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Colored Vertex Models and Interacting Reverse Plane Partitions

Jonah Guse, David Jiang, David Keating

TL;DR

This work connects the combinatorics of reverse plane partitions (RPP) with integrable multicolored vertex models by constructing a 2-color Yang–Baxter framework that encodes interacting pairs of RPPs of the same shape. The authors establish a product-form generating function for volume-weighted interacting RPPs with interaction parameter $t$, and show that setting $t=0$ collapses the model to single RPPs, with a bijection proving equinumerosity by total volume. A detailed development of white/gray row weights, border-strip path formulations, and border-strip–based bijections underpins the main theorem and its $t\to 0$ limit. The results provide explicit generating functions and a concrete combinatorial mechanism (via border-strip slides) to relate coupled RPPs to ordinary RPPs, linking lozenge tilings and symmetric-function–oriented methods to integrable vertex-model techniques. Overall, the paper advances the interplay between combinatorial representation theory, tiling models, and exactly solvable lattice models, with potential implications for Schur processes and related areas of mathematical physics.

Abstract

We study the coupling of pairs of reverse plane partitions of the same shape by assigning a certain local interaction between the reverse plane partitions. We show that they are in bijection with a certain Yang-Baxter integrable colored vertex model. By utilizing the Yang-Baxter equation for this colored vertex model, we are able to compute the generating function for the interacting pairs of reverse plane partitions. We also give a bijection between the coupled pairs of reverse plane partitions with the interaction strength set to zero and a single reverse plane partition of the same shape.

Colored Vertex Models and Interacting Reverse Plane Partitions

TL;DR

This work connects the combinatorics of reverse plane partitions (RPP) with integrable multicolored vertex models by constructing a 2-color Yang–Baxter framework that encodes interacting pairs of RPPs of the same shape. The authors establish a product-form generating function for volume-weighted interacting RPPs with interaction parameter , and show that setting collapses the model to single RPPs, with a bijection proving equinumerosity by total volume. A detailed development of white/gray row weights, border-strip path formulations, and border-strip–based bijections underpins the main theorem and its limit. The results provide explicit generating functions and a concrete combinatorial mechanism (via border-strip slides) to relate coupled RPPs to ordinary RPPs, linking lozenge tilings and symmetric-function–oriented methods to integrable vertex-model techniques. Overall, the paper advances the interplay between combinatorial representation theory, tiling models, and exactly solvable lattice models, with potential implications for Schur processes and related areas of mathematical physics.

Abstract

We study the coupling of pairs of reverse plane partitions of the same shape by assigning a certain local interaction between the reverse plane partitions. We show that they are in bijection with a certain Yang-Baxter integrable colored vertex model. By utilizing the Yang-Baxter equation for this colored vertex model, we are able to compute the generating function for the interacting pairs of reverse plane partitions. We also give a bijection between the coupled pairs of reverse plane partitions with the interaction strength set to zero and a single reverse plane partition of the same shape.

Paper Structure

This paper contains 22 sections, 17 theorems, 58 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions
  4. Integer partitions and Young Diagram
  5. Interlacing Partitions
  6. Maya Diagram
  7. Hook lengths
  8. Plane Partitions
  9. Reverse Plane Partitions
  10. Five Vertex Models
  11. Yang Baxter Equation
  12. A Vertex Model for Reverse Plane Partition
  13. The Generating Function
  14. An alternate path formulation
  15. Multicolored vertex models and interacting RPP
  16. ...and 7 more sections

Key Result

Theorem 2.15

If we fix $\lambda$ the the generating function of volume weighted RPP of shape $\lambda$ is given by where the product is over all cells of the Young diagram of $\lambda$ and $h_{\lambda}(c)$ is the hook length of the cell $c$ in $\lambda$.

Figures (5)

  • Figure 1: Allowed vertices and their weights for the white vertices.
  • Figure 2: The allowed vertices and their weights for the gray vertices.
  • Figure 3: Cross Weights
  • Figure 4: The allowed local configurations and the corresponding weights for the 2-color white vertices.
  • Figure 5: The allowed local configurations and the corresponding weights for the 2-color gray vertices.

Theorems & Definitions (60)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Definition 2.9
  • Example 2.10
  • ...and 50 more