Seminar by Misha Mazin (Kansas University)

Diagonal Coinvariants, Procesi Bundle and Shuffle Conjecture

2015.12.03 | Christine Dilling

 Date Tue 08 Dec Time 15:15 — 16:15 Location D4

Abstract
The ring of diagonal coinvariants was studied by A. Garsia, M. Haiman, F. Bergeron and others in relation to the Macdonald positivity conjecture. In 2001 Haiman proved Macdonald's conjecture. The key part of the proof was constructing the so-called Procesi bundle on the Hilbert scheme of points on the complex plane. Haiman showed that the space of diagonal coinvariants is isomorphic to the space of global sections of the Procesi bundle, while Macdonald polynomials correspond to the fibers of the Procesi bundle over the torus fixed points of the Hilbert scheme. The localization techniques then allows one to express the Frobenious characteristic of the space of diagonal coinvariants as a linear combination of Macdonald polynomials with rational coefficients. Shuffle conjecture provides a positive combinatorial formula for the Frobenious characteristic of the space of diagonal coinvariants. Eric Carlsson and Anton Mellit recently posted a preprint claiming to prove Shuffle conjecture.

In this talk, I will sketch Haiman's constructions of the Procesi bundle, formulate Shuffle conjecture, and, time permitting, say few words about Carsson-Mellit's proof of the Shuffle conjecture.

Seminar