Filip Jonsson Kling: The strong Lefschetz property via Gröbner bases and lattice paths

Abstract.

In this talk, we will sketch part of a new proof that an algebra defined by the ideal generated by the squares of all variables of a polynomial ring do have the strong Lefschetz property. This will be done by finding a nicely described Gröbner bases for a family of related ideals. One surprising fact about these Gröbner basis is that all elements have 0,1-coefficients, allowing for combinatorial interpretations to be made. In particular, we will show how the initial terms of the Gröbner basis elements enumerate a family of lattice paths and how this enumeration will aid us in establishing the strong Lefschetz property. Finally, we will show how the Catalan numbers and related sequences appear everywhere when studying these ideals.

This is based on joint work with Samuel Lundqvist, Fatemeh Mohammadi, Matthias Orth and Eduardo Sáenz-de-Cabezón.