-Vinberg变性对于从群集结构中获得的变性是同构的。
We study the relation between the integer tropical points of a cluster variety (satisfying the full Fock-Goncharov conjecture) and the totally positive part of the tropicalization of an ideal presenting the corresponding cluster algebra. Suppose we are given a presentation of the cluster algebra by a Khovanskii basis for a collection of ${\bf g}$-vector valuations associated with several seeds related by mutations. In presence of a full rank fully extended exchange matrix we construct the rays of a subfan of the totally positive part of the tropicalization of the ideal that coincides combinatorially with the subgraph of the exchange graph of the cluster algebra corresponding to the collection of seeds. Moreover, geometric information about Gross-Hacking-Keel-Kontsevich's toric degenerations associated with seeds gets identified with the Gröbner toric degenerations obtained from maximal cones in the tropicalization. As application we prove a conjecture about the relation between Rietsch-Williams' valuations for Grassmannians arising from plabic graphs \cite{RW17} to Kaveh-Manon's work on valuations from the tropicalization of an ideal \cite{KM16}. In a second application we give a partial answer to the question if the Feigin-Fourier-Littelmann-Vinberg degeneration of the full flag variety in type $\mathtt A$ is isomorphic to a degeneration obtained from the cluster structure.
