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Double boxes and double dimers

Tatyana Benko, Benjamin Young

Abstract

We give a combinatorial proof of a result in rank 2 Donaldson-Thomas theory, which states that the generating function for certain plane-partition-like objects, called double-box configurations, is equal to a product of MacMahon's generating function for (boxed) plane partitions. In our proof, we first give the correspondence between double-box configurations and double-dimer configurations on the hexagon lattice with a particular tripartite node pairing. Using this correspondence, we apply graphical condensation and double-dimer condensation to prove the result.

Double boxes and double dimers

Abstract

We give a combinatorial proof of a result in rank 2 Donaldson-Thomas theory, which states that the generating function for certain plane-partition-like objects, called double-box configurations, is equal to a product of MacMahon's generating function for (boxed) plane partitions. In our proof, we first give the correspondence between double-box configurations and double-dimer configurations on the hexagon lattice with a particular tripartite node pairing. Using this correspondence, we apply graphical condensation and double-dimer condensation to prove the result.

Paper Structure

This paper contains 6 sections, 2 theorems, 19 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Double-box configurations
  4. Tripartite double-dimer configurations
  5. Mapping double-box to double-dimer
  6. Condensation relation

Key Result

Theorem 1

Let $a,b,c \in \mathbb{N}$, then where $Z_{DBC}^{a,b,c}(q)$ denotes the generating function for double-box configurations, and is MacMahon's generating function for plane partitions, and is MacMahon's generating function for boxed $a \times b \times c$ plane partitions.

Figures (7)

  • Figure 1: Folklore bijection between plane partitions (left-most) and dimer configurations on the hexagon graph (right-most)
  • Figure 2: Double-dimer configuration on the hexagon graph (right-most) from two single-dimer configurations
  • Figure 3: Basepoints of plane partitions $\eta_1, \eta_2, \eta_3$ in $\mathbb{R}^3$
  • Figure 4: Example of $\eta \neq \tilde{\eta}$ with $[\eta] = [\tilde{\eta}]$.
  • Figure 5: Example of a double-box configuration.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 1
  • Example 2
  • Definition 5
  • Definition 6
  • Theorem 2