Nancy Abdallah: Lefschetz Properties and Almkvist's Conjecture in two variables

Abstract: Let A(m,n)=R/I, where R is the algebra generated by the elementary symmetric functions in n variables, and I is the ideal generated by the mth power elementary symmetric functions. A(m,n) is then a two-parameter family of non-standard graded complete intersection. In 1989, Almkvist conjectured that the Hilbert function of A(m,n) is unimodal for n odd and sufficiently large m, and for n even and any m. Since unimodality is a necessary condition for Lefschetz properties, we conjecture that A(m, n) has the strong Lefschetz property for sufficiently large m, and for even n. In this talk, we consider the case n = 2, and we show that A(m, 2) has the strong Lefschetz property and the complex Hodge-Riemann property if and only if m is even. This gives an answer to Almkvist's conjecture in the two variables case.