Tue, December 29, 2020
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6:00pm [6:30pm] Anthony Iarrobino, Northeastern University, Boston, MA
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).