# Hilbert schemes, polygraphs and the Macdonald positivity conjecture

## Abstract

We study the *isospectral Hilbert scheme* defined as the reduced fiber product of , with the Hilbert scheme of points in the plane over the symmetric power , By a theorem of Fogarty, . is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over We derive two important consequences. .

(1) We prove the strong form of the *conjecture* of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients This establishes the .*Macdonald positivity conjecture*, namely that .

(2) We show that the Hilbert scheme is isomorphic to the * scheme -Hilbert* of Nakamura, in such a way that is identified with the universal family over From this point of view, . describes the fiber of a *character sheaf* at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements called ,*polygraphs*, whose coordinate rings carry geometric information about The key result is that . is a free module over the polynomial ring in one set of coordinates on This is proven by an intricate inductive argument based on elementary commutative algebra. .

## 1. Introduction

The Hilbert scheme of points in the plane is an algebraic variety which parametrizes finite subschemes of length in To each such subscheme . corresponds an multiset, or unordered -element with possible repetitions, -tuple of points in where the , are the points of repeated with appropriate multiplicities. There is a variety , finite over , whose fiber over the point of , corresponding to consists of all ordered -tuples whose underlying multiset is We call . the *isospectral Hilbert scheme*.

By a theorem of Fogarty Reference 14, the Hilbert scheme is irreducible and nonsingular. The geometry of is more complicated, but also very special. Our main geometric result, Theorem 3.1, is that is normal, Cohen-Macaulay and Gorenstein.

Earlier investigations by the author Reference 24 unearthed indications of a far-reaching correspondence between the geometry and sheaf cohomology of and on the one hand, and the theory of *Macdonald polynomials* on the other. The Macdonald polynomials

are a basis of the algebra of symmetric functions in variables *Hall-Littlewood polynomials* and the *Jack polynomials* (for a thorough treatment see Reference 40). It promptly became clear that the discovery of Macdonald polynomials was fundamental and sure to have many ramifications. Developments in the years since have borne this out, notably, Cherednik’s proof of the Macdonald constant-term identities Reference 9 and other discoveries relating Macdonald polynomials to the representation theory of quantum groups Reference 13 and affine Hecke algebras Reference 32Reference 33Reference 41, the Calogero-Sutherland model in particle physics Reference 35, and combinatorial conjectures on diagonal harmonics Reference 3Reference 16Reference 22.

The link between Macdonald polynomials and Hilbert schemes comes from work by Garsia and the author on the *Macdonald positivity conjecture*. The Schur function expansions of Macdonald polynomials lead to transition coefficients *Kostka-Macdonald coefficients*. As defined, they are rational functions of

The positivity conjecture has remained open since Macdonald formulated it at the time of his original discovery. For

In Reference 15, Garsia and the author conjectured an interpretation of the Kostka-Macdonald coefficients .

It develops that these conjectures are closely tied to the geometry of the isospectral Hilbert scheme. Specifically, in Reference 24 we were able to show that the Cohen-Macaulay property of

Another consequence of our results, equivalent in fact to our main theorem, is that the Hilbert scheme *generalized McKay correspondence*, which says that if *crepant* resolution of singularities, then the sum of the Betti numbers of

We wish to say a little at this point about how the discoveries presented here came about. It has long been known Reference 27Reference 45 that the

In the spring of 1992, we discussed our efforts on the

The connection between

The remainder of the paper is organized as follows. In §2 we give the relevant definitions concerning Macdonald polynomials and state the positivity,

The proof of the main theorem uses a technical result, Theorem 4.1, that the coordinate ring of a certain type of subspace arrangement we call a *polygraph* is a free module over the polynomial ring generated by some of the coordinates. Section 4 contains the definition and study of polygraphs, culminating in the proof of Theorem 4.1. At the end, in §5, we discuss other implications of our results, including the connection with

## 2. The and Macdonald positivity conjectures

### 2.1. Macdonald polynomials

We work with the *transformed integral forms*

where

(not to be confused with

The square brackets in Equation 3 stand for *plethystic substitution*. We pause briefly to review the definition of this operation (see Reference 24 for a fuller discussion). Let

Hence there is a unique

In general we write

There is a simple direct characterization of the transformed Macdonald polynomials

We set

It is known that

### 2.2. The and graded character conjectures

Let

be the polynomial ring in

If *diagram* is the set

(Note that in our definition the rows and columns of the diagram

The polynomial

Given a partition

the space spanned by all the iterated partial derivatives of

The

The ring *i.e.*, we have

by

We write

Macdonald had shown that

In Reference 24 the author showed that Conjecture 2.2.2 would follow from the Cohen-Macaulay property of

## 3. The isospectral Hilbert scheme

### 3.1. Preliminaries

In this section we define the isospectral Hilbert scheme *nested Hilbert scheme*

The main technical device used in the proof of Theorem 3.1 is a theorem on certain subspace arrangements called *polygraphs*, Theorem 4.1. The proof of the latter theorem is lengthy and logically distinct from the geometric reasoning leading from there to Theorem 3.1. For these reasons we have deferred Theorem 4.1 and its proof to the separate §4.

Throughout this section we work in the category of schemes of finite type over the field of complex numbers, *variety* is a reduced and irreducible scheme.

Every locally free coherent sheaf

A scheme *Cohen-Macaulay* or *Gorenstein* if its local ring

### 3.2. The schemes and

Let *length* of

The Hilbert scheme *universal family*.

To see how

We have the following fundamental theorem of Fogarty Reference 14.

The generic examples of finite closed subschemes

The most special closed subschemes in a certain sense are those defined by monomial ideals. If *standard monomials*

The algebraic torus

acts on

We write