squish 映射与 $\ ext{SL}_2$ 双二聚体模型
The squish map and the $\text{SL}_2$ double dimer model
AI总结：
本文将平面分拆的粗化操作联系于 squish 映射，构造蜂窝图二聚体模型到粗图 $\ ext{SL}_2$ 双二聚体模型的保测映射，并据此得到平面分拆对的 $(-1)$-枚举生成函数。
摘要：
平面分拆的三维 Young 图由单位立方体组成，它可以由边长为 2 的立方体构成的“更粗”平面分拆来逼近。事实上，通过向最近立方体“向上取整”或“向下取整”，可以得到两种这样的逼近。我们将这种粗化（或下采样）操作与第二作者在早期工作中引入的 squish 映射联系起来。我们展示了蜂窝图上的二聚体模型与更粗蜂窝图上的 $\ ext{SL}_2$ 双二聚体模型之间一个相关的保测映射；我们计算了该映射最有趣的特殊情形，它与具有 2-周期权重的平面分拆 $q$-枚举有关。作为应用，我们将权重特殊化为某些单位根，得到了新颖的生成函数（有些已知，有些全新，有些则是猜想性的），这些函数根据二聚体构型如何相互作用，对某些类别的平面分拆对进行 $(-1)$-枚举。
英文摘要：
A plane partition, whose 3D Young diagram is made of unit cubes, can be approximated by a ``coarser" plane partition, made of cubes of side length 2. Indeed, there are two such approximations obtained by ``rounding up" or ``rounding down" to the nearest cube. We relate this coarsening (or downsampling) operation to the squish map introduced by the second author in earlier work. We exhibit a related measure-preserving map between the dimer model on the honeycomb graph, and the $\text{SL}_2$ double dimer model on a coarser honeycomb graph; we compute the most interesting special case of this map, related to plane partition $q$-enumeration with 2-periodic weights. As an application, we specialize the weights to be certain roots of unity, obtain novel generating functions (some known, some new, and some conjectural) that $(-1)$-enumerate certain classes of pairs of plane partitions according to how their dimer configurations interact.
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