[Paper Review] Enumerations of lozenge tilings, lattice paths, and perfect matchings and the weak Lefschetz property

David Cook, Uwe Nagel|arXiv (Cornell University)|May 6, 2013

Advanced Combinatorial Mathematics57 references4 citations

TL;DR

This paper establishes signed bijections between lozenge tilings, perfect matchings, and non-intersecting lattice paths in balanced triangular regions, proving the equivalence of two sign definitions via a novel resolution of punctures technique. The key contribution is a combinatorial foundation for two determinantal enumerations and their application to the weak Lefschetz property in Artinian monomial algebras over polynomial rings.

ABSTRACT

MacMahon enumerated the plane partitions in an $a imes b imes c$ box. These are in bijection to lozenge tilings of a hexagon, to certain perfect matchings, and to families of non-intersecting lattice paths. In this work we consider more general regions, called triangular regions, and establish signed versions of the latter three bijections. Indeed, we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. A combinatorial argument involving a new method, called resolution of a puncture, then shows that the signs are in fact equivalent. This provides in particular two different determinantal enumerations of these families. These results are then applied to study the weak Lefschetz property of Artinian quotients by monomial ideals of a three-dimensional polynomial ring. We establish sufficient conditions guaranteeing the weak Lefschetz property as well as the semistability of the syzygy bundle of the ideal, classify the type two algebras with the weak Lefschetz property, and study monomial almost complete intersections in depth. Furthermore, we develop a general method that often associates to an algebra that fails the weak Lefschetz property a toric surface that satisfies a Laplace equation. We also present examples of toric varieties that satisfy arbitrarily many Laplace equations. Our combinatorial methods allow us to address the dependence on the characteristic of the base field for many of our results.

Motivation & Objective

• To extend MacMahon's enumeration of plane partitions in $a\times b\times c$ boxes to more general triangular regions beyond hexagons.
• To define and compare two distinct sign assignments for lozenge tilings via perfect matchings and non-intersecting lattice paths.
• To establish a combinatorial equivalence between the two sign definitions using a new technique called 'resolution of a puncture'.
• To apply the resulting signed combinatorics to study the weak Lefschetz property and syzygy bundle stability in Artinian monomial algebras.
• To investigate the dependence of these algebraic properties on the characteristic of the base field using combinatorial tools.

Proposed method

• Define triangular subregions $T \subset \mathcal{T}_d$ by removing upward-pointing triangles (punctures), with balanced regions having equal numbers of up- and down-pointing unit triangles.
• Introduce two sign assignments for lozenge tilings: one via perfect matchings and one via families of non-intersecting lattice paths.
• Develop the 'resolution of a puncture' technique to transform a punctured region into a larger balanced region, enabling a direct comparison of the two sign definitions.
• Use the Gessel-Viennot-Lindström-Stembridge-Krattenthaler theory to express enumerations of signed families of non-intersecting lattice paths as determinants.
• Relate the combinatorial data to algebraic invariants of Artinian monomial algebras, particularly the weak Lefschetz property and syzygy bundle stability.
• Construct a toric surface from algebras failing the weak Lefschetz property that satisfies a Laplace equation, generalizing classical results.

Experimental results

Research questions

• RQ1 Are the two sign definitions for lozenge tilings—via perfect matchings and via non-intersecting lattice paths—equivalent in general triangular subregions?
• RQ2 Can the signed enumeration of lozenge tilings be expressed as a determinant in a way that generalizes MacMahon’s formula beyond hexagonal regions?
• RQ3 What conditions guarantee the weak Lefschetz property for Artinian monomial algebras in three variables?
• RQ4 How does the failure of the weak Lefschetz property relate to geometric structures such as toric surfaces satisfying Laplace equations?
• RQ5 To what extent do the algebraic properties of monomial algebras depend on the characteristic of the base field?

Key findings

• The two sign definitions for lozenge tilings—based on perfect matchings and non-intersecting lattice paths—are equivalent via the resolution of punctures method.
• The number of signed perfect matchings and signed non-intersecting lattice paths in a balanced triangular region are both enumerated by the same determinant, providing two distinct determinantal formulas.
• For Artinian monomial algebras of type two in three variables, the paper classifies all such algebras with the weak Lefschetz property.
• Sufficient conditions are established for the weak Lefschetz property and for the syzygy bundle of the ideal to be semistable.
• When an algebra fails the weak Lefschetz property, a toric surface satisfying a Laplace equation can be canonically associated, and examples of such surfaces satisfying arbitrarily many Laplace equations are constructed.
• The results are characteristic-dependent: the paper explicitly analyzes the role of the base field’s characteristic in the validity of the weak Lefschetz property and related invariants.