Fall 2018
Time & Location: All talks are on Monday in Gibson Hall 325 at 3:00 P.M. unless otherwise noted.
Organizer: Mahir Can
September 15
Topic
Yildiray OzanInstitution
Abstract: TBA
Gibson Hall 325
Time 11:00am
September 17
Variations on the $S_n$-module $Lie_n$
Sheila SundaramPierrepont School
Abstract:
October 29
Catalan Functions and k-Schur functions
Anna PunDrexel University
Abstract:
Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase (n-1, ..., 1,0) (of which there are Catalan many) and a composition weight of length n. They include the Schur functions ,the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to GL-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott theorem. We have discovered various properties of Catalan functions, providing a new insight on the existing theorems and conjectures inspired by Macdonald positivity conjecture.
A key discovery in our work is an elegant set of ideals of roots that the associated Catalan functions are k-Schur functions and proved that graded k-Schur functions are G-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded k-Schur functions and resolved the Schur positivity and k-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are k-Schur positive which strengthens the Schur positivity of Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems.
This is joint work with Jonah Blasiak, Jennifer Morse and Daniel Summers. https://arxiv.org/abs/1804.03701
October 22
The Variety of Polarizations
Aram BinghamTulane University
Abstract:
