## Combinatorial Aspects of Nilpotent Orbits

Friday April 20, 2018
Northeastern University
One-day conference preceding the AMS special session with same title
Organizers: Leila Khatami, Juianna Tymoczko, Anthony Iarrobino

All talks will take place in 544 Nightingale Hall, which is around the corner from 553 Lake Hall

9:30 AM: Gathering: tea, crumpets (553 Lake Hall)
10 - 10:45 AM: Discussion: short intros/problems by participants (544 Nightingale Hall)

11 AM - 12:15 PM: Survey on the questions related to coordinate rings of nilpotent orbit closures
Jerzy Weyman (University of Connecticut)
Abstract: In this talk I will survey the results and open problems on the normality and defining equations of coordinate rings of nilpotent orbit closures. I will also discuss similar questions for Vinberg representations, i.e. a class of representations with finitely many orbits.

12:30 - 1 PM: K-theoretic crystals on Grothendieck polynomials
Oliver Pechenik (University of Michigan)
Abstract: Classically, Schur polynomials simultaneously represent both irreducible representations of GL_n and Schubert classes in the cohomology of Grassmannians. The analogous objects, moving from cohomology to K-theory, are Grothendieck polynomials; however, we don't have a corresponding representation-theoretic interpretation of them. Using the set-valued tableaux of Buch, we build a crystal structure on Grothendieck polynomials, yielding a new combinatorial formula for their Schur expansions. We consider how to extend this crystal by extra "K-theoretic" Kashiwara operators to study the K-theoretic deformations of Demazure characters introduced by Lascoux and Kirillov. (Joint work with Cara Monical and Travis Scrimshaw.)

2:30 - 3:30 PM: Remarks on combinatorial and algebro-geometric properties of orbit closures of Dynkin quivers
Abstract: Let A be a graded Artinian algebra. The Jordan type of a linear form $\ell in A_1$ is the partition $P_\ell$ whose parts are the block sizes in the Jordan canonical form for its multiplication map m: A to A. The generic Jordan type of A is the largest occurring Jordan type $P_\ell$ among all $\ell\in A_1$ with respect to the dominance order on partitions, and $A$ has the strong Lefschetz property if its generic Jordan type is as large as possible. Given graded Artinian algebras A, B, C, we say C is an A-module tensor product product if C if free over A and is isomorphic to the tensor product $A\otimes B$ as A modules (but not necessarily as rings). We show that the generic Jordan type of the A-module tensor product C is bounded below by the generic Jordan type of the actual tensor product $A\otimes B$. A corollary is that the strong Lefschetz property for the actual tensor product implies the strong Lefschetz property for the A-module tensor product. We will also give examples from invariant theory showing this implication is strict. (Work joint with S. Chen, A. Iarrobino, P. Marques)