Lefschetz Properties and $h$-vectors of Graded Artinian Algebras

  • Uwe Nagel (University of Kentucky, Lexington, USA)
E1 05 (Leibniz-Saal)


A standard graded algebra $A$ over a field is said to have the weak Lefschetz property (WLP) if it contains a degree one element $\ell$ such that multiplication by $\ell$ from one degree component of $A$ to the next always has maximal rank. In this case the Hilbert function or $h$-vector of $A$ is unimodal. This vector records the vector space dimensions of the graded components of $A$. Even in the case that the relations of $A$ are given by monomials such an algebra may fail the WLP or have a non-unimodal $h$-vector. Computer experiments suggests that algebras with these properties are rather rare. We discuss classes of randomly generated monomial algebras whose expected $h$-vectors are unimodal.

01.07.19 04.07.19

Summer School on Randomness and Learning in Non-Linear Algebra

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Felix-Klein-Hörsaal

Saskia Gutzschebauch

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Paul Breiding

Technische Universität Berlin

Jesus De Loera

University of California at Davis

Despina Stasi

Illinois Institute of Technology

Sonja Petrovic

Illinois Institute of Technology