Talk 1*Speaker: *Anthony Iarrobino, Northeastern University, Boston, MA *
Date/Time: *29 December 2020, 6:30pm IST/ 1:00pm GMT/ 8:00am EST* (joining
time: 6:15 pm IST - 6:30 pm IST).
Google meet link:
https://meet.google.com/tih-ghos-zzg
Title: *Jordan type and Lefschetz properties for Artinian algebras*
Abstract: The Jordan type of a pair (A,x), where x is in the maximum ideal
of a standard graded Artinian algebra A, is the partition P giving the
Jordan block decomposition of the multiplication map by x on A. When A is
Artinian Gorenstein, we say that (A,x) is weak Lefschetz if the number of
parts in the Jordan type P_x is the Sperner number of A – the highest value
of the Hilbert function H(A). We say that (A,x) is strong Lefschetz if P_x
is the conjugate of the Hilbert function.
Weak and strong Lefschetz properties of A for a generic choice of x have
been studied, due to the connection with topology and geometry, where A is
the cohomology ring of a topological space or a variety X. We discuss some
of the properties of Jordan type, and its use as an invariant of A, its
behavior for tensor products and free extensions (defined by T. Harima and
J. Watanabe).
If there is time, we will discuss an application to the study of local
Artinian Gorenstein algebras of fixed Hilbert function H; in recent work
with Pedro Macias Marques we show that in codimension three the properties
of Jordan type and of symmetric decompositions show that certain families
Gor(H) in codimension three or greater have several irreducible components.
The first part of the talk is based on work with Chris McDaniel and Pedro
Marques (arXiv:math.AC/1802.07383, to appear JCA).