Séminaire Lotharingien de Combinatoire, 93B.51 (2025), 12 pp.

Filip Jonsson Kling, Samuel Lundqvist, Fatemeh Mohammadi, Matthias Orth and Eduardo Sáenz-de-Cabezón

Gröbner Bases and the Lefschetz Properties for Powers of a General Linear Form in the Squarefree Algebra

Abstract. For the almost complete intersection ideals (x₁², ..., x_n², (x₁ + ... + x_n)^k), we compute their reduced Gröbner 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.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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