Gr\"obner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra

TL;DR

This paper analyzes the algebraic and combinatorial structure of certain ideals in squarefree algebra, classifies their Lefschetz properties, and constructs minimal free resolutions using Gr"obner bases and lattice path combinatorics.

Contribution

It provides explicit Gr"obner bases, classifies Lefschetz properties, and constructs minimal free resolutions for a class of ideals in squarefree algebra, linking algebraic and combinatorial techniques.

Findings

Computed reduced Gr"obner bases revealing lattice path structures.

Classified weak Lefschetz property for these ideals.

Constructed minimal free resolutions via Mayer-Vietoris trees.

Abstract

For the almost complete intersection ideals $(x_1^2, \dots, x_n^2, (x_1 + \cdots + x_n)^k)$, we compute their reduced Gr\"obner basis for any term ordering, revealing a combinatorial structure linked to lattice paths, elementary symmetric polynomials, and Catalan numbers. Using this structure, we classify the weak Lefschetz property for these ideals. Additionally, we provide a new proof of the well-known result that the squarefree algebra satisfies the strong Lefschetz property. Finally, we compute the Betti numbers of the initial ideals and construct a minimal free resolution using a Mayer-Vietoris tree approach.