THE n! CONJECTURE AND A VECTOR BUNDLE ON THE HILBERT SCHEME OF n POINTS IN THE PLANE

CAROL CHANG

Abstract. The focus of this paper is on algebraic vector bundles over ℙ^{n} and their applications to the Garsia-Haiman representation theoretic interpretation of the Macdonald symmetric polynomials. This interpretation involves a certain bigraded S_{n}-module, H_{μ}, indexed by partitions μ of n.

F. Bergeron and Garsia [1] consider the relationships between H_{μ}, for μ a partition of n + 1, and the spaces H_{μ}_{i}, for μ_{i} a partition of n contained in μ. They formulated conjectures regarding the sums and intersections of these spaces. This paper provides a geometric interpretation of these conjectures.

Résumé. Dans cet article on met l’accent sur les fibrés vectoriels (algébriques) sur ℙ^{n} et leur application à l’interprétation en termes de réprésentations des polynômes symétriques de Macdonald, due à Garsia-Haiman. Cette interprétation utilise certains S_{n}-modules bigradués, H_{μ}, indexés par les partitions μ of n.

La relation entre H_{μ} pour une partition μ de n + 1 et les espaces H_{μ}_{i} pour les partitions μ_{i} de n incluses dans μ a été étudiée par F. Bergeron et Garsia dans [1], où ils proposent des conjectures concernant la somme et l’intersection de tels espaces. Dans le présent article on fournit une interprétation géométrique de ces conjectures.

1. Introduction

Macdonald introduced a two parameter family of symmetric functions, J_{μ}(x; q, t), now known as the integral form Macdonald polynomials. Expanding in terms of certain modified Schur symmetric functions, we obtain the Macdonald q, t-Kostka coefficients K_{λμ}(q, t):

J_{μ}(x; q, t) = ∑_{λ} K_{λμ}(q, t)s_{λ}[X(1 − t)].

The notation s_{λ}[X(1 − t)] is defined in Section 2. One of the major open problems regarding the Macdonald polynomials is to show that the K_{λμ}(q, t), which are polynomials in the two parameters, have non-negative integer coefficients. This is known as Macdonald’s positivity conjecture.

Date: November 1997. 1991 Mathematics Subject Classification. Primary 05E05, Secondary 14F05, 05E10. Key words and phrases. Macdonald polynomials, Hilbert scheme, projective varieties, vector bundles. Produced by AMS-L^{A}TEX.

In order to find a combinatorial and representation theoretic interpretation of the positivity conjecture, Garsia and Haiman [5] define a doubly graded S_{n} module, H_{μ}, indexed by a partition of n. They conjecture that the dimension of this module is n!, an assertion known as the n! conjecture. Haiman [12] has shown that the n! conjecture implies Macdonald’s positivity conjecture. Specifically, he shows that if the dimension of H_{μ} is n!, then its characteristic series (the generating function giving the characters and multiplicities in each bigraded component) is a simple transformation of the Macdonald polynomial indexed by μ.

Many computer experiments and some partial results point toward the validity of the n! conjecture. With the intention of proving the n! conjecture by induction, F. Bergeron and Garsia [1] consider the relationship between H_{μ}, for μ a partition of n + 1, and the spaces H_{μ}_{i}, for μ_{i} a partition of n contained in μ. They conjecture that the space H_{μ} has a particular decomposition and the pieces of this decomposition come from the lattice of spaces generated by the H_{μ_{i}}’s according to a rule due to N. Bergeron and Haiman. As a result, F. Bergeron and Garsia develop conjectures regarding the sums and intersections of the spaces H_{μ_{i}}.

This paper gives an explanation of these conjectures of F. Bergeron and Garsia in an algebraic geometric setting that was originally proposed by M. Haiman. In Section 3, we describe the geometric interpretation in the context of the Hilbert scheme of n points in the plane. We construct a coherent sheaf P on the Hilbert scheme and show that the n! conjecture is true if and only if P is a locally free sheaf, i.e., a vector bundle (necessarily of rank n!). Since the intention is ultimately to prove the n! conjecture for n + 1 by induction, we assume the n! conjecture for partitions of n. We then work with the fact that the fibers of P over special points are isomorphic to the Garsia-Haiman modules, H_{μ_{i}}.

Section 4 outlines our main geometric results. We study the restriction of P to subvarieties isomorphic to ℙ^{k−1} contained in the Hilbert scheme. We consider this to be the geometric analog of the collection of the spaces H_{μ_{i}}, when the given partition μ has k corners. We reduce the series of conjectures of F. Bergeron and Garsia to one simple conjecture on the structure of this vector bundle, restricted to a projective space ℙ^{k} embedded in the Hilbert scheme. We finally reinterpret geometric statements combinatorially.

1.1. Acknowledgements. The author would like to thank M. Haiman for suggesting the problem and for his assistance in forming the link to algebraic geometry. Thanks also to A. Garsia for his personal explanation of his conjectures.

2. Background

We begin this section by giving a definition of a slightly non-traditional version of the Macdonald polynomials. Then we proceed to give a description of the Garsia-Haiman S_{n} module and the inductive approach of Bergeron and Garsia.

2.1. Macdonald Polynomials. Our notation will be compatible with that of Macdonald in [18]. We let μ ⊢ n be the notation for μ a partition of n, where μ = (μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{l} > 0), and μ′ denotes its conjugate.

THE n! CONJECTURE AND A VECTOR BUNDLE 3

First we recall the plethystic notation used by Garsia and Haiman in [5]. Let A be a formal power series in the variables, {a_{1}, a_{2}, … }. Define p_{k}[A] to be the expression that replaces each a_{i} by a_{i}^{k}. Then we extend so that p_{k} ↦ p_{k}[A] is a ring homomorphism on the whole ring of symmetric functions. For example, if X = x_{1} + x_{2} + ⋯, then f[X] = f(x_{1}, x_{2}, …) and f[\frac{X}{1−q}] = f(x_{1}, x_{2}, …, qx_{1}, qx_{2}, …).

Macdonald [18] introduced a basis for the ring of symmetric functions, {J_{μ}(x; q, t)}, in the variables x = {x_{1}, x_{2}, … } indexed by partitions μ ⊢ n, with coefficients in the field of rational functions ℚ(q, t). Expanding these in terms of modified Schur functions, we obtain

J_{μ}[X; q, t] = ∑_{λ} K_{λμ}(q, t)s_{λ}[X(1 − t)]. (2.1)

The coefficients K_{λμ}(q, t), which are called Macdonald’s q, t-Kostka coefficients, are polynomials in the two parameters with integer coefficients. This was proved independently by various authors [7],[8],[16],[17],[20]. Macdonald’s positivity conjecture states that these polynomials have non-negative integer coefficients.

In this paper, we refer to the variant basis, first introduced in [5],

H̃_{μ}[X; q, t] = t^{n(μ)}J_{μ}[\frac{X}{1 − t^{−1}} ; q, t^{−1}] (2.2)

where n(μ) = ∑_{i}(i − 1)μ_{i}. Then we have

H̃_{μ}[X; q, t] = ∑_{λ} K̃_{λμ}(q, t)s_{λ}[X] (2.3)

with

K̃_{λμ}(q, t) = t^{n(μ)}K_{λμ}(q, t^{−1}). (2.4)

2.2. The n! conjecture. Garsia and Haiman reformulate Macdonald’s positivity conjecture as the problem of showing that H̃_{μ}(X; q, t) is the Frobenius series of some finite dimensional doubly graded S_{n}-module.

We recall the definition of the Frobenius series. Let H = ⊕_{h,k}(H)_{h,k} be a doubly graded S_{n}-module. Recall that the Frobenius map, ϕ, is a linear map from S_{n} characters to the symmetric functions taking the irreducible S_{n} character χ_{λ} to the Schur function s_{λ}. Then the Frobenius series of H is defined to be the generating function for the image by ϕ of the characters in the various bihomogeneous components:

F(H) = ∑_{h,k} ϕ(char(H)_{h,k})t^{h}q^{k}. (2.5)

Thus F(H) is a symmetric function with coefficients in ℚ[q, t].

Garsia and Haiman found a candidate for a doubly graded S_{n}-module that would explain Macdonald’s positivity conjecture. The module is defined as follows. Let Δ_{μ} be the polynomial analogous to the Vandermonde determinant:

Δ_{μ}(x_{1}, …, x_{n}; y_{1}, …, y_{n}) = det[x_{i}^{p_{j}}y_{i}^{q_{j}}]_{i,j=1,...,n} (2.6) where the pairs (p_{j}, q_{j}) = (row(c_{j}), col(c_{j})) run through the coordinates of the cells c_{j} in the Ferrers diagram of μ.

Example: μ = (2, 1) 00 10 01

Δ_{(2,1)}(x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}) = det [ \begin{matrix}1 & y_{1} & x_{1} \\ 1 & y_{2} & x_{2} \\ 1 & y_{3} & x_{3} \end{matrix} ] . (2.7)

Definition 2.1. H_{μ} is the ℚ-linear span of all partial derivatives of all orders of Δ_{μ}.

With this module defined, there is the following formulation of the positivity conjecture by Garsia and Haiman.

Conjecture 2.2.

F(H_{μ}) = H̃_{μ}(X; q, t).

Since it is known that K̃_{λμ}(1, 1) is the dimension of the irreducible representation V_{λ}, it follows that if the Frobenius series of H_{μ} is H̃_{μ}(X, q, t), then H_{μ} affords the regular representation of S_{n}. Thus in particular Conjecture 2.2 implies that the dimension of the space H_{μ} is n!, a conjecture of Garsia and Haiman known as the n! conjecture. In addition, Haiman proved the following theorem.

Theorem 2.3. (Haiman) If dim H_{μ} = n! then H̃_{μ}(X, q, t) is the Frobenius series of H_{μ}.

We give an alternate description of H_{μ}, which is more natural in the algebraic geometry setting. First we describe the setting in terms of ℚ. Let J_{μ} ⊆ ℚ[x_{1}, y_{1}, …, x_{n}, y_{n}] be the ideal of polynomials p(x_{1}, y_{1}, …, x_{n}, y_{n}) with the property that p(∂_{x_{1}}, ∂_{y_{1}}, …, ∂_{x_{n}}, ∂_{y_{n}}) annihilates Δ_{μ}. Define R_{μ}^{ℚ} = ℚ[x_{1}, y_{1}, …, x_{n}, y_{n}]/J_{μ} with its natural structure of an S_{n} module. The ring R_{μ}^{ℚ} is isomorphic as a doubly graded S_{n} module to H_{μ} see [5] for details. From a geometric point of view it is more natural to work with R_{μ} = ℂ ⊗ R_{μ}^{ℚ}, which in any event has the same decomposition as a doubly graded S_{n}-module.

2.3. Conjectures of Bergeron and Garsia. Let μ be a partition of (n+1) with k corners and let μ_{i} be the partition of n obtained by removing the ith corner of μ. F. Bergeron and Garsia studied the relationships that hold between the space H_{μ} and the spaces H_{μ_{i}}. The idea is to construct a basis for H_{μ} using information about the H_{μ_{i}}’s.

In particular, they conjecture together with N. Bergeron and Haiman that for each cell (i, j) ∈ μ there is a particular subspace H_{ij} of ∑_{i} H_{μ_{i}}, such that

H_{μ} = ⨁_{(i,j)∈μ} H_{ij}(∂)∂_{x_{n+1}}^{i}∂_{y_{n+1}}^{j}Δ_{μ}, (2.8)

where H_{ij}(∂) is obtained by substituting the partial derivative operator ∂_{x_{p}}, ∂_{y_{p}} for the variables x_{p}, y_{p} in the polynomials belonging to H_{ij}.

M. Haiman and N. Bergeron gave a conjectured algorithm for choosing the modules H_{ij}. The modules H_{ij} are elements of the lattice of subspaces of ∑_{i} H_{μ_{i}} generated by the spaces H_{μ}_{1}, …, H_{μ}_{k}. The algorithm is as follows.

First for every (i, j) in the top row of μ, let H_{ij} = H_{μ}_{1}. The assignments for subsequent rows are obtained inductively by the following procedure. Let A be a row whose assignments have been made, and B be the next row. There are two cases.

1. If the length of B is equal to the length of A, then the assignment for each cell in row B is that of the cell above it in row A.

2. If the lengths of the rows differ, let a be the length of row A, b be the length of row B, and c = b − a ≥ 1. Let

A_{1}, A_{2}, …, A_{a}

be the assignments to the cells in row A. Set

(2.9) A′_{s} = \begin{cases}A_{s} & for 1 ≤ s ≤ a, \\ H_{μ}_{1} + ⋯ + H_{μ}_{k} & for s ≤ 0, \\ 0 & for a + 1 ≤ s ≤ b. \end{cases}

Then the sth cell of row B is assigned

B_{s} = A′_{s} + (A′_{s−c} ∩ H_{μ_{i}}), (2.10)

where the last cell in row B is the ith corner of μ.

Example: For μ = (4, 3, 1), the filling is as illustrated in Figure 1.

H_{μ}^{1} H_{μ}_{1} + H_{μ}_{2}  ^{H_{μ}^{2}}  H_{μ}_{1} ∩ H_{μ}_{2} H_{μ}_{1} + H_{μ}_{2} + H_{μ}_{3} H_{μ}_{2} + (H_{μ}_{1} ∩ H_{μ}_{3})H_{μ}_{2} ∩ (H_{μ}_{1} + H_{μ}_{3}) H_{μ}_{1} ∩ H_{μ}_{2} ∩ H_{μ}_{3}

Figure 1

To complete the story, F. Bergeron and Garsia developed conjectures on the lattice of subspaces generated by the spaces H_{μ_{i}}. The first of these conjectures says the space H_{μ}_{1} + ⋯ + H_{μ}_{k} has a direct sum decomposition in which each space H_{μ_{i}} is a partial sum.

Conjecture 2.4. There exists subspaces H_{S} for ∅ ≠ S ⊆ {1, …, k} such that

∑_{i=1}^{k} H_{μ}_{i} = ⨁ H_{S}, and H_{μ}_{i} = ⨁_{i∈S} H_{S}.

This is equivalent to saying the subspaces H_{μ_{i}}’s generate a distributive lattice. A further conjecture relates the Frobenius series of the summands. First let Ψ_{i} be the operator on symmetric functions with coefficients in ℚ(q, t) defined by

Ψ_{i}(f[X; q, t]) = ωf[X; q^{−1}, t^{−1}]t^{n(μ_{i})}q^{n(μ′_{i})}, (2.11)

where ω is the involution that sends the Schur function s_{λ} to s_{λ′}.

Conjecture 2.5. If i ∈ S then

F(H_{S}) = Ψ_{i}(F(H_{S^{C}∪{i}})),

where S^{C} is the complement of S in {1, …, k}.

Using the theory of Macdonald polynomials, specifically that the Frobenius series of H_{μ_{i}} is given by H̃_{μ_{i}}[X; q, t], this conjecture determines the dimensions of the summands of the direct sum decomposition in Conjecture 2.4. In particular, it implies the following conjecture.

Conjecture 2.6. Let μ ⊢ (n + 1) with k corners. Let μ_{1}, μ_{2}, ..., μ_{k} ⊢ n, where μ_{i} is μ with the ith corner removed. Then for {i_{1}, …, i_{m}} ⊆ {1, …, k},

dim H_{μ}_{i_{1}} ∩ H_{μ}_{i_{2}} ∩ ⋯ ∩ H_{μ}_{i_{m}} = \frac{n!}{m}.

We illustrate with an example. Let each circle in Figure 2 represent a corner removed module, H_{μ}_{i}.

Figure 2

Then, we have the decomposition

H_{μ}_{1} + H_{μ}_{2} + H_{μ}_{3} = ϕ^{100} ⊕ ϕ^{010} ⊕ ϕ^{001}⊕ ϕ^{101} ⊕ ϕ^{110} ⊕ ϕ^{011} ⊕ ϕ^{111},

where, for example,

H_{μ}_{1} = ϕ^{100} ⊕ ϕ^{101} ⊕ ϕ^{110} ⊕ ⊕ ϕ^{111}.

We also have an identification of regions in the Venn diagram, for example,

F(ϕ^{111}) ≅ Ψ_{1}(F(ϕ^{100})) ≅ Ψ_{2}(F(ϕ^{010})) ≅ Ψ_{3}(F(ϕ^{001}))

and

F(ϕ^{110}) ≅ Ψ_{1}(F(ϕ^{101})).

It can be shown that Conjectures 2.4 and 2.5, together imply that dim H_{μ} = (n + 1)!. For μ having only 2 corners, Garsia and Bergeron proved Conjectures 2.4 and 2.5, assuming that dim H_{μ}_{1} ∩ H_{μ}_{2} = \frac{n!}{2} (as predicted by Conjecture 2.6).

Theorem 2.7. Let μ ⊢ (n + 1) with 2 corners. Let μ_{1}, μ_{2} ⊢ n be obtained by removing each corner respectively. Assume that dim H_{μ}_{1} = dim H_{μ}_{2} = n! and dim H_{μ}_{1} ∩ H_{μ}_{2} = \frac{n!}{2}. Then dim H_{μ} = (n + 1)!.

Assuming Conjectures 2.4 and 2.5 along with the n! conjecture, one can write down a “Pieri rule” for the Macdonald polynomials H̃_{μ}[X; q, t] which can be shown remarkably to agree with the known one given by Macdonald.

For our purposes we need to reformulate the statements about the lattice of subspaces generated by the H_{μ_{i}}’s in terms of the lattice of ideals generated by the J_{μ_{i}}’s.

Definition 2.8. Let f, g ∈ ℚ[x_{1}, y_{1}, …, x_{n}, y_{n}]. Then ⟨f, g⟩ is defined to be the constant term of f(∂_{x_{1}}, ∂_{y_{1}}, …, ∂_{x_{n}}, ∂_{y_{n}})g(x_{1}, y_{1}, …, x_{n}, y_{n}).

In [6], Garsia and Haiman showed that J_{μ}^{⊥} ≅ H_{μ} and (J_{μ}^{⊥})^{⊥} ≅ J_{μ}. Note that J_{μ_{1}}^{⊥} ∩ J_{μ_{2}}^{⊥} ≅ (J_{μ_{1}} + J_{μ_{2}})^{⊥} and J_{μ_{1}}^{⊥} + J_{μ_{2}}^{⊥} ≅ (J_{μ_{1}} ∩ J_{μ_{2}})^{⊥}. Thus, for example, we have with R^{ℚ} = ℚ[x_{1}, y_{1}, …, x_{n}, y_{n}], the isomorphisms R^{ℚ}/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}) ≅ H_{μ}_{1} + ⋯ + H_{μ}_{k} and R^{ℚ}/(J_{μ_{1}} + ⋯ + J_{μ_{k}}) ≅ H_{μ}_{1} ∩ ⋯ ∩ H_{μ}_{k}. Replacing R^{ℚ} by R = ℂ[x_{1}, y_{1}, …, x_{n}, y_{n}] and J_{μ_{i}} by ℂ ⊗ J_{μ_{i}} does not alter the characters of these rings as doubly graded S_{n} modules. Throughout what follows we always work over ℂ.

3. Geometric Interpretation

In an effort to prove the conjectures of Bergeron and Garsia, we translate their statements into an algebraic geometric setting. Assuming the n! conjecture for partitions of n, we construct a vector bundle on the Hilbert scheme of n points in the plane. Using the combinatorics of the global sections of this bundle, we explain the distributive lattice structure. We also give a geometric version of Conjecture 2.5.

We begin by reviewing the description of the Hilbert scheme of n points in the plane (see [11] for details).

3.1. The Hilbert Scheme of n Points in the Plane. Let 𝔸^{2} = Spec ℂ[x, y] be the affine plane. The Hilbert scheme of n points in the plane, Hilb^{n}(𝔸^{2}), is the set of all ideals I ⊆ ℂ[x, y] such that dim_{ℂ}ℂ[x, y]/I = n. There is a natural action of the torus T^{2} = (ℂ^{∗})^{2} on Hilb^{n}(𝔸^{2}), given by (t, q)·x = tx and (t, q)·y = qy. Under this action, an ideal I ∈ Hilb^{n}(𝔸^{2}) is a T^{2} fixed point if it is of the form

I_{μ} := (x^{i}y^{j} | (i, j) ∉ μ),

for μ ⊢ n. Every ideal I ∈ Hilb^{n}(𝔸^{2}) has a torus fixed point in the closure of its orbit.

3.2. Special Subvarieties of Hilb^{n}(𝔸^{2}). For present purposes, we want to restrict our attention to certain projective subvarieties of the punctual Hilbert scheme. To describe them, let μ be a partition of (n+1) with k corners, i.e. μ = (l_{1}^{m_{1}}, …, l_{k}^{m_{k}}) where m_{1}l_{1}+⋯+m_{k}l_{k} = n + 1. Let (α_{i}, β_{i}) be the coordinates of the ith corner cell. We illustrate below.

μ = (α_{1}, β_{1}) (α_{k}, β_{k}) Figure 3

The geometric analog of the inductive approach would be to consider all the ideals I in Hilb^{n}(𝔸^{2}) which contain I_{μ} for this μ. Since dim ℂ[x, y]/I_{μ} = n + 1, the only way to obtain such an ideal I of codimension n is if I = I_{μ} + (f), where f is a linear combination of corner monomials. Specifically, f must be of the form f = z_{1}x^{α_{1}}y^{β_{1}} + ⋯ + z_{k}x^{α_{k}}y^{β_{k}}, where the z_{i}’s are scalars not all zero. Thus, the ideals I ∈ Hilb^{n}(𝔸^{2}) that contain I_{μ} form a projective space ℙ^{k−1} ⊆ Hilb^{n}(𝔸^{2}) with coordinates (z_{1} : ⋯: z_{k}). For I in this family, we use the notation

I(z_{1} : ⋯: z_{k}) = (I_{μ} + (z_{1}x^{α_{1}}y^{β_{1}} + ⋯ + z_{k}x^{α_{k}}y^{β_{k}})). (3.1)

We note that I_{μ_{i}} = I_{μ} + (x^{α_{i}}y^{β_{i}}) = I(0 : ⋯: 1 : ⋯: 0) with 1 in position i, is the ideal that corresponds to the predecessor μ_{i} of μ.

3.3. Constructing our Bundle. Let B be the tautological vector bundle of rank n over Hilb^{n}(𝔸^{2}), whose fiber at I is the vector space ℂ[x, y]/I. In [11], Haiman proved that the highest exterior power of this tautological bundle is O(1), for a natural embedding of Hilb^{n}(𝔸^{2}) as a quasi-projective variety. Note that B^{⊗n} can be identified as the bundle whose fiber at I is ℂ[x_{1}, y_{1}, … x_{n}, y_{n}]/(I(x_{1}, y_{1}) + ⋯ + I(x_{n}, y_{n})). Note that B and B^{⊗n} are bundles of ℂ-algebras.

There is a natural action of S_{n} on B^{⊗n} which permutes the factors. We then have the map:

(3.2)  B^{⊗n} ⊗ B^{⊗n} ⟶ B^{⊗n} ⟶ ∧^{n}B ≅ O(1)
f ⊗ g ⟼ fg ⟼ Alt(fg),

where the first map is multiplication and the second is the S_{n} alternation operator defined by

Alt(f) = \frac{1}{n!}_{∑w∈S_{n}}ε(w)w(f). (3.3)

This induces a map:

ϕ : B^{⊗n} ⟶ Hom(B^{⊗n}, ∧^{n}B) ≅ (B^{⊗n})^{∗} ⊗ O(1). (3.4)

Let J := ker ϕ. Define P to be B^{⊗n}/J, or equivalently, im ϕ. A priori, the subsheaf J might not be a subbundle and thus P is only a coherent sheaf. We show that the n! conjecture holds for partitions of n if and only P is locally free, i.e., J is a subbundle and P is the corresponding quotient bundle.

Lemma 3.1. Let ϕ_{I} : B_{I}^{⊗n} → ((B^{⊗n})^{∗}⊗O(1))_{I} be the map induced on the fiber at I by (3.4). Let I ∈ Hilb^{n}(𝔸^{2}) be a generic point, i.e., I = I(S) where S = {p_{1}, …, p_{n}} ⊆ 𝔸^{2}. Then rank ϕ_{I} = n!.

Proof. The fibers of B^{⊗n} at I can be thought of as the set of ℂ-valued functions on the set of sequences (a_{1}, …, a_{n}) for a_{i} ∈ S. If f ∈ B_{I}^{⊗n} is alternating then f is zero on any sequence with a repeat. Let N be the set of sequences which are permutations of S, i.e., sequences with distinct entries. Thus the complement of N consists of the sequences with repeats.

We claim that ker ϕ_{I} = {f ∈ B_{I}^{⊗n} : f|_{N} = 0}. Suppose f|_{N} = 0. Then for all g ∈ B_{I}^{⊗n}, fg|_{N} = 0. Thus, Alt(fg)|_{N} = 0, which implies f ∈ ker ϕ_{I}. Conversely, suppose f|_{N} ≠ 0.

Then choose a = (a_{1}, …, a_{n}) ∈ N such that f(a) ≠ 0. Multiplying f by a suitable g, we obtain a function such that fg(a) ≠ 0 and fg(b) = 0 for all b ≠ a. Then Alt(fg) ≠ 0.

Thus, B_{I}^{⊗n}/ker ϕ_{I} are exactly the functions on N, which must be n! dimensional; and, in particular this verifies that the rank of ϕ_{I} is generically n!.

Lemma 3.2. Let ϕ_{μ} : B_{μ}^{⊗n} → ((B^{⊗n})^{∗}⊗O(1))_{μ} be the map induced on the fiber at I_{μ} by B^{⊗n} → (B^{⊗n})^{∗}⊗O(1). Then ker ϕ_{μ} = J_{μ}. Thus in particular, rank ϕ_{μ} = dim H_{μ}.

Proof. Let R = ℂ[x_{1}, y_{1}, …, x_{n}, y_{n}]. Consider the linear functional

l : R → Λ^{n}B_{μ} ≅ ℂ (3.5)

given by l(f) ≡ Alt(f) modulo I_{μ}(x_{1}, y_{1}) + ⋯ I_{μ}(x_{n}, y_{n}). The functional kills all f in the nonalternating S_{n} isotypic components. In addition, it kills every alternation of a monomial except Δ_{μ}, since a monomial alternant Alt(x_{1}^{h_{1}}y_{1}^{k_{1}} ⋯ x_{n}^{h_{n}}y_{n}^{k_{n}}) is killed whenever any (h_{i}, k_{i}) lies outside the diagram of μ.

Now, this functional l is, up to a scalar multiple, the same as the functional given by ψ_{μ}(f) = ⟨f, Δ_{μ}⟩, where ⟨ , ⟩ is given in Definition 2.9. Then f ∈ J_{μ} if and only if ⟨g, f(∂_{X}, ∂_{Y})Δ_{μ}⟩ = 0 for all g ∈ R. But the latter is equivalent to ⟨fg, Δ_{μ}⟩ = 0 for all g ∈ R. Thus ψ_{μ}(fg) = 0 for all g ∈ R if and only if l(fg) = 0 for all g ∈ R which is equivalent to f ∈ ker ϕ_{μ}.

In particular, assuming the n! conjecture holds for partitions of n, the rank of ϕ_{μ} is n!.

Corollary 3.3. Assuming the n! conjecture, the fiber of P at the torus fixed point I_{μ} ∈ Hilb^{n}(𝔸^{2}) is isomorphic to

ℂ[x_{1}, y_{1}, …, x_{n}, y_{n}]/J_{μ}

where J_{μ} is the ideal of polynomials which annihilates Δ_{μ}.

Theorem 3.4. The n! conjecture is true if and only if P is a locally free sheaf or equivalently if the rank of ϕ in (3.4) is constant.

Proof. The generic points are dense in Hilb^{n}(𝔸^{2}), and the generic rank of ϕ_{I} is n! from Lemma 3.1. Also, every ideal I ∈ Hilb^{n}(𝔸^{2}) has a torus fixed point, I_{μ} in the closure of its orbit [11]. Assuming the n! conjecture, the rank of ϕ at the torus fixed points I_{μ} is also n!. Thus, the rank of ϕ must be constant, and we have that P is locally free.

Recall we are assuming by induction that the n! conjecture is true for partitions of n; thus from here on we assume that P is a bundle. We also mention some properties of P which will be useful in proving its decomposition when restricted to certain subvarieties of Hilb^{n}(𝔸^{2}).

Proposition 3.5. P has the following properties:

1. P is generated by global sections.

2. Assuming the n! conjecture, P is a vector bundle such that P ≅ P^{∗} ⊗ O(1) where P^{∗} is the dual of P.

Proof. The first part follows from the fact that P is is a quotient of B^{⊗n}, and B^{⊗n} is a quotient of a trivial bundle.

For the second part, observe that the pairing B^{⊗n} ⊗ B^{⊗n} ⟶ ∧^{n}B ≅ O(1) induces a nondegenerate pairing P⊗P ⟶ O(1). Thus, in particular, P ≅ Hom(P, O(1)) ≅ P^{∗}⊗O(1).

4. Main Results

Throughout this section we assume the n! conjecture for partitions of a given n, and thus also assume that P is a vector bundle.

We consider the restriction of P to the subvarieties ℙ^{k−1}. Let R = ℂ[x_{1}, y_{1}, …, x_{n}, y_{n}]. Since B^{⊗n} is the tautological quotient bundle of the trivial sheaf O⊗R, and P is a quotient of B^{⊗n}, we have the following surjective map of vector bundles,

O⊗R ↠ P, (4.1)

and corresponding map on global sections,

θ : R → Γ(ℙ^{k−1}, P). (4.2)

Conjecture 4.1. The map θ given by (4.2) is surjective.

4.1. Two Corner Case. When μ is a partition with 2 corners, we are dealing with a projective line ℙ^{1} ⊆ Hilb^{n}(𝔸^{2}).

Theorem 4.2. (Birkhoff [2]) Every algebraic vector bundle V over ℙ^{1} has a unique decomposition of the form V ≅ ⨁_{k} O(k)^{r_{k}}.

Corollary 4.3. P|_{ℙ^{1}} ≅ O^{\frac{n!}{2}} ⊕O(1)^{\frac{n!}{2}} .

Proof. In the decomposition of Theorem 4.2 we cannot have any terms O(k) in P|_{ℙ^{1}} with k < 0, since P is generated by global sections. Thus, P ≅ ⊕_{k≥0}O(k)^{r_{k}}. We also have from the duality property that P ≅ P^{∗} ⊗ O(1) ≅ ⊕_{k≥0}O(−k + 1)^{r_{k}}. This implies that r_{k} = 0 for all k ≥ 2, and r_{0} = r_{1}. Since P is of rank n!, r_{0} +r_{1} = n!. Thus we obtain the decomposition P|_{ℙ^{1}} ≅ O^{\frac{n!}{2}} ⊕ O(1)^{\frac{n!}{2}} .

We now have the correct setting to prove the following result.

Proposition 4.4. If the map θ of (4.2) is surjective, then Γ(ℙ^{1}, P) ≅ R/(J_{μ_{1}} ∩ J_{μ_{2}}).

Proof. Recall we are assuming that θ : R ↠ Γ(ℙ^{1}, P) = Γ(ℙ^{1}, O^{\frac{n!}{2}} ⊕O(1)^{\frac{n!}{2}} ) is surjective.

First we notice that sections of O and O(1) are completely determined by their value at any one point or respectively any two points. The sections of P|_{ℙ^{1}} are thus determined by their value on any two points. Given a polynomial f ∈ J_{μ_{1}} ∩ J_{μ_{2}} (representing a section) then f evaluates to zero in the fibers P_{I_{μ_{1}}} ≅ R/J_{μ_{1}} and P_{I_{μ_{2}}} ≅ R/J_{μ_{2}}. Since f evaluates to zero at two points, it must be the zero section. Thus J_{μ_{1}} ∩ J_{μ_{2}} ⊆ ker θ.

Conversely, if f ∈ ker θ then f is the zero section, evaluating to zero in every fiber. In particular, f ∈ J_{μ_{1}} ∩ J_{μ_{2}}.

In particular from Lemma 4.4, it follows that dim(H_{μ}_{1} + H_{μ}_{2}) = dim R/(J_{μ_{1}} ∩ J_{μ_{2}}) = dim Γ(ℙ^{1}, P), i.e.,

dim(H_{μ}_{1} + H_{μ}_{2}) = \frac{3n!}{2}. (4.3)

Furthermore, assuming the inductive hypothesis that dim H_{μ}_{1} = dim H_{μ}_{2} = n!, it necessarily follows that

dim(H_{μ}_{1} ∩ H_{μ}_{2}) = \frac{n!}{2}. (4.4)

4.2. Multiple Corner Case. In the general case when μ has k corners, we look at the restriction of P to ℙ^{k−1}. Recall that ℙ^{k−1} parametrizes all ideals I ∈ Hilb^{n}(𝔸^{2}) that contain I_{μ} in the following way:

I(z_{1} : ⋯: z_{k}) := (I_{μ} + (z_{1}x^{α_{1}}y^{β_{1}} + ⋯ + z_{k}x^{α_{k}}y^{β_{k}})), (4.5)

where (α_{i}, β_{i}) is the coordinate of a corner cell of μ.

Another description of this particular subvariety of Hilb^{n}(𝔸^{2}) is the following. Let

V = span_{ℂ}{x^{α_{1}}y^{β_{1}}, …, x^{α_{k}}y^{β_{k}}}, (4.6)

be the linear span of the corner monomials of μ. Notice that ℙ(V) = ℙ^{k−1}. We then have a one to one correspondence between the set of I ∈ Hilb^{n}(𝔸^{2}) such that I_{μ} ⊆ I and points L ∈ ℙ(V), i.e.,

I(z_{1} : ⋯: z_{k}) ⟷ the line L = span_{ℂ}{z_{1}x^{α_{1}}y^{β_{1}} + ⋯ + z_{k}x^{α_{k}}y^{β_{k}}}.

Let O(−1) be the tautological bundle on ℙ(V) whose fiber at L ∈ ℙ(V) is L. Let Q be the quotient bundle whose fiber at L is V/L. Then Q fits into the exact sequence:

0 → O(−1) → O⊗V → Q → 0, (4.7)

where O⊗V is a trivial bundle of rank k. We also have for the jth exterior power of Q the sequence:

0 → O(−1)⊗Λ^{j−1}Q → Λ^{j}(O⊗V) → Λ^{j}Q → 0. (4.8)

We can now give the conjecture regarding the decomposition of P|_{ℙ^{k−1}}.

Conjecture 4.5. (The Decomposition Conjecture)
P|_{ℙ^{k−1}} ≅ (Λ^{0}Q)^{\frac{n!}{k}} ⊕ ⋯ ⊕ (Λ^{i}Q)^{\frac{n!}{k\binom{k−1}{i}}} ⊕ ⋯ ⊕ (Λ^{k−1}Q)^{\frac{n!}{k}} .

Corollary 4.3 implies that the Decomposition Conjecture is true for ℙ^{1}. In a separate paper [4] we show that a slight strengthening of Conjecture 4.1 would imply the Decomposition Conjecture. Here we will only discuss its implications.

In order to identify the global sections of P|_{ℙ^{1}} with H_{μ}_{1} + H_{μ}_{2} we made use of its direct sum decomposition and the global sections of the pieces of the direct sum. We must do the same for P|_{ℙ^{k−1}}. First we describe the global sections of the conjectured decomposition.

The following well known fact is a consequence of the Borel-Weil-Bott theorem for line bundles on the flag variety.

Proposition 4.6. Γ(ℙ^{k−1}, Λ^{j}Q) ≅ Λ^{j}V.

The next corollary follows immediately from this proposition.

Corollary 4.7. Assuming the Decomposition Conjecture, we have that

Γ(ℙ^{k−1}, P) = (Λ^{0}V)^{\frac{n!}{k}} ⊕ ⋯ ⊕ (Λ^{i}V)^{\frac{n!}{k\binom{k−1}{i}}} ⊕ ⋯ ⊕ (Λ^{k−1}V)^{\frac{n!}{k}} . (4.9)

We can choose a basis of Γ(ℙ^{k−1}, P) according to the basis on each summand in its direct sum decomposition. Recall V is the linear span of the corner monomials of μ as defined in (4.6). Thus an appropriate basis of V is

B_{1} = {b_{1}, …, b_{k} | b_{i} = x^{α_{i}}y^{β_{i}}}. (4.10)

This induces the basis

B_{j} = {b_{i_{1}} ∧ ⋯ ∧ b_{i_{j}} | i_{1} < ⋯ < i_{j}} (4.11)

of Λ^{j}V. Note that Λ^{0}V ≅ ℂ with basis B_{0} = {1}. Fixing the direct sum decomposition in (4.9) and using the basis B_{j} in each summand we obtain a basis for Γ(ℙ^{k−1}, P).

Next, we describe the fibers of P with respect to the conjectured decomposition. Set L_{i} = span_{ℂ}{x^{α_{i}}y^{β_{i}}} = span_{ℂ}{b_{i}}. L_{i} can be considered as the point in ℙ(V) which corresponds to I_{μ_{i}} = I_{μ} + (x^{α_{i}}y^{β_{i}}) ∈ ℙ^{k−1} ⊆ Hilb^{n}(𝔸^{2}). Then the space V of (4.6) can be identified with V ≅ L_{1}⊕ ⋯ ⊕L_{k}. Furthermore, let M_{i} = ⊕_{j≠i}L_{i}. We can now identify the fibers of P.

1. The fiber of O⊗V at I_{μ_{i}} is identified with V ≅ L_{i}⊕M_{i}.

2. The fiber of O(−1) at I_{μ_{i}} is identified with L_{i}.

3. The fiber of Q at I_{μ_{i}} is identified with V/L_{i} ≅ M_{i}.

4. The fiber of Λ^{j}Q at I_{μ_{i}} is identified with Λ^{j}(M_{i}) ≅ Λ^{j}V/(L_{i}⊗Λ^{j−1}(M_{i})).

5. Thus, the fiber of P at I_{μ_{i}} is identified with

P_{I_{μ_{i}}} ≅ Λ^{0}(M_{i})^{\frac{n!}{k}} ⊕ ⋯ Λ^{j}(M_{i})^{\frac{n!}{k\binom{k−1}{j}}} ⊕ ⋯ ⊕Λ^{k−1}(M_{i})^{\frac{n!}{k}} . (4.12)

We also remark that for any j < k,

(L_{1}⊗Λ^{j−1}(M_{1})) ∩ ⋯ ∩ (L_{k}⊗Λ^{j−1}(M_{k})) = 0,

since Λ^{j}V is the direct sum of the 1-dimensional spaces L_{i_{1}} ⊗ ⋯ ⊗ L_{i_{j}} for i_{1} < ⋯ < i_{j}, and the rth factor in the above intersection consists of those for which r ∈ {i_{1}, ⋯, i_{j}}.

Lemma 4.8. Assuming the Decomposition Conjecture, the map

i : Γ(ℙ^{k−1}, P) → P_{I_{μ_{1}}}⊕ ⋯ ⊕P_{I_{μ_{k}}}
s ↦ s(I_{μ_{1}}) + ⋯ + s(I_{μ_{k}}),

is injective.

Proof. If v ∈ Λ^{j}V (i.e. in Γ(ℙ^{k−1}, P)) is mapped to zero under i, then

v ∈ (L_{1}⊗Λ^{j−1}(M_{1})) ∩ ⋯ ∩ (L_{k}⊗Λ^{j−1}(M_{k}),

since it is zero in each of the fibers at I_{μ_{1}}, …, I_{μ_{k}}. Thus v = 0.

We make the following identification of the global sections of P restricted to ℙ^{k−1}.

Proposition 4.9. Assuming the Decomposition Conjecture and that θ defined in (4.2) is surjective, then

Γ(ℙ^{k−1}, P) ≅ R/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}).

Proof. Recall we are assuming that θ : R ↠ Γ(ℙ^{k−1}, P) is surjective.

Given f ∈ R, if f ∈ ker θ then f is the zero section, evaluating to zero in every fiber. In particular, f ∈ J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}.

To prove the converse, first we note that from Lemma 4.8 we have that the map

i : Γ(ℙ^{k−1}, P) → P_{I_{μ_{1}}}⊕ ⋯ ⊕P_{I_{μ_{k}}}

is injective. If f ∈ J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}} then f is zero in each of the fibers P_{I_{μ_{1}}}, …, P_{I_{μ_{k}}}. Thus f = 0 in Γ(ℙ^{k−1}, P), which implies that f ∈ ker θ.

We now discuss the distributive lattice structure of the spaces H_{μ_{i}}. Recall we are still using the assumption that all global sections of P|_{ℙ^{k−1}} come from polynomials in R. Thus, we make use of the isomorphism in Proposition 4.9 between the space of global sections, Γ(ℙ^{k−1}, P), and R/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}) ≅ H_{μ}_{1} + ⋯ + H_{μ}_{k}. Furthermore, to describe the distributive lattice structure of H_{μ}_{1} + ⋯ + H_{μ}_{k}, we prove equivalently, in the algebraic geometric setting, that there is a basis of

Γ(ℙ^{k−1}, P) ≅ R/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}})

such that each space J_{μ_{i}}/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}) is spanned by a subset of this basis. This is equivalent since under the surjective map of global sections onto a fiber P_{I_{μ_{i}}} ≅ R/J_{μ_{i}}, i.e.,

Γ(ℙ^{k−1}, P)  ⟶^{α_{i}} P_{I_{μ_{i}}}
↓≅  ↓≅
R/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}) ⟶^{α_{i}} R/J_{μ_{i}},

the kernel of α_{i} is exactly J_{μ_{i}}/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}).

Recall that the fiber of P at I_{μ_{i}} can be identified as

P_{I_{μ_{i}}} ≅ ℂ^{\frac{n!}{k}} ⊕ ⋯ ⊕Λ^{j}(M_{i})^{\frac{n!}{k\binom{k−1}{j}}} ⊕ ⋯ ⊕Λ^{k−1}(M_{i})^{\frac{n!}{k}} ,

where M_{i} = span_{ℂ}{b_{j} : j ≠ i}, and b_{j} is the jth corner monomial as defined earlier in (4.10). We can clearly see that a basis for M_{i} is B \ {b_{i}}, where elements of B are all the corner monomials. For Γ(ℙ^{k−1}, P), we use the basis chosen earlier. Thus, looking at the image of our chosen basis elements of Γ(ℙ^{k−1}, P) under the map α_{i} above, we see that any exterior products involving b_{i} must be in the kernel of α_{i}, i.e., must be in J_{μ_{i}}/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}). Since dim R/J_{μ_{i}} is assumed to be n! and the number of basis elements in Γ(ℙ^{k−1}, P) involving a b_{i} is equal to the dim Γ(ℙ^{k−1}, P) − n!, these basis elements involving a b_{i} must span J_{μ_{i}}/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}). Thus we arrive at the distributive lattice structure.

Proposition 4.10. Assuming the hypothesis of Proposition 4.9, there is a basis for

Γ(ℙ^{k−1}, P) ≅ R/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}})

such that each space J_{μ_{i}}/(J_{μ_{1}} ∩ ⋯ ∩ J_{μ_{k}}) is spanned by a subset of this basis.

We additionally make the following observations. We notice that certain pieces of the direct sum decomposition of Γ(ℙ^{k−1}, P) correspond to certain exterior powers of Q, in other words to certain intersections of the spaces H_{μ}_{1}, …, H_{μ}_{k}. For example, the basis elements which belong to the trivial piece, Γ(ℙ^{k−1}, Λ^{0}Q^{\frac{n!}{k}}) = (Λ^{0}V)^{\frac{n!}{k}}, correspond to R/(J_{μ_{1}} + ⋯ + J_{μ_{k}}) ≅ H_{μ}_{1} ∩ ⋯ ∩ H_{μ}_{k} since the basis elements coming from the trivial piece remain linearly independent in every fiber, P_{I_{μ_{1}}}, …, P_{I_{μ_{k}}}. A basis element coming from Γ(ℙ^{k−1}, Λ^{1}Q) = V has the property that it is mapped to zero in eactly one of the fibers P_{I_{μ_{1}}}, …, P_{I_{μ_{k}}}, e.g., b_{i} ∈ V is mapped to zero in the fiber P_{I_{μ_{i}}} and no other fiber P_{I_{μ_{j}}} for j ≠ i. Thus, these basis elements represent elements in the intersection of k − 1 modules. In general, a basis element coming from Γ(ℙ^{k−1}, Λ^{j}Q) = Λ^{j}V corresponds to the intersection of k − j of these spaces. For example, the basis element of the form b_{i_{1}} ∧ ⋯ ∧ b_{i_{j}} ∈ Λ^{j}V maps to zero in exactly the fibers with the same indices, P_{I_{μ_{i_{1}}}} , …, P_{I_{μ_{i_{j}}}}.

In order to explain the geometric version of Conjecture 2.5 first we observe that there is an equivariant S_{n} action on P induced from the action on B^{⊗n}, and an equivariant T^{2} action coming from the action on Hilb^{n}(𝔸^{2}). These actions commute. Thus, P is vector bundle of S_{n} modules and the action of T^{2} gives it a double grading. We can write P|_{ℙ^{k−1}} as

P|_{ℙ^{k−1}} ≅ (Λ^{0}Q ⊗ V_{(0)}) ⊕ (Λ^{1}Q ⊗ V_{(1)}) ⊕ ⋯ ⊕ (Λ^{k−1}Q ⊗ V_{(k−1)}), (4.13)

where in any particular fiber, V_{(i)} is an _{\frac{n!}{k\binom{k−1}{i}}} dimensional doubly graded S_{n} module. On

the level of polynomials in R representing elements of Γ(ℙ^{k−1}, P), the T^{2} action gives the bidegrees and the S_{n} action permutes the indices.

Now, recall the pairing B^{⊗n}⊗B^{⊗n} → O(1) given in (3.2) induces a perfect pairing P ⊗P → O(1), which is equivariant under the two group actions. This pairing also gives an equivariant isomorphism P ≅ P^{∗} ⊗ O(1). Furthermore, it can be shown that the perfect pairing, P ⊗ P → O(1), must induce a perfect pairing

Λ^{j}Q ⊗ Λ^{k−1−j}Q → O(1)|_{ℙ^{k−1}},

where Λ^{j}Q and Λ^{k−1−j}Q appear as summands in the decomposition of P|_{ℙ^{k−1}}.

We observe that the T^{2} character of (Λ^{n}B)_{I_{μ_{i}}} ≅ O(1)_{I_{μ_{i}}} is t^{n(μ_{i})}q^{n(μ′_{i})} and S_{n} character is the sign character. Thus, to explain the Ψ_{i} operator defined in (2.11), we note that the perfect pairing defines an isomorphism

(Λ^{j}Q)_{I_{μ_{i}}} ≅ ((Λ^{k−1−j}Q)^{∗} ⊗ Λ^{n}B)_{I_{μ_{i}}},

where the dual corresponds to inverting the parameters q and t, and tensoring with Λ^{n}B ≅ O(1) corresponds to multiplying by t^{n(μ_{i})}q^{n(μ′_{i})} and applying the operator ω to the S_{n} character. Thus it follows on the Frobenius characteristic level that

F((Λ^{j}Q)_{I_{μ_{i}}}) = Ψ_{i}(F((Λ^{k−1−j}Q)_{I_{μ_{i}}})).

In summary, the Γ(ℙ^{k−1}, P) represents ∑ H_{μ_{i}} and each fiber P_{I_{μ_{i}}} corresponds to one of the modules H_{μ_{i}}. Partial fibers of a particular summand in P|_{ℙ^{k−1}} correspond to the modules H_{S}. The pairing implies that for i ∈ S,

F(H_{S}) = Ψ_{i}(H_{S^{C}∪{i}}).

This relation is with respect to the T^{2} × S_{n} representation of Λ^{n}B ≅ O(1) whose character is given by t^{n(μ_{i})}q^{n(μ′_{i})} for T^{2} and the sign character for S_{n}.

To explain Conjecture 2.6, we observe that this is a simple consequence of the isomorphism in Proposition 4.9.

Corollary 4.11. Let μ ⊢ (n + 1) with k corners. Let μ_{1}, μ_{2}, ..., μ_{k} ⊢ n, where μ_{i} is μ with the ith corner removed. Then (under the hypothesis of Proposition 4.9) for {i_{1}, …, i_{m}} ⊆ {1, …, k},

dim(H_{μ}_{i_{1}} ∩ ⋯ ∩ H_{μ}_{i_{m}}) = \frac{n!}{m}.

Finally, we mention that the Decomposition Conjecture, which gave the multiplicities of each exterior power of Q, geometrically explained the conjectures of Bergeron and Garsia. However, we note that the pairing relation, or equivalently, P ≅ P^{∗} ⊗O(1) always holds if P is assumed to be a vector bundle. Assuming the decomposition of P into exterior powers of Q without knowing the multiplicities, we can use the pairing relation along with the theory of Macdonald polynomials to still arrive at the multiplicities. We plan to discuss this in a separate paper.

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Department of Mathematics, U.C. San Diego, CA 92093-0112

E-mail address: cchang@math.ucsd.edu