Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)
Title: Combinatorial Aspects of the rational Shuffle Conjecture
Speaker: Emily Leven (formerly Sergel), University of California, San Diego
Date: Friday, October 3, 2014 12:00pm** **(note special date and time, joint with Lie Groups seminar)
Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ
In this talk, we will review the history and recent extensions of the Classical Shuffle Conjecture. This conjecture equates two symmetric polynomials, one of which is known to give the Frobenius characteristic of the space DHn of diagonal harmonics. The other side of the conjecture is purely combinatorial, showing the remarkable ability of certain symmetric function operators to control combinatorial objects, such as Dyck paths and parking functions. This branch of algebraic combinatorics was created to explore the representation-theoretical aspects of Macdonald polynomials. This led to the n! conjecture and the introduction of the space of Diagonal Harmonics. A program outlined by Procesi led to the proof by Mark Haiman of the n!-conjecture by algebraic geometrical tools. There has recently been a flood of new operators and conjectures created, in our subject, by algebraic geometers. This talk covers some of the new results and conjectures obtained by a continuing effort to translate these developments back into the original algebraic-combinatorial setting. Our presentation should be accessible to the general mathematical audience.