1. Introduction

This article is an explication of some remarkable connections between the two-parameter symmetric polynomials discovered by Macdonald [1988] and the geometry of certain algebraic varieties, notably the Hilbert scheme Hilbn( ℂ2) of points in the plane and the variety Cn of pairs of commuting n x n matrices (“commuting variety”, for short). The conjectures on diagonal harmonics introduced in [Haiman 1994; Garsia and Haiman 1996a] also relate to this geometric setting.

I have sought to give a reasonably self-contained treatment of these topics, by providing an introduction to the theory of Macdonald polynomials, to the “plethystic substitution” notation for symmetric functions (which is invaluable in dealing with them), and to the conjectures and related phenomena that we aim to explain geometrically. The geometric discussion is less self-contained, it being unavoidable to use scheme-theoretic language, constructions such as blowups, and some sheaf cohomological arguments. I do however give geometric descriptions in elementary terms of the various algebraic varieties encountered, and review whatever of their special features we might use, so as to orient the reader not previously familiar with them.

The linchpin of the geometric connections we consider is the so-called “n! conjecture” [Garsia and Haiman 1993; 1996b], which remains unproved at present.