# Nasrin Altafi: Jordan types for Artinian algebras of codimension two

Time: Wed 2021-01-13 10.15 - 11.15

Location:

Lecturer: Nasrin Altafi

Abstract: Multiplication by a linear form $$\ell$$ on graded Artinian algebra A determines a nilpotent linear operator on A, the Jordan type partition of this operator is an integer partition of the dimension of A as a vector space. The Jordan type partition is a finer invariant than the weak and strong Lefschetz properties of A. A graded Artinian algebra A is said to satisfy the Lefschetz properties if multiplication map by powers of a linear form $$\ell\in$$ A has maximal rank in various degrees. In this talk, I will describe the connection between the Jordan type and the Lefschetz properties for graded Artinian algebras having arbitrary codimension. In polynomial ring R with two variables, I will describe how we determine which partitions of n may occur as the Jordan type for some linear form $$\ell$$ on a graded complete intersection Artinian quotient A = R/(f, g). I will explain how we obtain such partitions combinatorially and algebraically. As a generalization of this result, I will explain how the combinatorial properties of a fixed partition P, namely the hook code, can be applied to determine the minimal number of generators of a generic Artinian algebra of codimension two with P as its Jordan type for some linear form. This is joint work with A. Iarrobino, L. Khatami, and J. Yam´eogo.