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The characterization of an Alternating Sign Matrix using a triplet

Toyokazu Ohmoto

TL;DR

The paper establishes a cohesive framework to characterize Alternating Sign Matrices (ASMs) by linking them to the six-vertex model and to height functions through a per-entry triplet $({a}_{i,j},{c}_{i,j},{r}_{i,j})$. It then derives a bijection between ASM data and height functions via the explicit formula ${h}_{i,j}=i+j-2\sum_{1\le k\le i,1\le l\le j}{a}_{k,l}$ with an explicit inverse, enabling a transfer of combinatorial structure between ASMs, SV configurations, and height-function matrices. The work further integrates the Fully Packed Loop (FPL) perspective, defining link patterns and operators on them, and develops Wieland gyration and dihedral-symmetry theory to study symmetry-induced refinements of counts. Finally, it constructs a vector-space framework on link patterns, introduces projection and Hamiltonian-like operators, and outlines future research directions including half-turn ASMs and potential connections to root systems. Overall, the paper provides a concrete, triplet-based blueprint to translate ASM enumeration problems into combinatorial and algebraic settings linked to SV, FPL, and height-function formalisms, with symmetry principles guiding structure and potential extensions.

Abstract

Alternating Sign Matrix(ASM for short) is a square matrix which is consist of 0, 1 and -1. In this paper, we characterize an ASM by showing a bijection between alternating sign matrix and six vertex model, and a bijection between six vertex model and height function. In order to show these bijections, we define a triplet $({a}_{i, j}, {c}_{i, j}, {r}_{i, j})$ for each entry of an ASM. We also define a \textsl{track} for each index of height function, and state more properties of height function.

The characterization of an Alternating Sign Matrix using a triplet

TL;DR

The paper establishes a cohesive framework to characterize Alternating Sign Matrices (ASMs) by linking them to the six-vertex model and to height functions through a per-entry triplet . It then derives a bijection between ASM data and height functions via the explicit formula with an explicit inverse, enabling a transfer of combinatorial structure between ASMs, SV configurations, and height-function matrices. The work further integrates the Fully Packed Loop (FPL) perspective, defining link patterns and operators on them, and develops Wieland gyration and dihedral-symmetry theory to study symmetry-induced refinements of counts. Finally, it constructs a vector-space framework on link patterns, introduces projection and Hamiltonian-like operators, and outlines future research directions including half-turn ASMs and potential connections to root systems. Overall, the paper provides a concrete, triplet-based blueprint to translate ASM enumeration problems into combinatorial and algebraic settings linked to SV, FPL, and height-function formalisms, with symmetry principles guiding structure and potential extensions.

Abstract

Alternating Sign Matrix(ASM for short) is a square matrix which is consist of 0, 1 and -1. In this paper, we characterize an ASM by showing a bijection between alternating sign matrix and six vertex model, and a bijection between six vertex model and height function. In order to show these bijections, we define a triplet for each entry of an ASM. We also define a \textsl{track} for each index of height function, and state more properties of height function.
Paper Structure (24 sections, 9 theorems, 47 equations, 25 figures, 1 table)

This paper contains 24 sections, 9 theorems, 47 equations, 25 figures, 1 table.

Table of Contents

  1. Introduction
  2. Alternating sign matrix
  3. Alternating sign matrix (ASM)
  4. Six vertex model and Fully packed loop model
  5. Six vertex model
  6. Fully packed loop model
  7. Plaquette
  8. The correpondence between six vertex model and ASM
  9. Height function
  10. Properties of a boundary condition of six vertex model
  11. The map from six vertex model to matricies
  12. height function
  13. The bijection between ASM and height function
  14. Properties of height functions
  15. Perfect matching and link pattern
  16. ...and 9 more sections

Key Result

Lemma 3.1

Let $m$ and $n$ be positive integers. For any state of six vertex model on ${L}_{m, n}$, the number of bounday edge which comes in the boundary vertex equals $m + n$.

Figures (25)

  • Figure 1: ${L}_{3}$
  • Figure 2: An example of boundary condition
  • Figure 3: An example of a state of Six vertex model
  • Figure 4: an example of FPL on ${L}_{3}$
  • Figure 5: ${\tau}_{+}$ when $n = 3$
  • ...and 20 more figures

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Corollary 3.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 14 more