AI 中文总结
该研究针对A型双Bruhat胞上的约化词Berenstein-Fomin-Zelevinsky丛图谱深轨迹展开，推导其结构定理与递推公式，完成低秩算例并探讨其一般性质。
AI 中文摘要
我们研究A型双Bruhat胞上约化词Berenstein-Fomin-Zelevinsky丛图谱的深轨迹。根据CGSS24中的定义，对于丛簇X上的一个种子集合，深轨迹是所有对应丛环面并集的补集。我们聚焦于G=SL(n)时的双Bruhat胞G(u,v)。我们的主要结果是若干描述这些深轨迹结构的定理，包括它们与Poisson结构的相互作用、不同胞的深轨迹之间的关系，以及一个递推公式，该公式允许我们从低维胞的约化词深轨迹计算出目标胞的约化词深轨迹。本文还包含SL(3)中双Bruhat胞以及SL(4)中Borel双Bruhat胞的显式低秩计算结果。最后，我们讨论了这些结果与深轨迹若干一般性质之间的关系。
英文摘要
We study deep loci for the reduced-word Berenstein-Fomin-Zelevinsky cluster atlas on type-A double Bruhat cells. As defined in CGSS24, for a seed collection on a cluster variety X, the deep locus is the complement of the union of all corresponding cluster tori. We focus on double Bruhat cells G(u,v) for G = SL(n). Our main results are several theorems describing the structure of these deep loci, including their interaction with the Poisson structure, the relationships between the deep loci of different cells, and a tracking formula that allows us to compute the reduced-word deep locus of a cell from a lower-dimensional one. The paper also includes explicit low-rank calculations for double Bruhat cells in SL(3) and Borel double Bruhat cells in SL(4). Finally, we discuss the relationship between these results and some general properties of deep loci.
