Lattice Paths, Lefschetz Properties, and Almkvist’s Conjecture in Two Variables Nancy Abdallah, Chris McDaniel Algebraic Combinatorics, 2025 We study a certain two-parameter family of non-standard graded complete intersection algebras A(m,n). In case n=2, we show that if m is even then A(m,2) has the strong Lefschetz property and satisfies the complex Hodge–Riemann relations, while if m is odd then A(m,2) satisfies these properties only up to a certain degree. This supports a strengthening of a conjecture of Almkvist on the unimodality of the Hilbert function of A(m,n).
A note on Artin Gorenstein algebras with Hilbert function (1,4,k,k,4,1) Nancy Abdallah Contemporary Mathematics, 2024 We study the free resolutions of some Artin Gorenstein algebras of Hilbert function(1,4,k,k,4,1)(1,4,k,k,4,1)and we prove that all such algebras have the strong Lefschetz property if they have the weak Lefschetz property. In the casek=4k=4we prove that the Hilbert function alone fixes the Betti table. For higherkkstronger conditions on the algebras are needed to fix the Betti table. In particular, if the algebra is a complete intersection or if it is defined by an equigenerated ideal then the Betti table is unique.
