The squish map and the $\text{SL}_2$ double dimer model
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910439110082560 |
|---|---|
| author | Foster, Leigh Young, Benjamin |
| author_facet | Foster, Leigh Young, Benjamin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_03230 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The squish map and the $\text{SL}_2$ double dimer model Foster, Leigh Young, Benjamin Combinatorics 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. |
| title | The squish map and the $\text{SL}_2$ double dimer model |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.03230 |