The squish map and the $\text{SL}_2$ double dimer model

Abstract

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.