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 SL2 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.
@article{10_37236_12447,
author = {Leigh Foster and Benjamin Young},
title = {The squish map and the {\(\mathrm{SL}_2\)} double dimer model},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {1},
doi = {10.37236/12447},
zbl = {1536.05035},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12447/}
}
TY - JOUR
AU - Leigh Foster
AU - Benjamin Young
TI - The squish map and the \(\mathrm{SL}_2\) double dimer model
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/12447/
DO - 10.37236/12447
ID - 10_37236_12447
ER -
%0 Journal Article
%A Leigh Foster
%A Benjamin Young
%T The squish map and the \(\mathrm{SL}_2\) double dimer model
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/12447/
%R 10.37236/12447
%F 10_37236_12447
Leigh Foster; Benjamin Young. The squish map and the \(\mathrm{SL}_2\) double dimer model. The electronic journal of combinatorics, Tome 31 (2024) no. 1. doi: 10.37236/12447
