发表机构
Department of Mathematics, University of Michigan（密歇根大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究双重 Bott-Samelson 簇上的开簇环面，将其等同于带权 plabic 栅栏，将簇变量解释为广义最小匹配，并推导出 Chamber Ansatz 公式。
AI 中文摘要
固定两个正辫子词 $\eta_+,\eta_-$，并令 $\ ext{Conf}(\eta_+,\eta_-)$ 为相应的（A 型）双重 Bott-Samelson 簇。设 $\eta$ 为一个包含 $\eta_+,\eta_-$ 作为顶部和底部词的双重辫子词。我们考虑与 arXiv:1904.07992 中 $\ ext{Conf}(\eta_+,\eta_-)$ 的三角剖分 $C_\eta$ 相关联的开簇环面 $T(C_\eta)$，并将它们等同于带权 plabic 栅栏，其中平面二分图上的通常局部移动自然对应于双重 Bott-Samelson 簇中环面坐标的变化。此外，$T(C_\eta)$ 可以通过某些矩阵乘积显式参数化，并且还具有由簇变量给出的单项式坐标，这些簇变量是矩阵子式。我们将这些子式解释为带权 plabic 栅栏上完美匹配的广义概念，这些栅栏可能不是约化的 plabic 图。利用这一解释，我们证明簇变量由“广义最小匹配”给出，推广了 arXiv:1606.08383 中的最小匹配。最后，我们通过二聚体理论中的面交替积推导出双重 Bott-Samelson 簇的 Chamber Ansatz 公式。一般而言，plabic 栅栏不是约化的 plabic 图，但我们能够将标准工具（如 plabic 图上的局部移动、trips 和最小匹配）扩展并应用于它们。
英文摘要
Fix two positive braid words $β_+,β_-$, and let $\text{Conf}(β_+,β_-)$ be the corresponding (type A) double Bott-Samelson variety. Let $β$ be a double braid word containing $β_+,β_-$ as the top, bottom words. We consider the open cluster torus $T(C_β)$ associated to a triangulation $C_β$ in $\text{Conf}(β_+,β_-)$ from arXiv:1904.07992, and we identify these with weighted plabic fences, where the usual local moves on planar bipartite graphs naturally correspond to change of torus coordinates in double Bott-Samelson variety. Moreover, $T(C_β)$ can be parametrized explicitly by certain matrix products, and also has monomial coordinates given by the cluster variables, which are matrix minors. We interpret these minors as a generalized notion of perfect matchings on weighted plabic fences, which may not be reduced plabic graphs. Using this interpretation, we show that the cluster variables are given by "generalized minimal matchings", extending the minimal matchings from arXiv:1606.08383. Lastly, we derive a Chamber Ansatz formula for double Bott-Samelson varieties via face alternating products from dimer theory. In general, plabic fences are not reduced plabic graphs, yet we are able to extend and apply the standard tools such as local moves on plabic graphs, trips, and minimal matchings to them.
Comments33 pages, 36 figures, comments welcome
