Combinatorics, symmetric functions and Hilbert schemes

Abstract

We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the "n!" and "(n+1) n-1 " conjectures relating Macdonald polynomials to the characters of doubly-graded Sn modules. To make the treatment self-contained, we include background material from combinatorics, symmetric function theory, representation theory and geometry. At the end we discuss future directions, new conjectures and related work of Ginzburg, Kumar and Thomsen, Gordon, and Haglund and Loehr. Contents 1. Introduction 39 2. Background from combinatorics 41 3. Background from symmetric function theory 49 4. The n! and (n + 1) n-1 conjectures 73 5. Hilbert scheme interpretation 78 6. Discussion of proofs of the main theorems 89 7. Current developments 94