The Lefschetz property for an algebra defined by matchings

TL;DR

This paper investigates an algebra constructed from matchings in a complete graph, proving it possesses the strong Lefschetz property, which has implications in algebraic combinatorics.

Contribution

It introduces a new algebra defined via the generating function of matchings and proves it has the strong Lefschetz property, a significant algebraic feature.

Findings

The algebra is Artinian Gorenstein.

The algebra satisfies the strong Lefschetz property.

The generating function is central to the algebra's structure.

Abstract

In this article, we consider the weighted generating function of matchings in the complete graph. We define an Artinian Gorenstein algebra as the quotient ring of a polynomial ring by the annihilator of the generating function. We show the strong Lefschetz property of the algebra.