Combinatorics Seminar


Speaker: Nasrin Altafi

Title: Jordan types for Artinian algebras of codimension two

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. Yaméogo.


Speaker: Wencin Poh

Title: Characterization of queer supercrystals

Abstract: We provide a characterization of the crystal bases for the queer Lie superalgebra introduced by Grantcharov et al.. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for simply-laced root systems, which were introduced by Assaf and Oguz, with further axioms and a new graph G characterizing the relations of the type A components of the queer supercrystal. We provide a counterexample to Assaf's and Oguz' conjecture that the local queer axioms uniquely characterize the queer supercrystal. We obtain a combinatorial description of the graph G on the type A components by providing explicit combinatorial rules for the odd queer operators on certain highest weight elements. This talk is based on the joint work with Maria Gillespie, Graham Hawkes and Anne Schilling.