圆盘中平面网络的内缘向量与Talaska公式的推广
Internal edge vectors on plabic networks in the disk and a generalization of Talaska formula
摘要：
在[42]之后，完全非负格拉斯曼粒子${Gr^{\mbox{TNN}} (k,n)}$中的正向细胞${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$允许在圆盘中的平面双色有向完美网络上通过正权重进行参数化。在[49]中，根据这种网络上的流，得到了表示${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$中的点的矩阵元素的显式公式。[42，49]中的公式定义在边界边缘向量上。在本文中，我们提出了一个扩展这些公式的内部边缘定义为求和路径上给定的有向网络的射线方向的选择衡量的向量。这种规范选择不影响边界边缘矢量，边界边缘矢量生成Postnikov边界测量图。对应于不同选择的规范射线方向的内棱矢量系在符号上一致，符号规则有一个简单明确的描述。我们证明了这些边缘向量的分量是合理的权重与减-自由的分解。此外，这些组件表示的内部边缘流，这些公式扩展了原来的Talaska的内部边缘。这些向量还解决了与相应网络相关联的几何关系系统。这些关系是满秩的，并尊重全正向胞的总非负性。我们还提供了明确的公式的边向量的变换规则的方向，并为他们的转换，由于移动和减少的网络。
英文摘要：
Following [42], positroid cells ${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$ in totally non-negative Grassmannians ${Gr^{\mbox{TNN}} (k,n)}$ admit parametrizations by positive weights on planar bicolored directed perfect networks in the disk. An explicit formula for elements of matrices representing the points in ${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$ was obtained in [49] in terms of flows on such networks. The formulas from [42,49] are defined on the boundary edge vectors. In this paper we propose an extension of these formulas for vectors on internal edges defined as summations over paths on the given directed network gauged by the choice of a ray direction. This gauge choice does not affect the boundary edge vectors, which generate the Postnikov boundary measurement map. The systems of internal edge vectors corresponding to different choices of gauge ray directions coincide up to sign, the sign rule admits a simple explicit description. We prove that the components of these edge vectors are rational in the weights with subtraction--free denominators. Moreover, these components are expressed in terms of internal edge flows; these formulas extend the original Talaska ones to the internal edges. These vectors also solve the system of geometric relations associated to the corresponding network. These relations are full rank and respect the total non--negativity property on the full positroid cell. We also provide explicit formulas both for the transformation rules of the edge vectors with respect to the orientation, and for their transformations due to moves and reductions of networks.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
