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Block diagonally symmetric lozenge tilings

Seok Hyun Byun, Yi-Lin Lee

TL;DR

The paper addresses refined symmetry classes for lozenge tilings and boxed plane partitions, introducing the $\mathbf{r}$-block diagonally symmetric class and proving a closed product formula for its weighted generating function $\operatorname{M}_{q,t}^{\mathbf{r}}(H(m,m,n))$, from which the volume generating function for $\mathbf{r}$-block symmetric plane partitions follows. It extends to $(\mathbf{r},\mathbf{r}')$-block diagonally symmetric tilings on a cylinder and provides a signed-enumeration identity. Two complementary proofs are developed: (i) a modified non-intersecting lattice-path method yielding a determinant evaluation, and (ii) a Schur-polynomial approach via the dual Pieri rule, linking tilings to (skew) Schur functions. Together, these results deepen the connections between symmetric tilings, plane partitions, and representation-theoretic objects, and generalize MacMahon-type product formulas to new symmetry classes.

Abstract

We introduce a new symmetry class of both boxed plane partitions and lozenge tilings of a hexagon, called the $\mathbf{r}$-block diagonal symmetry class, where $\mathbf{r}$ is an $n$-tuple of non-negative integers. We prove that the tiling generating function of this symmetry class under a certain weight assignment is given by a simple product formula. As a consequence, the volume generating function of $\mathbf{r}$-block symmetric plane partitions is obtained. Additionally, we consider $(\mathbf{r},\mathbf{r^{\prime}})$-block diagonally symmetric lozenge tilings by embedding the hexagon into a cylinder and present an identity for the signed enumeration of this symmetry class in specific cases. Two methods are provided to study this symmetry class: (1) the method of non-intersecting lattice paths with a modification, and (2) interpreting weighted lozenge tilings algebraically as (skew) Schur polynomials and applying the dual Pieri rule.

Block diagonally symmetric lozenge tilings

TL;DR

The paper addresses refined symmetry classes for lozenge tilings and boxed plane partitions, introducing the -block diagonally symmetric class and proving a closed product formula for its weighted generating function , from which the volume generating function for -block symmetric plane partitions follows. It extends to -block diagonally symmetric tilings on a cylinder and provides a signed-enumeration identity. Two complementary proofs are developed: (i) a modified non-intersecting lattice-path method yielding a determinant evaluation, and (ii) a Schur-polynomial approach via the dual Pieri rule, linking tilings to (skew) Schur functions. Together, these results deepen the connections between symmetric tilings, plane partitions, and representation-theoretic objects, and generalize MacMahon-type product formulas to new symmetry classes.

Abstract

We introduce a new symmetry class of both boxed plane partitions and lozenge tilings of a hexagon, called the -block diagonal symmetry class, where is an -tuple of non-negative integers. We prove that the tiling generating function of this symmetry class under a certain weight assignment is given by a simple product formula. As a consequence, the volume generating function of -block symmetric plane partitions is obtained. Additionally, we consider -block diagonally symmetric lozenge tilings by embedding the hexagon into a cylinder and present an identity for the signed enumeration of this symmetry class in specific cases. Two methods are provided to study this symmetry class: (1) the method of non-intersecting lattice paths with a modification, and (2) interpreting weighted lozenge tilings algebraically as (skew) Schur polynomials and applying the dual Pieri rule.

Paper Structure

This paper contains 10 sections, 60 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. The off-diagonal symmetry class
  4. $\mathbf{r}$-block diagonally symmetric lozenge tilings
  5. $\mathbf{r}$-block symmetric plane partitions
  6. $(\mathbf{r},\mathbf{r}^{\prime})$-block diagonally symmetric lozenge tilings
  7. The method of non-intersecting lattice paths
  8. Schur polynomials and the dual Pieri rule
  9. Skew Schur polynomials and lozenge tilings of trapezoidal regions with dents
  10. Proof of Theorem \ref{['thm:2']}

Figures (11)

  • Figure 1: The correspondence between a boxed plane partition, its 3D realization as a pile of unit cubes, and a lozenge tiling of $H(3,4,4)$ (from left to right).
  • Figure 2: (a) $AD(6)$ with the six cells marked. (b) An off-diagonally symmetric domino tilings of $AD(6)$.
  • Figure 3: (a) $H(6,6,6)$ with six $1$-cells marked. (b) An off-diagonally symmetric lozenge tiling of $H(6,6,6)$.
  • Figure 4: (a) A symmetric hexagon $H(8,8,5)$ with a $2$-cell, a $0$-cell, a $2$-cell, a $1$-cell, and a $3$-cell marked (from bottom to top). (b) A $(2,0,2,1,3)$-block diagonally symmetric lozenge tiling of $H(8,8,5)$.
  • Figure 5: (a) A $(2,0,2,1,3)$-block diagonally symmetric lozenge tiling of $H(8,8,5)$. Due to the symmetry, the tiling is uniquely determined by the lozenges inside the marked subregion. (b) The corresponding lozenge tiling of the region $T(5,8;P)$ where $P=\{1,3,5,7,8,10,12,13\}$. From right to left in each column, there are $3$, $3$, $3$, $2$, $3$, $2$, $4$, and $3$ (shaded) lozenges that are weighted by $x_1,x_2,\dots,x_8$, respectively. All the other lozenges are weighted by $1$.
  • ...and 6 more figures

Theorems & Definitions (6)

  • proof
  • proof
  • proof : First proof of Theorem \ref{['thm:1']}
  • proof
  • proof : Second proof of Theorem \ref{['thm:1']}
  • proof : Proof of Theorem \ref{['thm:2']}