SOME NATURAL BIGRADED S_{n}-MODULES and q,t-KOSTKA COEFFICIENTS.

A. M. Garsia ^{(∗)} and M. Haiman ^{(∗)}

This work is gratefully dedicated to Dominique Foata for his inspiring and pioneering work in algebraic combinatorics. We hope that he will find it to be in harmony with the Lotharingian spirit which he has nurtured for so many years.

Submitted: May 15, 1995; Accepted: January 26, 1996

ABSTRACT. We construct for each μ ⊢ n a bigraded S_{n}-module H_{μ} and conjecture that its Frobenius characteristic C_{μ}(x; q, t) yields the Macdonald coefficients K_{λμ}(q, t). To be precise, we conjecture that the expansion of C_{μ}(x; q, t) in terms of the Schur basis yields coefficients C_{λμ}(q, t) which are related to the K_{λμ}(q, t) by the identity C_{λμ}(q, t) = K_{λμ}(q, 1/t)t^{n(μ)}. The validity of this would give a representation theoretical setting for the Macdonald basis {P_{μ}(x; q, t)}_{μ} and establish the Macdonald conjecture that the K_{λμ}(q, t) are polynomials with positive integer coefficients. The space H_{μ} is defined as the linear span of derivatives of a certain bihomogeneous polynomial Δ_{μ}(x, y) in the variables x_{1}, x_{2}, …, x_{n}, y_{1}, y_{2}, …, y_{n}. On the validity of our conjecture H_{μ} would necessarily have n! dimension. We refer to the latter assertion as the n!-conjecture. Several equivalent forms of this conjecture will be discussed here together with some of their consequences. In particular, we derive that the polynomials C_{λμ}(q, t) have a number of basic properties in common with the coefficients K̃_{λμ}(q, t) = K_{λμ}(q, 1/t)t^{n(μ)}. For instance, we show that C_{λμ}(0, t) = K̃_{λμ}(0, t), C_{λμ}(q, 0) = K̃_{λμ}(q, 0) and show that on the n! conjecture we must also have the equalities C_{λμ}(1, t) = K̃_{λμ}(1, t) and C_{λμ}(q, 1) = K̃_{λμ}(q, 1). The conjectured equality C_{λμ}(q, t) = K_{λμ}(q, 1/t)t^{n(μ)} will be shown here to hold true when λ or μ is a hook. It has also been shown (see [9]) when μ is a 2-row or 2-column partition and in [18] when μ is an augmented hook.

Introduction

Throughout this writing μ will be a partition of n and μ′ will denote its conjugate. We shall also identify μ with its Ferrers’ diagram. As customary, for μ = (μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{k} > 0), we let

n(μ) = ∑_{i=1}^{k} (i − 1) μ_{i} .  I.1

We draw μ as a set of successive left justified rows of lattice squares by the French convention, with row lengths decreasing from bottom to top. We also let the coordinates (i, j) of a cell s ∈ μ respectively measure the height of s and the position of s in its row. We shall set

B_{μ}(q, t) = ∑_{(i,j)∈μ} t^{i−1}q^{j−1} .  I.2

(∗) Work carried out under NSF grant support.

1 The pairs (i−1,j−1) occurring in this sum will be briefly referred to as the biexponents of μ and B_{μ}(q, t) itself will be called the biexponent generator of μ. Now let (p_{1}, q_{1}), …, (p_{n}, q_{n}) denote the set of biexponents arranged in lexicographic order and set

Δ_{μ}(x, y) = Δ_{μ}(x_{1}, …, x_{n}; y_{1}, …, y_{n}) = det ∥x_{i}^{p_{j}}y_{i}^{q_{j}}∥_{i,j=1..n} .  I.3

This given we let H_{μ} be the collection of polynomials in the variables x_{1}, …, x_{n}; y_{1}, …, y_{n}obtained by taking the linear span of all the partial derivatives of Δ_{μ}. In symbols we may write

H_{μ} = L{∂_{x}^{p}∂_{y}^{q} Δ_{μ}(x; y)}  I.4

where ∂_{x}^{p} = ∂_{x_{1}}^{p_{1}} ⋯ ∂_{x_{n}}^{p_{n}} and ∂_{y}^{q} = ∂_{y_{1}}^{q_{1}} ⋯ ∂_{y_{n}}^{q_{n}}. Sometimes we may write H_{μ}[X_{n}, Y_{n}] for H_{μ} to distinguish it from the singly graded module H_{μ}[X_{n}] studied in [12]. Here X_{n} and Y_{n} denote the two alphabets X_{n} = {x_{1}, …, x_{n}}, Y_{n} = {y_{1}, …, y_{n}}.

The natural action of a permutation σ = (σ_{1}, …, σ_{n}) on a polynomial P (x_{1}, …, x_{n}; y_{1}, …, y_{n}) is the so called diagonal action which is defined by setting

σ P (x_{1}, …, x_{n}; y_{1}, …, y_{n}) = P (x_{σ_{1}}, …, x_{σ_{n}}; y_{σ_{1}}, …, y_{σ_{n}}) .

Since σΔ_{μ} = ±Δ_{μ} according to the sign of σ, the space H_{μ}[X_{n}; Y_{n}] necessarily remains invariant under this action. Our main object of study here is the character of the corresponding S_{n} representation. Note that, since Δ_{μ} is bihomogeneous of degree n(μ) in x and n(μ′) in y, we also have the direct sum decomposition

H_{μ}[X_{n}; Y_{n}] = ⨁_{h=0}^{n(μ)} ⨁_{k=0}^{n(μ′)} H_{h,k}(H_{μ}) ,

where H_{h,k}(H_{μ}) denotes the subspace of H_{μ} spanned by its bihomogeneous elements of degree h in x and degree k in y. Since the diagonal action clearly preserves bidegree, each of the subspaces H_{h,k}(H_{μ}) is also S_{n}-invariant. Thus we see that H_{μ} has the structure of a bigraded module. The generating function of the characters of its bihomogeneous components, which we shall refer to as the bigraded character of H_{μ}, may be written in the form

π^{μ}(q, t) = ∑_{h=0}^{n(μ)} ∑_{k=0}^{n(μ′)} t^{h}q^{k} char H_{h,k}(H_{μ}) .  I.5

We then let C_{μ}(x; q, t) denote the Frobenius characteristic of H_{μ} that is

C_{μ}(x; q, t) = \frac{1}{n!} ∑_{σ∈S_{n}} π^{μ}(σ; q, t)p_{λ(σ)}(x) ,  I.6

where π^{μ}(σ; q, t) denotes the value of π^{μ}(q, t) at σ and p_{λ(σ)}(x) is the power symmetric function indexed by the shape of σ. Expanding C_{μ}(x; q, t) in terms of the Schur basis we get

C_{μ}(x; q, t) = ∑_{μ⊢n} S_{λ}(x) C_{λμ}(q, t) .  I.7

where the coefficients C_{λμ}(q, t) are themselves polynomials with positive integer coefficients. Indeed, the coefficient of t^{h}q^{k} in C_{λμ}(q, t) gives the multiplicity of the irreducible S_{n} character χ^{λ} in the character of the submodule H_{h,k}(H_{μ}).

There is no question that the intimate relationship of the polynomial Δ_{μ} and the corresponding module H_{μ} to the partition μ makes the character π_{μ}(q, t) a very natural object of study. However, some deep developments are actually involved here which may not be apparent from the simplicity of our construction of H_{μ}. Indeed, in all the cases that we have been able to compute them, the polynomials C_{λμ}(q, t) are in remarkable agreement with the available tables of the Macdonald q, t-Kostka coefficients. To make this connection precise we recall that Macdonald in [17] established the existence of a symmetric function basis {P_{μ}(x; q, t)}_{μ} uniquely characterized by the following conditions

\begin{matrix}a) P_{λ} = S_{λ} + ∑_{μ<λ} S_{μ} ξ_{μλ}(q, t) \\ b) ⟨P_{λ} , P_{μ}⟩_{q,t} = 0 for λ ≠ μ \end{matrix}  I.8

Where ⟨ , ⟩_{q,t} denotes the scalar product of symmetric polynomials defined by setting for the power basis {p_{ρ}}

⟨p_{ρ_{1}} , p_{ρ_{2}}⟩_{q,t} = \{ \begin{matrix}z_{ρ} p_{ρ}[\frac{1−q}{1−t} ] & if ρ_{1} = ρ_{2} = ρ and \\ 0 & otherwise \end{matrix}  I.9

Here we use λ-ring notation and z_{ρ} is the integer that makes n!/z_{ρ} the number of permutations with cycle structure ρ. There are a number of outstanding conjectures concerning these polynomials (see [17]). We are dealing here with those involving the so called integral forms J_{μ}(x; q, t) and their associated Macdonald-Kostka coefficients K_{λμ}(q, t). We shall use the same notation as in [17]. In particular {Q_{λ}(x; q, t)} denotes the basis dual to {P_{λ}(x; q, t)} with respect to the scalar product ⟨ , ⟩_{q,t}. Clearly, I.8 b) gives

Q_{λ}(x; q, t) = d_{λ}(q, t) P_{λ}(x; q, t) ,  I.10

for a suitable rational function d_{λ}(q, t). However in [17] it is shown that

d_{λ}(q, t) = \frac{h_{λ}(q, t)}{h′_{λ}(q, t)}

with

h_{λ}(q, t) = ∏_{s∈λ}(1 − q^{a_{λ}(s)}t^{l_{λ}(s)+1}) , h′_{λ}(q, t) = ∏_{s∈λ}(1 − q^{a_{λ}(s)+1}t^{l_{λ}(s)})

where s denotes a generic lattice square and a_{λ}(s), l_{λ}(s) respectively denote the arm and the leg of s in the Ferrers’ diagram of λ.

We recall from [17] that

J_{μ}(x; q, t) = h_{μ}(q, t)P_{μ}(x; q, t) = h′_{μ}(q, t)Q_{μ}(x; q, t) ,  I.11

and the coefficients K_{λμ}(q, t) are defined through an expansion which in λ-ring notation may be written as

J_{μ}(x; q, t) = ∑_{λ} S_{λ}[X(1 − t)] K_{λμ}(q, t) ,  I.12

where for an alphabet of n letters we set X = x_{1} + x_{2} + ⋯ + x_{n}. Macdonald conjectures that these coefficients are polynomials in q and t with non-negative integer coefficients. We shall refer to this here and after as the MPK conjecture. Now, as we shall see, we can calculate by hand our coefficients C_{λμ}(q, t) without difficulty for all partitions of n ≤ 6 and for most partitions of 7. In all these cases we invariably verify that

C_{λμ}(q, t) = K_{λμ}(q, 1/t) t^{n(μ)} .  I.13

Our investigations have yielded overwhelming evidence of the general validity of this identity. In terms of C_{μ}(x; q, t), I.13 can be written in the form

C_{μ}(x; q, t) = H̃_{μ}(x; q, t) ,  I.14

where we have set

H̃_{μ}(x; q, t) = H_{μ}(x; q, 1/t) t^{n(μ)}  I.15

with

H_{μ}(x; q, t) = ∑_{λ} S_{λ} K_{λμ}(q, t) = J_{μ}[X/(1 − t); q, t] .  I.16

We shall here and after refer to I.14 as the C = H̃ conjecture. Macdonald in [17] derives a number of properties of the K_{λμ}(q, t), in particular he shows that for any partition μ

K_{λμ}(1, 1) = f_{λ} ,  I.17

where as customary f_{λ} denotes the number of standard tableaux of shape λ. Thus the validity of I.14 requires that our module H_{μ} should yield a bigraded version of the left regular representation. Then a fortiori H_{μ} should have n! dimension. It is easy to see that for μ = 1^{n} and μ = (n) Δ_{μ} reduces to the Vandermonde determinant in x and y respectively. In these cases it is a classical result (see [21]) that dim H_{μ} = n!. Surprisingly, so far even this simpler identity has been difficult to establish in full generality. We shall here and after refer to it as the n! conjecture. This conjecture has been verified (by computer) for all μ ⊢ n ≤ 8. Here we shall deal only with the case when μ is a hook. A proof of C = H̃ for the case that μ is a two-column or a two-row partition may be found in [9].

The contents of this paper are divided into five sections. In the first section we show that dim H_{μ} ≤ n! and show that when equality holds, H_{μ} is, as desired, a bigraded version of the left regular representation. We also derive there the identities C_{λμ}(0, t) = K̃_{λμ}(0, t) and C_{λμ}(q, 0) = K̃_{λμ}(q, 0) mentioned above. In the second section we show that on the n! conjecture, we must necessarily have also the equalities C_{λμ}(1, t) = K̃_{λμ}(1, t) and C_{λμ}(q, 1) = K̃_{λμ}(q, 1). We do this by introducing two additional S_{n}-modules M_{μ}^{x} and M_{μ}^{y} whose x-graded and y-graded characters respectively have the expansions

char_{t} M_{μ}^{x} = ∑_{λ⊢n} χ^{λ} K̃_{λμ}(1, t)

and

char_{q} M_{μ}^{y} = ∑_{λ⊢n} χ^{λ} K̃_{λμ}(q, 1) .

Then we show that on the n! conjecture both of them must be equal to H_{μ}. In the third section, we show that conversely the equality of M_{μ}^{x} and M_{μ}^{y} implies the n! conjecture. The developments in this section should throw some light on the true nature of the n! conjecture.

In the fourth section we present some further remarkable conjectures regarding our space H_{μ}. We show there that their validity not only implies the n!-conjecture but reveals that several properties of the coefficients K̃_{λμ} (which may be observed from the computer constructed tables) are due to a very surprising algebraic combinatorial mechanism. Finally in the fifth section we give our original (1992) proof of the C = H̃ conjecture for μ a hook. We should mention that since then E. Allen (see [1]) found a completely different proof which uses Rota’s straightening in biletter algebras.

Our crucial tools in this paper are some general results on orbit harmonics which are developed in [9]. The reader will also find in [9] a leisurely presentation of all the identities on Macdonald polynomials we shall have to take for granted in the present treatment. We should also mention that in [10] we have shown that the polynomials C_{μ}(x; q, t) satisfy Pieri rules which are consistent with C = H̃. Further evidence supporting our conjectures may be found in [11].

1. The singly graded versions.

To proceed we need some material from the theory of orbit harmonics. We shall only review the definitions and state the basic results to the extent needed to make the presentation here selfcontained. The reader is referred to [9] for a fully detailed, elementary treatment of the topic and further information.

Let R = Q[x] denote the ring of polynomials in x_{1}, …, x_{m} with rational coefficients. We can introduce a scalar product in R by setting for any pair P, Q ∈ R

⟨P , Q⟩ = L_{o}P (∂_{x})Q(x)  1.1

where P (∂_{x}) denotes the differential operator obtained by replacing the variable x_{i} in P by the corresponding partial derivative ∂_{x_{i}}, and L_{o} represents evaluation at the origin. If A is an m × m matrix we can make A act on m-dimensional row vectors by right multiplication and set for each polynomial P (x) = P (x_{1}, …, x_{m})

T_{A} P (x) = P (xA) .  1.2

Note that this action preserves degree as well homogeneity. It is easy to show that, if A is orthogonal, then this action also preserves our scalar product. We shall need a more general grading of R defined by setting for any monomial x^{p} = x_{1}^{p_{1}} ⋯ x_{m}^{p_{m}}

d_{w}(x^{p}) = ∑_{i=1}^{m} w_{i}p_{i}

where w_{1}, …, w_{m} are given positive weights. For any polynomial P = ∑_{p} c_{p} x^{p} and any integer k we shall set

π_{k}^{w} P = ∑_{d_{w}(x^{p})=k} c_{p} x^{p}

and call it the w-homogeneous component of w-degree k.(*) This given we shall say that a subspace V of R is w-homogeneous if π_{k}^{w}V ⊆ V for all k. Since our scalar product makes the monomials into an orthogonal set, we necessarily have that components of different w-degree will also be orthogonal. In particular the orthogonal complement of a homogeneous subspace of R is also homogeneous. Here and after we shall use the symbol ⊥ to denote orthogonal complementation with respect to the scalar product in 1.1. We should note that as long as it is applied to homogeneous spaces, this operation behaves like orthogonal complementation in a finite dimensional space. To be precise we have

Proposition 1.1 If V is a w-homogeneous subspace then so is V^{⊥} and

(V^{⊥})^{⊥} = V .  1.3

If J ⊆ R is an ideal then

J^{⊥} = { P ∈ Q[X] : f(∂_{x})P (x) = 0 ∀ f ∈ J } ,  1.4

in particular J^{⊥} is closed under differentiation.

Proof

The first assertion and identity 1.3 are immediate since ⊥ acts separately on each w-homogeneous component of R, and each component is finite-dimensional. As for 1.4 we note that if P ∈ R is orthogonal to J then, since J is an ideal, we must necessarily have

L_{o}∂_{x}^{p}f(∂_{x})P (X) = 0 ( ∀f ∈ J and ∀p ) .

However, this means that f(∂_{x})P (x) and all its derivatives vanish at the origin. Taylor’s theorem then gives that this polynomial must vanish identically. Thus 1.4 must hold true as asserted. The last assertion is a trivial consequence of 1.4. Q.E.D.

We should note that if J is not a homogeneous ideal the collection in 1.4 may consist only of {0} (for example let J be the ideal (x − a) for any non vanishing scalar a). Thus 1.3 fails when J is not homogeneous.

(*) In a context when the grading is kept fixed we will omit the prefix w.

Now let J ⊆ R be any ideal and set R_{J} = R/J. The associated w-graded ideal of J is by definition the ideal

gr_{w} J = ( h_{w}(P) : P ∈ J ) ,

where for each P ∈ R we let h_{w}(P) denote the w-homogeneous component of P of highest w-degree. This given, the quotient ring R/gr_{w} J is referred to as the w-graded version of R_{J} and denoted by gr_{w}R_{J}.

Let now G be a finite group of orthogonal matrices. In the present context there is no loss in assuming that the matrices in G have rational entries. Given a point ρ = (ρ_{1}, …, ρ_{m}) its G-orbit is the set

[ρ]_{G} = { ρA : A ∈ G }.

We also set

R_{[ρ]_{G}} = R/J_{[ρ]_{G}}

where

J_{[ρ]_{G}} = { P ∈ R : P (x) = 0 ∀ x ∈ [ρ]_{G} }

We may view R_{[ρ]_{G}} as the coordinate ring of [ρ]_{G} (considered as an algebraic variety). Since the ideal J_{[ρ]_{G}} is clearly G-invariant, it follows that G acts on R_{[ρ]_{G}}. In fact, it is easy to see that the corresponding representation is equivalent to the action of G on left cosets of the stabilizer of ρ. In particular, when ρ is a regular point (i.e. when its stabilizer is trivial), then R_{[ρ]_{G}} is simply a version of the left regular representation of G. If, for each A ∈ G, the action in 1.2 preserves the given w-degree then we can associate to [ρ]_{G} two further G-modules. Namely, gr_{w} R_{[ρ]_{G}} and the orthogonal complement

H_{[ρ]_{G}} = (gr_{w} J_{[ρ]_{G}})^{⊥} .  1.5

Note that if f(x) is any G-invariant polynomial then f(x) − f(ρ) belongs to the ideal J_{[ρ]_{G}} and hence if f(x) is also w-homogeneous then f(x) itself must belong to gr_{w}J_{[ρ]_{G}}. This implies that any element P ∈ H_{[ρ]_{G}} must satisfy the differential equation

f(∂_{x}) P = 0 .

Since G is assumed to consist of orthogonal matrices, so that the polynomial x_{1}^{2} + ⋯ + x_{m}^{2} is Ginvariant, we see that the elements of H_{[ρ]_{G}} must be harmonic polynomials. For this reason we shall refer to them as the harmonics of the orbit [ρ]_{G}. Clearly, gr_{w} R_{[ρ]_{G}} and H_{[ρ]_{G}} are equivalent w-graded modules since the elements of H_{[ρ]_{G}} may be taken as representatives of the classes of gr_{w} R_{[ρ]_{G}}. More importantly, it develops that these two spaces realize a graded version of the representation corresponding to R_{[ρ]_{G}}. This is not difficult to show and we refer the reader to [9] for a proof. We see then that in this manner we have a mechanism for producing a variety of graded versions of the regular representation of G. We shall see shortly that the S_{n}-module H_{μ} defined in the introduction can be constructed by means of this mechanism.

To do this we need to introduce a few more ingredients . Given a k-part partition μ = (μ_{1} ≥ μ_{2}, … ≥ μ_{k} > 0) let h = μ_{1} denote the number of parts of its conjugate μ′ and let {α_{1}, …, α_{k}}, {β_{1}, …, β_{h}} be distinct rational numbers. Alternatively, the α’s and β’s may be taken to be additional indeterminates. Recall that an injective tableau T of shape μ ⊢ n is a labelling of the cells of μ by the numbers {1, 2, …, n}. The collection of all such tableaux will be denoted by IT (μ). For each T ∈ IT (μ) and l ∈ {1, …, n} we let s_{T} (l) = (i_{T} (l), j_{T} (l)) denote the cell of T which contains the label l. This given we construct a point ρ(T) = (a(T), b(T)) in 2n-dimensional space by setting

a_{l}(T) = α_{i_{T} (l)} , b_{l}(T) = β_{j_{T} (l)}  (for l = 1, …, n ) .  1.6

Note that the collection

{ ρ(T) : T ∈ IT (μ) }  1.7

consists of n! distinct points. Indeed, since the α′s and the β ′s are assumed to be all distinct, we can reconstruct the position of any label l in T by simply looking at the coordinates a_{l} and b_{l} of ρ(T). Note that the collection in 1.7 is simply the S_{n}-orbit under the diagonal action of any one its elements, where the diagonal action of S_{n} on 2n-dimensional space is defined by setting for σ ∈ S_{n}

σ(x_{1}, …, x_{n}; y_{1}, …, y_{n}) = (x_{σ_{1}}, …, x_{σ_{n}}; y_{σ_{1}}, …, y_{σ_{n}}) .

We can thus construct graded versions of the left regular representation of S_{n} by the mechanism described above following Proposition 1.1. In this instance we take G to be the group of permutation matrices giving the diagonal action of S_{n} on the vectors (x_{1}, …, x_{n}; y_{1}, …, y_{n}). The number of variables m is here equal to 2n and R is the ring of polynomials

R = R[X, Y] = Q[x_{1}, …, x_{n}; y_{1}, …, y_{n}] .

As for the w-degree, we still have some freedom since any vector w = (w_{1}, …, w_{2n}) which gives a weight w_{x} to all the x′_{i}s and another weight w_{y} to all the y′_{j}s will produce a grading that is invariant under the diagonal action. To be consistent with the notation introduced earlier we shall denote the orbit in 1.7 by [ρ_{μ}]. We shall also denote by R_{[ρ_{μ}]}, gr_{w}R_{[ρ_{μ}]} and H_{[ρ_{μ}]} the corresponding coordinate ring, its graded version and the corresponding space of harmonics. Here we omit the subscript G, as the group will remain fixed throughout the rest of our treatment. The definition of these spaces suggests that they might depend on our choice of the α_{i}’s and β_{j}’s. This is certainly so for the coordinate ring R_{[ρ_{μ}]}. Nevertheless, the next two results give strong evidence that the space of harmonics H_{[ρ_{μ}]} as well as the ideal gr_{w}J_{[ρ_{μ}]} and the quotient ring gr_{w} R_{[ρ_{μ}]} only depend on the choice of the partition μ.

The point of departure is the following remarkable fact.

Proposition 1.2

If (i, j) is a cell outside μ then for any s ∈ {1, …, n} the monomial x_{s}^{i−1}y_{s}^{j−1} belongs to the ideal gr_{w} J_{[ρ_{μ}]}. In particular if a monomial x^{p}y^{q} = x_{1}^{p_{1}} ⋯ x_{n}^{p_{n}}y_{1}^{q_{1}} ⋯ y_{n}^{q_{n}} does not vanish in gr_{w} R_{[ρ_{μ}]} then all the pairs (p_{s}, q_{s}) must be biexponents of μ, and every polynomial in H_{[ρ_{μ}]} must be a linear combination of monomials satisfying the same condition.

Proof We claim that the polynomial

f(x, y) = ∏_{i′=1}^{i−1}(x_{s} − α_{i′}) ∏_{j′=1}^{j−1}(y_{s} − β_{j′})

must necessarily vanish on all of [ρ_{μ}]. Indeed, if f were non-zero at a point

(a, b) = (a_{1}, …, a_{n}, b_{1}, …, b_{n})

then the a_{s} could not take any of the values α_{1}, α_{2}, …, α_{i−1}. If T is the injective tableau giving (a, b) via the construction in 1.6, then the label s could not fall in any of the first i − 1 rows of T. Similarly, the non-vanishing of f at (a, b) would force the label s out of the first j − 1 columns of T. However, this would force s out of the diagram of μ altogether, an impossibility. We must therefore conclude that f ∈ J_{[ρ_{μ}]} and a fortiori its highest w-homogeneous component, x_{s}^{i−1}y_{s}^{j−1}, must lie in gr_{w} J_{[ρ_{μ}]}. This verifies our first assertion.

We deduce that a monomial x^{p}y^{q} = x_{1}^{p_{1}} ⋯ x_{n}^{p_{n}}y_{1}^{q_{1}} ⋯ y_{n}^{q_{n}} is in gr_{w} J_{[ρ_{μ}]} if it contains any x_{s}^{i−1}y_{s}^{j−1} as a factor. If the monomial does not vanish in gr_{w} R_{[ρ_{μ}]}, this forces all the pairs (p_{s} + 1, q_{s} + 1) inside the diagram of μ. But this is the same as saying that all the pairs (p_{s}, q_{s}) must be biexponents of μ. Finally, note that using Proposition 1.1 (equation 1.4) we deduce that every polynomial P ∈ H_{[ρ_{μ}]} must satisfy the differential equations

∂_{x_{s}}^{i−1}∂_{y_{s}}^{j−1}P (x, y) = 0 .

Therefore any monomial which contains x_{s}^{i−1}y_{s}^{j−1} as a factor must have zero coefficient in P. This completes the proof.

Theorem 1.1

For any choice of the α_{i}’s and β_{j}’s and any diagonally invariant w-grading we have the containment

H_{μ} ⊆ H_{[ρ_{μ}]}  1.8

Proof

Since H_{[ρ_{μ}]} as an S_{n}-module is, for any choice of our parameters, a version of the left regular representation of S_{n}, it must contain the alternating representation with multiplicity 1. This means that H_{[ρ_{μ}]} contains a polynomial Δ(x, y), unique up to a scalar factor, which alternates under the diagonal action. Such a polynomial is necessarily a combination of monomials x^{p}y^{q} = x_{1}^{p_{1}} ⋯ x_{n}^{p_{n}}y_{1}^{q_{1}} ⋯ y_{n}^{q_{n}} with distinct biexponent pairs (p_{1}, q_{1}), (p_{2}, q_{2}), …, (p_{n}, q_{n}). On the other hand, by the last assertion of Proposition 1.2 all these pairs must also be biexponents of μ. Since μ has altogether only n distinct biexponents, we must conclude that these pairs must be a permutation of the biexponents of μ. This forces Δ(x, y) to be a scalar multiple of the diagonal alternation of a tableau monomial

m_{T} (x, y) = ∏_{l=1}^{n} x_{l}^{i_{T} (l)−1}y_{l}^{j_{T} (l)−1} .  1.9

In other words Δ(x, y) must be a multiple of the polynomial Δ_{μ}(x, y) defined in I.3. This places Δ_{μ}(x, y) in H_{μ}. Since, by Proposition 1.1, H_{[ρ_{μ}]} is closed under differentiation, the entire space H_{μ} must be contained in H_{[ρ_{μ}]}. Q.E.D.

We are thus led to the following remarkable corollary.

Theorem 1.2

On the n! conjecture, for any S_{n}-invariant choice of w and any specialization of the α_{i}’s and β_{j}’s, we must have the equalities

H_{[ρ_{μ}]} = H_{μ} ,  1.10

and

gr_{w}J_{[ρ_{μ}]} = I_{μ} ,  1.11

where I_{μ} is the ideal of polynomials f(x, y) which kill Δ_{μ}, that is

I_{μ} = ( f ∈ R : f(∂_{x}, ∂_{y}) Δ_{μ}(x, y) = 0 ) .  1.12

In particular, H_{μ} must be a bigraded version of the left regular representation of S_{n}.

Proof

We only need to verify 1.11. To this end note that since gr_{w}J_{[ρ_{μ}]} is a w-homogeneous ideal we can use Proposition 1.1 and deduce from 1.10 that

gr_{w}J_{[ρ_{μ}]} = ( H_{μ} )^{⊥} .  1.13

Now for a polynomial f(x, y) to be orthogonal to H_{μ} it is necessary and sufficient that for all p amd q we have

L_{o} f(∂_{x}, ∂_{y}) ∂_{x}^{p}∂_{y}^{q}Δ_{μ} = 0 .

this means that the polynomial f(∂_{x}, ∂_{y})Δ_{μ} and all its derivatives must vanish at the origin. Taylor’s theorem then yields that it must vanish identically. Conversely, any element of I_{μ} is trivially orthogonal to H_{μ}. Thus 1.11 must hold true as asserted. The last assertion is an immediate consequence of 1.10. Q.E.D.

The module H_{μ} has a linear automorphism that complements bidegree and conjugates representations, which we shall refer to as the flip. To define it we simply set for any ϕ ∈ H_{μ}

flip ϕ = ϕ(∂_{x}, ∂_{y}) Δ_{μ}(x, y) .  1.14

We can easily see that if ϕ is of bidegree (h, k) then flip ϕ is of bidegree (h′, k′) with

h + h′ = n(μ) and k + k′ = n(μ′)  1.15

Moreover, because of the alternating property of Δ_{μ} the isotypic component of H_{h,k}(H_{μ}) corresponding to character χ^{λ} is mapped by flip into the isotypic component of H_{h′,k′}(H_{μ}) corresponding to χ^{λ′}. This simple observation translates into the following basic identities.

Theorem 1.3 For any two partitions λ, μ we have

C_{λμ}(q, t) = C_{λ′μ}(1/q, 1/t)q^{n(μ′)}t^{n(μ)} .  1.16

Equivalently, for all μ

C_{μ}(x; q, t) = ω C_{μ}(x; 1/q, 1/t)q^{n(μ′)}t^{n(μ)}  1.17

where ω denotes the involution which sends the homogeneous basis of symmetric polynomials into the elementary basis.

The corresponding identity to 1.16 for the Macdonald coefficients K_{λμ}(q, t) is among the properties derived by Macdonald in [17]; thus Theorem 1.3 is our first evidence for the C = H̃ conjecture. Further evidence comes from a relationship between the C_{λμ}(q, t), specialized to q = 0, and the Kostka-Foulkes coefficients K_{λμ}(t).

To obtain this relationship by the shortest and most elementary path we shall have to rely on results from [2] and [12]. To this end we need to review some definitions and recall some basic theorems.

If A = {a_{1}, a_{2}, …, a_{m}} is an ordered subset of {1, 2, …, n} let Δ(X_{A}) denote the Vandermonde determinant in the subalphabet X_{A} = {x_{a_{1}}, x_{a_{2}}, …, x_{a_{m}}}. To be definite (since order matters), we set

Δ(X_{A}) = det∥x_{a_{i}}^{j−1}∥_{i,j=1}^{m} ,

Let μ = (μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{k}) be a partition of n and μ′ = (μ′_{1} ≥ μ′_{2} ≥ ⋯ ≥ μ′_{h} > 0) denote its conjugate. Let T be a tableau of shape μ with entries 1, 2, …, n and let C_{1}, C_{2}, …, C_{h} denote the columns of T ordered from left to right. Note, that with this notation, column C_{i} is necessarily of length μ′_{i}. We shall also agree to order the elements of C_{i} as they occur in T from bottom to top. This given, we set

Δ_{T} (x) = Δ(X_{C_{1}})Δ(X_{C_{2}}) … Δ(X_{C_{h}}) ,  1.18

and refer to it as the Garnir polynomial corresponding to T. We shall also say that Δ_{T} (x) is standard, of shape μ, etc., if the same holds true for T itself.

For example if

T_{1} = 1 2 3 4 5 6 7  1.19

Then

Δ_{T_{1}}(x) = det ( \begin{matrix}1 & x_{5} & x_{5}^{2} \\ 1 & x_{2} & x_{2}^{2} \\ 1 & x_{1} & x_{1}^{2} \end{matrix} ) × det ( \begin{matrix}1 & x_{6} \\ 1 & x_{3} \end{matrix} ) × det ( \begin{matrix}1 & x_{7} \\ 1 & x_{4} \end{matrix} ) .  1.20

These polynomials were used by Garnir in [14], in his reconstruction of Young’s natural representation (see also [19]). For convenience let CI(μ) denote the collection of all column increasing injective tableaux of shape μ with entries 1, 2, …, n. This given it is not difficult to show (see [2]) that the space

Γ_{μ} = L{Δ_{T} : T ∈ CI(μ) } = L{Δ_{T} : T ∈ IT (μ) } ,  1.21

the linear span of the Garnir polynomials of shape μ, is an irreducible S_{n}-module with character given by χ^{μ} in the Young indexing.

Let us split ρ_{μ} into its x and y parts by writing ρ_{μ} = (a_{μ}, b_{μ}). Recall that a_{μ} = a(T_{o}) and b_{μ} = b(T_{o}) according to 1.6 applied to T_{o} (any fixed tableau of shape μ). We may then apply the orbit harmonics construction separately to each of the points a_{μ} and b_{μ}. Of course here we should take the number of variables m to be n, and G to be the group of permutation matrices giving the standard action of S_{n} on n-component vectors (x_{1}, …, x_{n}). If we do this we are led to two additional spaces of orbit harmonics. To distinguish them from each other and from the space H_{μ} defined by I.4, we shall denote the latter by H_{μ}[X_{n}, Y_{n}], H_{[a_{μ}]} by H_{μ}[X_{n}] and H_{[b_{μ}]} by H_{μ′}[Y_{n}]. The reasons for this last piece of notation are clear. We can easily see that b_{μ} is precisely what a_{μ′} would be if we replaced the α_{i}’s by the β_{j}’s. Moreover, to study how these three spaces fit together we need to consider the elements of H_{[b_{μ}]} as polynomials in the variables y_{1}, …, y_{n}.

Contrarily to the present situation concerning our space H_{μ}[X_{n}, Y_{n}], these last two spaces of harmonics are well understood. Indeed, there are very natural monomial bases for the corresponding coordinate rings, and very useful sets of generators are available for the graded versions of the corresponding defining ideals (see [12]). We shall review the definitions of all these ingredients as our developments will require. For the moment we only need to recall the following basic facts.

Theorem 1.4 For any choices of the α′_{i}s we have

H_{μ}[X_{n}] = L{∂_{x}^{p}Δ_{T} (x) : T ∈ CI(μ) } .  1.22

Moreover, if π^{μ}(t) denotes the (singly) graded character of H_{μ}[X_{n}], we have the expansion

π^{μ}(t) = ∑_{λ⊢n} χ^{λ} K̃_{λμ}(t)  1.23

where the coefficients K̃_{λμ}(t) are related to the Kostka-Foulkes polynomials by the identity

K̃_{λμ}(t) = K_{λμ}(1/t) t^{n(μ)} .  1.24

.

Elementary proofs of 1.22 and 1.23 can be found in [2]. Here we shall only add a few comments. Essentially, 1.22 and 1.23 may be interpreted as stating that the analogues of the n! and the C = H̃ conjectures hold true hold true for [a_{μ}]. The space H_{[a_{μ}]} is not a cone (i.e., the space of derivatives of a single polynomial) for the simple reason that a_{μ} is not a regular point with respect to the standard action of S_{n}. Nevertheless, it is the linear span of the derivatives of a very natural collection of polynomials. Clearly, similar relations hold for the space H_{[b_{μ}]} as well. We need only replace μ by μ′ and x_{1}, …, x_{n} by y_{1}, …, y_{n} in 1.22 and 1.23.

An immediate corollary of Theorem 1.4 is an identification of the marginal values of our coefficients C_{λμ}(q, t). Namely,

Theorem 1.5

The homogeneous components of 0 y-degree and 0 x-degree in H_{μ}[X_{n}, Y_{n}] are respectively equal to the spaces H_{μ}[X_{n}] and H_{μ′}[Y_{n}]. In particular we derive that

C_{λμ}(0, t) = K̃_{λμ}(t) = K̃_{λμ}(0, t) , C_{λμ}(q, 0) = K̃_{λμ′}(q) = K̃_{λμ}(q, 0) ,  1.25

and thus also

C_{λμ}(q, t)|_{q^{n(μ′)}} = K̃_{λμ}(q, t)|_{q^{n(μ′)}} , C_{λμ}(q, t)|_{t^{n(μ)}} = K̃_{λμ}(q, t)|_{t^{n(μ)}} ,  1.26

Proof

We can obtain our polynomial Δ_{μ}(x, y) by alternating the monomial in 1.9 corresponding to the superstandard tableau T_{o} of shape μ, that is, the tableau obtained by numberging the cells of μ from left to right within each row, one row at a time, from the bottom row to the top. Thus we have

Δ_{μ}(x, y) = ∑_{σ∈S_{n}} ϵ(σ) σm_{T_{o}}(x, y) .  1.27

where ϵ(σ) denotes the the sign of σ. Using notation which goes back to A. Young (see [23]) we let N(T) denote the formal signed sum of the elements of the column group of the tableau T. By breaking up the summation according to the left cosets of the column group of T_{o} we can rewrite 1.27 in the form

Δ_{μ}(x, y) = ∑_{τ∈S_{n}/N(T_{o})} ϵ(τ) τ ∏_{l=1}^{n} y_{l}^{j_{To} (l)−1} N(T_{o}) ∏_{l=1}^{n} x_{l}^{i_{To} (l)−1} ,  1.28

where with some abuse of notation S_{n}/N(T_{o}) is to denote a complete set of distinct representatives of the left cosets of the column group of T_{o}. It is easy to see that the expression N(T_{o}) ∏_{l=1}^{n} x_{l}^{i_{To} (l)−1} is none other than the Garnir polynomial corresponding to T_{o}. Now, for a given T ∈ CI_{(}μ), let τ(T) denote the permutation that sends Δ_{T_{o}}(x) into Δ_{T} (x) and let ϵ(T) denote its sign. This given, by chosing S_{n}/N(T_{o}), to be the collection {τ(T) : T ∈ CI(μ) }, 1.28 simplifies to the rather suggestive form

Δ_{μ}(x, y) = ∑_{T ∈CI(μ)} ϵ(T) ∏_{l=1}^{n} y_{l}^{j_{T} (l)−1} Δ_{T} (x) .  1.29

Note that the elements of H_{μ}[X_{n}, Y_{n}] which have no dependence on y may all be written in the form f(∂_{x}, ∂_{y})Δ_{μ}(x, y) with f(x, y) a polynomial which is homogeneous of degree n(μ′) in y_{1}, …, y_{n}. In view of 1.29 the resulting polynomial is then necessarily a linear combination of derivatives of Garnir polynomials for column increasing tableaux of shape μ. This immediately yields that the submodule of H_{μ}[X_{n}, Y_{n}] consisting of elements of degree zero in y is precisely given by the space H_{μ}[X_{n}] as asserted. By reversing the roles of x and y in the above argument we get that the submodule of elements of degree zero in x is given by H_{μ′}[Y_{n}]. Equation 1.23 then yields the equalities

C_{λμ}(0, t) = K̃_{λμ}(t) , C_{λμ}(q, 0) = K̃_{λμ′}(q) ,

The proof of 1.25 is completed by verifying that indeed we have

K̃_{λμ}(t) = K̃_{λμ}(0, t) , K̃_{λμ′}(q) = K̃_{λμ}(q, 0) .

However, this follows easily from the fact (proved by Macdonald in [17]) that the Macdonald polynomials reduce to the Hall-Littlewood polynomials when q = 0. The equalities in 1.26 then follow from those in 1.25 by a simple application of 1.16 and the corresponding identity

K̃_{λμ}(q, t) = K̃_{λ′μ}(1/q, 1/t)q^{n(μ′)}t^{n(μ)} .  1.30

for Macdonald coefficients, which may be straightforwardly derived from identities given in [17]. This completes our proof.

As an immediate application of 1.25 and 1.26 we obtain the following special cases of the C = H̃ conjecture.

Theorem 1.6

The identities

C_{λ,μ}(q, t) = K̃_{λ,μ}(q, t)  1.31

hold true for all λ whenever μ is of the form (2, 1^{n−2}) or (n − 1, 1). Proof

When μ = (2, 1^{n−2}) then n(μ′) = 1 so both C_{λ,μ}(q, t) and K̃_{λ,μ}(q, t) are linear in q and 1.31 is a consequence of the first equalities in 1.25 and 1.26. On the other hand if μ = (n − 1, 1) then n(μ) = 1 and 1.31 is a consequence of the last equalities in 1.25 and 1.26. Q.E.D.

We terminate the section with two further identities matching what is known about the coefficients K̃_{λμ}(q, t).

Theorem 1.7

For any pair λ, μ we have

C_{λμ′}(q, t) = C_{λμ}(t, q)  1.32

Proof

The identity

Δ_{μ′}(x, y) = Δ_{μ}(y, x)

is an immediate consequence of the definition I.3. Thus the elements of H_{μ′}[X_{n}, Y_{n}] are obtained by interchanging x′_{i}s with the y′_{i}s in each of the elements of H_{μ}[X_{n}, Y_{n}]. This clearly accounts for 1.32 since such an interchange does not affect the character of any of the isotypic components, but exchange x-degree with y-degree. Q.E.D.

In [17] Macdonald was able to determine the coefficients K_{λμ}(q, t) explicitly for arbitrary μ when λ is a hook. In the present notation, his result may be rewritten in the form

∑_{s=0}^{n} u^{s} K̃_{(1^{s}n−s),μ}(q, t) = ∏_{(i,j)∈μ}^{(o)} (1 − u t^{i−1}q^{j−1)}) ,  1.33

where the (o) superscript is to indicate that the product omits the (0, 0) biexponent of μ. We can show that our coefficients C_{(1^{s}n−s),μ}(q, t) are given by the same formula.

Our proof is based on a remark of A. Young who, with surprising historical premonition, admonished in [23], that some other approaches to the construction of the irreducible representations of S_{n}, might only be superficially different from his. Young’s observation is relevant not only to Garnir’s modules and the oft-cited Specht modules [16], [19] but it also provides precisely what is needed in our developments.

To see how this comes about, we shall construct irreducible components of H_{μ} corresponding to hook shapes by differentiating Δ_{μ} by certain polynomials Δ_{S,A} patterned upon Young’s idempotents. Then Young’s observation yields us that our procedure produces complete decompostions of the isotypic components indexed by hook shapes. The omitted proofs and details in the necessarily short treatment given here can be found in [9] or [13].

To prove that our coefficients C_{(1^{s},n−s),μ}(q, t) satisfy 1.33, we shall need to construct submodules of H_{μ} indexed by (s + 1)-element subsets of the biexponents, in which the first element is the biexponent (0, 0). It will be convenient thereore to let (p_{1}, q_{1}) = (0, 0), (p_{2}, q_{2}), …, (p_{n}, q_{n}) denote the biexponents of μ, and to consider index sets of the form

S = {1 = i_{1} < i_{2} < ⋯ < i_{s+1} ≤ n}.  1.34

Let also A be any (s + 1)-element set

A = {1 ≤ a_{1} < a_{2} < ⋯ < a_{s+1} ≤ n}  1.35

To the pair S, A we assign the (s + 1) × (s + 1) determinant

Δ_{S,A}(x, y) = det∥x_{a_{h}}^{p_{i_{k}}} y_{a_{h}}^{q_{i_{k}}} ∥_{h,k=1}^{s+1} .  1.36

Our submodule H_{S,μ}[X, Y] will now be the linear span of the derivatives

Δ_{S,A}(∂_{x}, ∂_{y})Δ_{μ},  1.37

as A ranges over the sets of labels in the first column of all standard tableaux of shape (1^{s}, n − s).

Young’s observation alluded to above can be stated in full as follows.

Lemma 1.1

Let T be a tableau of shape λ, λ a parition of n. As in 1.28, let N(T) denote the signed sum of the elements in the column group of T. Let m be an element of an S_{n}-module V which satisfies σm = m for all σ in the row group of T. Let Δ = N(T)m, and assume Δ ≠ 0. Then

(1) The linear span L{σΔ : σ ∈ S_{n}} is an irreducible submodule of V, with character χ^{λ};

(2) The set {σΔ : σ ∈ S_{n}, σ(T) standard} is a basis of the submodule in (1); and

(3) Denoting the elements of the basis in (2) by b_{1}, b_{2}, …, b_{f_{λ}}, there exists for each i, j an element E_{ij} of the group algebra of S_{n} such that E_{ij}b_{i} = b_{j} and E_{ij}b_{k} = 0 for all k ≠ i.

Now let T_{1}, …, T_{N}, for N = \binom{n−1}{s}, be all the standard tableaux of shape (1^{s}, n − s), and let A_{1}, …, A_{N} be their first columns. The determinant Δ_{S,A_{1}} is equal to N(T_{1})m_{T_{1}}, where

m_{T_{1}} = ∏_{j=1}^{s+1} x_{a_{j}}^{p_{i_{j}}} y_{b_{j}}^{q_{i_{j}}} .  1.38

Note that because we have set i_{1} = 1 in S and (p_{1}, q_{1}) = (0, 0), the monomial m_{T_{1}} involves only the variables x_{a_{j}} and y_{a_{j}} for j = 2, …, s + 1, and so is invariant for the row group of T_{1}.

We can apply the lemma with Δ = Δ_{S,A_{1}}, and we see that the polynomials Δ_{S,A_{1}}, …, Δ_{S,A_{N}} form the basis in (2). It also follows, since Δ_{μ} is S_{n}-alternating, that either

{Δ_{S,A_{i}}(∂_{x}, ∂_{y})Δ_{μ} : i = 1, …, N}  1.39

is a basis of an irreducible submodule of H_{μ} with character χ^{λ′}, or else all the polynomials Δ_{S,A}, for any A whatsoever, vanish identically. Now we are in position to prove our desired theorem.

Theorem 1.8 The subspaces

H_{S,μ}[X, Y] = L{Δ_{S,A}(∂_{x}, ∂_{y})Δ_{μ}(x, y) : A = {1 ≤ a_{1} < a_{2} < ⋯ < a_{s+1} ≤ n} }  1.40

obtained as S varies among the \binom{n−1}{s} subsets given by 1.34 are independent, irreducible submodules yielding a complete decomposition of the isotypic component of shape (1^{n−s−1}, s + 1) in H_{μ}[X, Y].

Proof

We must first show that the polynomials Δ_{S,A}(∂_{x}, ∂_{y})Δ_{μ}(x, y) do not all vanish identically. Suppose the contrary. The vanishing of these polynomials means the determinants Δ_{S,A}(x, y) lie in the ideal I_{μ} defined in 1.12. However with our choice of biexponents this cannot occur, for a very simple reason. Computing Δ_{μ}(x, y) by the Laplace expansion with respect to the columns indexed by the elements of S we derive that

Δ_{μ}(x, y) = ∑_{A={1≤a_{1}<a_{2}<···<a_{s+1}≤n}} Δ_{S,A}(x, y) ^{c}Δ_{S,A}(x, y) ,  1.41

where ^{c}Δ_{S,A}(x, y) denotes the complementary minor to Δ_{S,A}(x, y) in the matrix ∥x_{i}^{p_{j}}y_{i}^{q_{j}}∥_{i,j=1..n}. Thus if each Δ_{S,A}(x, y) lies in I_{μ} so must Δ_{μ}(x, y) as well. But then Δ_{μ} would be orthogonal to itself, a patent impossibility.

We are left to show that the constructed submodules are independent. Again suppose the contrary, so there is a linear dependence

\begin{matrix}( & {n−1 \atop s} & ) \\ ∑ \\ i,j=1 \end{matrix} c_{ij}Δ_{S_{j},A_{i}}(∂_{x}, ∂_{y})Δ_{μ} = 0,  1.42

or equivalently,

\begin{matrix}( & {n−1 \atop s} & ) \\ ∑ \\ i,j=1 \end{matrix} c_{ij}Δ_{S_{j},A_{i}} ∈ I_{μ}.  1.43

Applying the operator E_{i,1} from part (3) of the lemma, we obtain for each i

\begin{matrix}( & {n−1 \atop s} & ) \\ ∑ \\ j=1 \end{matrix} c_{ij}Δ_{S_{j},A_{1}} ∈ I_{μ}.  1.44

Finally, acting on this equation by σ ∈ S_{n}, we may replace A_{1} here by any A at all. However, we have again by the Laplace expansion that

∑_{A=\{1≤a_{1}<a_{2}<···<a_{s+1}≤n\}} Δ_{S,A}(x, y) ^{c}Δ_{S_{j},A}(x, y) = \{ \begin{matrix}Δ_{μ}(x, y) & if S_{j} = S and \\ 0 & otherwise \end{matrix} .  1.45

Indeed, the first case is again 1.41 and in the second case the sum on the left hand side is the Laplace expansion of a determinant with repeated rows. This given, multiplying 1.44 (with A in place of A_{1}) by ^{c}Δ_{S_{j},A}(x, y) and summing with respect to A we finally derive that

c_{ij} Δ_{μ}(x, y) ∈ I_{μ} .  1.46

Since as we have seen Δ_{μ}(x, y) cannot lie in I_{μ} the inevitable conclusion is that all the coefficients c_{ij} must be equal to zero.

To prove the last assertion of the theorem we need only observe that, in view of 1.8, the maximum number of irreducible constituents with character χ^{(n−s−1,s+1)} in H_{μ}[X, Y] is \binom{n−1}{s}. Thus our submodules necessarily must give a complete set. QED

As an immediate corollary of this theorem we obtain the equality of K̃_{λμ}(q, t) and C_{λμ}(q, t) when λ is a hook. Theorem 1.9

Setting

B_{μ}^{o}(q, t) = B_{μ}(q, t) − 1 ,

for all μ ⊢ n we have (in λ-ring notation)

C_{(1^{s},n−s),μ} (q, t) = e_{s}[B_{μ}^{o}(q, t)]  1.47

Proof Since all of the elements of H_{S,μ}[X, Y] are bihomogeneous polynomials of bidegree

( n(μ) − ∑_{i∈S} p_{i} , n(μ′) − ∑_{i∈S} q_{i} )

this submodule must contribute the term

t^{n(μ)}q^{n(μ′)} ∏_{i∈S} t^{−p_{i}}q^{−q_{i}} = ∏_{i∈S′} t^{p_{i}}q^{q_{i}}  1.48

to our coefficient C_{(1^{n−s−1},s+1),μ} (q, t). Here S′ denotes the complement of S in {1, 2, …, n}. Theorem 1.8 gives that this coefficient may be obtained by summing these monomials over all possible choices of S. This gives

C_{(1^{n−s−1},s+1),μ)} (q, t) = ∑_{{|S′|=n−s−1 \atop S′⊆\{2,3,...,n\}}} ∏_{i∈S′} t^{p_{i}}q^{q_{i}} ,  1.49

and this is another way of stating 1.47. Note that 1.33 is likewise just another way of stating the same equality for the Macdonald coefficients K̃_{(1^{s},n−s),μ}.

2. Calculation of the x-graded and y-graded characters of H_{μ}[X, Y]

The main object of this section is to show that, on the n! conjecture, we have the identities

a) C_{λμ}(q, 1) = K̃_{λμ}(q, 1) ,  b) C_{λμ}(1, t) = K̃_{λμ}(1, t) .  2.1

Our basic ingredients here are formal power series

f(x, y) = ∑_{p,q} c_{pq} x^{p}y^{q} ,  2.2

in the two sets of variables, x_{1}, …, x_{n}, y_{1}, …, y_{n} with the diagonal action of S_{n} defined (as was done for polynomials) by setting for σ ∈ S_{n}

σ f(x_{1}, …, x_{n}; y_{1}, …, y_{n}) = f(x_{σ_{1}}, …, x_{σ_{n}}; y_{σ_{1}}, …, y_{σ_{n}}) .

All standard operations such as linear combinations, products, differentiations by any of the variables are defined for formal power series as they are for polynomials. This will also be the case for any operations which, likewise, permit the computation of each of the coefficients of the resulting series in a finite number of steps and as such involve no convergence questions.

In this section we will use only the customary grading of polynomials and we shall denote total degree simply by deg. In particular, if f is given by 2.2 then the polynomial

f^{(m)}(x, y) = ∑_{|p|+|q|=m} c_{pq} x^{p}y^{q} ,  2.3

will be referred to as the homogeneous component of degree m in f or briefly the m^{th} component of f. The homogeneous component of f of minimum degree which does not vanish identically will be denoted by m(f). Note that the operation f → m(f) is not linear. Indeed, for m(f) to be homogeneous of degree m_{o} we need that

f^{(m)} = 0 for m < m_{o} and f^{(m_{o})} ≠ 0 .  2.4

Then and only then can we conclude that m(f) = f^{(m_{o})}. This defines m(f) as a non-zero polynomial for all but the zero formal power series. It will be convenient to set m(0) = 0. Clearly, if f_{1} and f_{2} are two formal power series, we shall have

m(f_{1} + f_{2}) = m(f_{1}) + m(f_{2}) ⇔ \{ \begin{matrix}a) deg(m(f_{1})) = deg(m(f_{2})) \\ b) m(f_{1}) + m(f_{2}) ≠ 0 \end{matrix} .  2.5

We shall use the symbol ( , ) to denote the ordinary scalar product of n-vectors. Thus for x = (x_{1}, …, x_{n}) and a = (a_{1}, …, a_{n}) we set

(x, a) = x_{1}a_{1} + x_{2}a_{2} + ⋯ + x_{n}a_{n} .

All the formal power series that occur in our treatment will be obtained by carrying out the above mentioned operations on polynomial multiples of the exponential series

e^{(x,a)} = ∑_{p_{1}≥0} ⋯ ∑_{p_{n}≥0} \frac{x_{1}^{p_{1}}a_{1}^{p_{1}}}{p_{1}!} ⋯ \frac{x_{n}^{p_{n}}a_{n}^{p_{n}}}{p_{n}!} = ∑_{p} \frac{x^{p}a^{p}}{p!} .

This given, for a partition μ = (μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{k} > 0) we set

Φ_{μ}(x, y) = ∑_{σ∈S_{n}} ϵ(σ) σ ( e^{(x,a(T_{o}))} m_{T′_{o}}(y)) ,  2.6

where, as in section 1, T_{o} denotes the superstandard tableau of shape μ, a(T_{o}) is the vector whose coordinates are given by the first equalities in 1.6 and m_{T′_{o}}(y) is the y-monomial corresponding to the conjugate of the tableau T_{o}. More precisely, using again the same notation as in 1.6 we have

m_{T′_{o}}(y) = ∏_{l=1}^{n} y_{l}^{j_{To} (l)−1} .  2.7

With these conventions and notations we are in a position to introduce two spaces which play a crucial role in our proof of the identities in 2.1. Namely, we define Γ_{μ} as the linear span of the derivatives of Φ_{μ} and m(Γ_{μ}) as the space spanned by the least non-vanishing homogeneous components of elements of Γ_{μ}. In symbols

Γ_{μ} = L{ ∂_{x}^{p}∂_{y}^{q} Φ_{μ} }  2.8

and

m(Γ_{μ}) = L{ m(f) : f ∈ Γ_{μ} } .  2.9

Our reasoning will proceed as follows. Without any additional assumptions we can compute the character of Γ_{μ} as a y-graded S_{n}-module. We next show that, on the n! conjecture, m(Γ_{μ}) and our module H_{μ}[X, Y] are one and the same. This result permits us to derive that Γ_{μ} and H_{μ}[X, Y] are equivalent as y-graded modules and thereby complete our identification of the y-graded character of H_{μ}[X, Y]. To carry this out in detail we need to establish a few auxiliary results.

Let RI_{μ} denote the collection of row increasing injective tableaux of shape μ. For a given T ∈ RI_{μ} let ϵ(T) denote the sign of the permutation that sends T_{o} into T. For any biexponent sequence (p_{1}, q_{1}), (p_{2}, q_{2}), … (p_{n}, q_{n}) set

Δ_{pq}(x, y) = det∥x_{j}^{p_{i}}y_{j}^{q_{i}}∥ .

Lemma 2.1 The formal power series Φ_{μ} has also the following two equivalent definitions

Φ_{μ} = ∑_{T ∈RI_{μ}} ϵ(T) e^{(x,a(T))} Δ_{T′}(y) ,  2.10

and

Φ_{μ} = ∑_{p} \frac{a(T_{o})^{p}}{p!} Δ_{p,q_{o}}(x, y)  2.11

where q_{o} is the exponent sequence of the monomial m_{T′_{o}}(y), i.e., m_{T′_{o}}(y) = y^{q_{o}}.

Proof

Let T_{1}, T_{2}, …, T_{N} (N = n!/μ!) be the elements of RI_{μ} in any total order that makes T_{1} = T_{o}. Let τ_{i} be the permutation that sends T_{o} into T_{i}. Let P (T_{o}) denote the row group of T_{o}. This given we have the left coset decomposition

S_{n} = τ_{1}P (T_{o}) + τ_{2}P (T_{o}) + ⋯ + τ_{N}P (T_{o}) .  2.12

In other words every σ ∈ S_{n} has a unique factorization of the form σ = τ_{i}α with α ∈ P (T_{o}). Since for any α ∈ P (T_{o}) we have

τ_{i}αe^{(x,a(T_{o}))} = e^{(x,a(T_{i}))} ,

we may rewrite 2.6 in the form

Φ_{μ}(x, y) = ∑_{i=1}^{N} ϵ(τ_{i})e^{(x,a(T_{i}))} τ_{i} ∑_{α∈P (T_{o})} ϵ(α) α m_{T′_{o}}(y) .  2.13

Now we can easily see that

∑_{α∈P (T_{o})} ϵ(α) α m_{T′_{o}}(y) = Δ_{T′_{o}}(y) ,

the latter being the Garnir polynomial (in the y variables) corresponding to the conjugate of the tableau T_{o}. Substituting in 2.13 and noting that

τ_{i}Δ_{T′_{o}}(y) = Δ_{T′_{i}}(y)

we finally get

Φ_{μ}(x, y) = ∑_{i=1}^{N} ϵ(τ_{i})e^{(x,a(T_{i}))} Δ_{T′_{i}}(y)  2.14

which is another way of writing 2.10.

To verify 2.11 we simply note that if we expand the exponential in 2.6, set m_{T′_{o}}(y) = y^{q_{o}} and change order of summation we obtain

Φ_{μ}(x, y) = ∑_{p} \frac{a(T_{o})^{p}}{p!} ∑_{σ∈S_{n}} ϵ(σ) σx^{p}y^{q_{o}} .

Since the inner summation yields the polynomial Δ_{pq_{o}}(x, y) this expression reduces to 2.11 as desired.

The expansion in 2.14 immediately yields us the following useful fact:

Lemma 2.2 The collection of formal power series

C_{μ} = { e^{(x,a(T_{j}))} ∂_{y}^{q}Δ_{T′_{j}}(y) : j = 1, 2, …, n!/μ! ; q = (q_{1}, …, q_{n}) }

spans Γ_{μ}.

Proof Note that if, for any T ∈ RI_{μ}, we set (using the notation of section 1)

ϕ_{T} (x) = ∏_{l=1}^{n} \frac{(x_{l} − α_{1})}{(α_{i_{T} (l)} − α_{1})} ⋯ \frac{(x_{l} − α_{i_{T} (l)−1})}{(α_{i_{T} (l)} − α_{i_{T} (l)−1})} .

Then we easily verify that

ϕ_{T_{j}}(x) = \{ \begin{matrix}1 & if x = a(T_{j}) \\ 0 & if x = a(T_{i}) with i ≠ j \end{matrix}  2.15

This given, applying the differential operator ϕ_{T_{j}}(∂_{x})∂_{y}^{q} to both sides of 2.14 we get

ϕ_{T_{j}}(∂_{x})∂_{y}^{q}Φ_{μ}(x, y) = e^{(x,a(T_{j}))} ∂_{y}^{q}Δ_{T′_{j}}(y) .  2.16

We thus obtain that all the elements of C_{μ} belong to Γ_{μ}. On the other hand, 2.14 gives that for any exponents p,q we have

∂_{x}^{p}∂_{y}^{q} Φ_{μ}(x, y) = ∑_{i=1}^{N} ϵ(T_{i}) a(T_{i})^{p} e^{(x,a(T_{i}))} ∂_{y}^{q}Δ_{T′_{i}}(y) ,

and this shows that every element of Γ_{μ} is a linear combination of the elements of C_{μ}. Q.E.D.

Let us recall that if G is a finite group of n × n matrices, then the polynomials in x_{1}, …, x_{n} which are killed by all non-trivial homogeneous G-invariant differential operators are referred to as the harmonics of G. They form a finite dimensional subspace which we shall denote here by H_{G}[X]. It is well known (see [21] or [9]) that, when G is generated by reflections, H_{G}[X] is the linear span of the derivatives of a single polynomial Δ_{G}(x). In symbols,

H_{G} = L{∂_{x}^{p}Δ_{G}(X) } .  2.17

It is also shown in [21] that

dim H_{G} = |G|  2.18

These two properties are, in fact, characteristic [21] of finite groups generated by reflections. It is well known that when G is the group of permutation matrices corresponding to the natural action of S_{n} on polynomials then Δ_{G}(x) reduces to the Vandermonde determinant in x_{1}, …, x_{n}. In the general case Δ_{G}(x) can be obtained by taking the product of the linear forms giving the equations of the reflecting hyperplanes. The latter is usually referred to as the discriminant of G. This given, it is not difficult to see that the Garnir polynomial Δ_{T′}(y) is also the discriminant of the row group P (T). This observation leads us immediately to the following basic fact.

Proposition 2.1 The space Γ_{μ} decomposes into the direct sum

Γ_{μ} = ⨁_{T ∈RI_{μ}} e^{(x,a(T))} H_{P (T)}[Y] .  2.19

In particular, we have

dim Γ_{μ} = n! .  2.20

Proof

The identity 2.17, for G = P (T), gives that for any fixed T ∈ RI_{μ}, the elements e^{(x,a(T))}∂_{y}^{p}Δ_{T′}(y) must necessarily span the subspace obtained by multiplying the harmonics of P (T) (in the variables y_{1}, …, y_{n}) by the exponential e^{(x,a(T))}. And this is what is meant by e^{(x,a(T))} H_{P (T)}[Y] in 2.19. The fact that the sum in 2.19 is direct then follows from the fact that exponentials e^{(x,a)} with different exponents are independent. The dimension of Γ_{μ} must then be n! since (by 2.18) each of the direct summands has dimension μ! and the collection RI_{μ} has precisely n!/μ! elements. Q.E.D.

Note further that the linear span in Γ_{μ} of those elements e^{(x,a(T))}∂_{y}^{q}Δ_{T′} which are of degree m in y is invariant under the diagonal action. Thus we have also a direct sum decomposition of Γ_{μ} as a y-graded S_{n}-module:

Γ_{μ} = ⨁_{m=0}^{n(μ′)} H_{m}(Γ_{μ}) ,  2.21

where

H_{m}(Γ_{μ}) = L{ e^{(x,a(T))} ∂_{y}^{q}Δ_{T′}(y) : T ∈ RI_{μ}; |q| = n(μ′) − m} = ⨁_{T ∈RI_{μ}} e^{(x,a(T))}H_{m}(H_{P (T)}[Y]).

2.22

In particular, the generating function

π_{y}^{μ}(q) = ∑_{m=0}^{n(μ′)} q^{m} char H_{m}(Γ_{μ}) .

may be viewed as the y-graded character of Γ_{μ}.

From 2.19 and the left coset decomposition 2.12 it is plain that the character in question is nothing but the induced character from P (T_{o}) to S_{n} of the character of H_{P (T_{o})}[Y]. By 2.22, this also holds separately in each y-degree.

Now P (T_{o}) is the Young subgroup S_{μ_{1}} × S_{μ_{2}} × ⋯ × S_{μ_{k}}. The multiplicative property of the usual Frobenius characteristic map F, namely F Ind_{S_{l}×S_{m}}^{S_{l+m}}(χ_{1} ⊗ χ_{2}) = F χ_{1} F χ_{2}, immediately implies the same for the Frobenius characterstic of a graded character. Finally, it is well known that the Frobenius characteristic of H_{S_{m}}[Y] is (1−q)(1−q^{2}) ⋯ (1−q^{m})h_{m}[X/(1−q)] in λ-ring notation, where h_{m} is the complete homogeneous symmetric function of degree m.

These observations demonstrate the following crucial result.

Theorem 2.1

When μ = (μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{k} > 0), the Frobenius image of the y-graded character of Γ_{μ} is given by the identity

F π_{y}^{μ}(q) = H̃_{μ_{1}}(x; q)H̃_{μ_{2}}(x; q) ⋯ H̃_{μ_{k}}(x; q) = H̃_{μ}(x; q, 1) ,  2.23

Where for an integer m we have (in λ-ring notation)

H̃_{m}(x, q) = (1 − q)(1 − q^{2}) ⋯ (1 − q^{m}) h_{m}[X/(1 − q)] .  2.24

In particular, the q-multiplicities of the irreducible character χ^{λ} in Γ_{μ} are given by the Macdonald coefficients, that is,

π_{y}^{μ}(q) = ∑_{λ⊢n} χ^{λ} K̃_{λμ}(q, 1) .  2.25

Proof

Only 2.25 remains to be verified. But Macdonald in [17] obtained explicit expressions for his polynomials when q or t are set equal to 1, and these expressions readily yield the equivalence of 2.23 and 2.25 (see [9]).

Before we can use this result to calculate the y-graded character of H_{μ}[X, Y] we must look a bit closely at the relationship between the m(Γ_{μ}) and Γ_{μ}, which is expressed by the following result.

Proposition 2.2

We have

dim m(Γ_{μ}) = dim Γ_{μ} = n! .  2.26

Moreover the two spaces m(Γ_{μ}) and Γ_{μ} have the same y-graded character.

Proof

To start, we show that the decomposition of m(Γ_{μ}) into y-homogeneous subspaces is given by

m(Γ_{μ}) = ⨁_{k} m(H_{k}(Γ_{μ})),

where H_{k}(Γ_{μ}) denotes the subspace of elements in Γ_{μ} homogeneous of degree k in y. What is clear is that m(H_{k}(Γ_{μ})) is homogeneous of degree k in y, so that the sum above is indeed direct, and that m(H_{k}(Γ_{μ})) ⊆ m(Γ_{μ}). Therefore we need to prove the reverse inclusion m(Γ_{μ}) ⊆ ⨁_{k} m(H_{k}(Γ_{μ})). To this end we let f be an arbitrary element of m(Γ_{μ}) and write m(f) as m(f) = g_{0}+g_{1}+⋯+g_{m_{o}}, where g_{k} is homogeneous of degree k in y (and therefore also of degree m_{o} −k in x, where m_{o} = deg m(f)). Also let us write f as the sum of its y-homogeneous componenents f = f_{0} + f_{1} + ⋯ f_{n(μ′)}, with f_{k} ∈ H_{k}(Γ_{μ}). Now clearly for each k, either g_{k} = m(f_{k}) or g_{k} = 0, so m(f) ∈ ⨁_{k} m(H_{k}(Γ_{μ})).

To complete the proof we must establish that for each k, the two spaces m(H_{k}(Γ_{μ})) and H_{k}(Γ_{μ}) have the same dimension and character. To simplify notation, set V = H_{k}(Γ_{μ}). For each d, the set V_{≤d} = {f ∈ V : deg m(f) ≥ d} is an S_{n}-invariant subspace of V, provided we employ the convention deg m(0) = ∞. The linear function ϕ_{d}: ϕ_{d}(f) = f^{(d)} maps V_{≥d} surjectively on H_{d}(m(V)), with kernel V_{≥d+1}, and commutes with the S_{n} action. (Here the symbol H_{d} refers to the customary total degree, not the y-degree). As S_{n} modules we therefore have H_{d}(m(V)) ≅ V_{≥d}/V_{≥d+1} and hence char m(V) = ∑_{d} char H_{d}(m(V)) = ∑_{d} char V_{≥d} − char V_{≥d+1} = char V_{≥0} = char V. Q.E.D.

The following result also holds without assuming the n! conjecture.

Proposition 2.3 Our determinant Δ_{μ}(x, y) is an element of m(Γ_{μ}). Indeed, we have the identity

Δ_{μ}(x, y) = \frac{1}{Δ_{T_{o}}(a(T_{o}))} m(Φ_{μ})  2.27

Proof To show 2.27 we resort to the identity in 2.11. Note that by definition

Δ_{p,q_{o}}(x, y) = ∑_{σ∈S_{n}} ϵ(σ) σ x^{p}y^{q_{o}} .

Moreover, we have observed (in the proof of Theorem 1.1) that

Δ_{μ}(x, y) = Δ_{p_{o},q_{o}}(x, y) ,  2.28

where p_{o} and q_{o} are the exponent sequences giving

m_{T_{o}}(x, y) = x^{p_{o}}y^{q_{o}}

(see 1.9). This given, we need only verify that the surviving terms of minimum degree in 2.11 add up to a non vanishing multiple of Δ_{p_{o},q_{o}}(x, y). However, we can easily see that Δ_{p,q_{o}}(x, y) does not vanish only if the exponent vector p is not stabilized by any of the elements of the stabilizer of q_{o}. Since the latter is the column group of T_{o}, this condition forces the components p_{i} to take different values as i varies in a column of T_{o}. A moment’s reflection should reveal that, under this condition, the smallest possible value of |p| is n(μ). Indeed, such a minimum value of |p| can be achieved only when for some β in the column group of T_{o} we have p = p_{o}β. But in that case

Δ_{p,q_{o}}(x, y) = ϵ(β) Δ_{μ}(x, y) .

This gives that all the homogeneous components of Φ_{μ}(x, y) of degree less than n(μ) + n(μ′) vanish identically and the component of degree n(μ) + n(μ′) reduces to the expression

∑_{β∈N(T_{o})} ϵ(β) a(T_{o})^{p_{o}β} Δ_{μ}(x, y) .

From our Remark 1.1 we see that this sum evaluates to

Δ_{T_{o}}(a(T_{o})) Δ_{μ}(x, y) .

where Δ_{T_{o}}(x) denotes the corresponding Garnir polynomial. Now by definition (see 1.18), Δ_{T_{o}}(x) factors into a product of the Vandermonde determinants corresponding to the columns of T_{o}. Thus Δ_{T_{o}}(x) does not vanish at x = (x_{1}, …, x_{n}) if and only if the components x_{i} take different values when i varies in a column of T_{o}. The very construction of a(T_{0}) guarantees this property, as our α_{1}, α_{2}, …, α_{k} are all distinct. Thus the constant Δ_{T_{o}}(a(T_{o})) is different from zero and 2.27 must hold true.

Q.E.D.

As immediate corollary we obtain our desired connection between m(Γ_{μ}) and H_{μ}[X, Y].

Theorem 2.2 We always have the containment

H_{μ}[X, Y] ⊆ m(Γ_{μ}) .  2.29

In particular, on the n! conjecture these two spaces must be one and the same.

Proof

Note that if Q = m(f) for some f ∈ Γ_{μ} and ∂_{x_{i}}Q ≠ 0 then ∂_{x_{i}}Q = m(∂_{x_{i}}f). Clearly, the same must be true for ∂_{y_{i}}. In other words m(Γ_{μ}) is closed under differentiation. Since we have just shown that Δ_{μ}(x, y) ∈ m(Γ_{μ}), the whole of H_{μ}[X, Y] must lie in m(Γ_{μ}) as well. The last assertion follows from the dimension equality in 2.26.

We are finally in a position to verify the equalities 2.1 (a) and (b).

Theorem 2.3

On the n! conjecture Γ_{μ} and H_{μ} have the same y-graded character. In particular we must have

C_{μ}(q, 1) = H̃_{μ}(x; q, 1) = H̃_{μ_{1}}(x; q)H̃_{μ_{2}}(x; q) ⋯ H̃_{μ_{k}}(x; q) .  2.30

Proof This follows directly from Proposition 2.2 and Theorem 2.2. Q.E.D.

Theorems 2.1 and 2.3 combined yield the validity of 2.1 a) on the n! conjecture. Clearly, the dual identity 2.1 b) can be established in an analogous manner by simply reversing the roles of x and y. We may be tempted to leave it at that and go on with our study of H_{μ}. However, we shall see that it is worth delving a bit further into the nature of the additional modules we have introduced in this section. To do this in a precise manner it will be convenient, (also in anticipation of the developments of the next section), to make some slight changes of notation. First of all we should distinguish the y-graded nature of the ingredients used in our proof of 2.1 a) from the x-graded nature of those needed to study 2.1 b). We shall do this by denoting the power series in 2.6 by Φ_{μ}^{y} rather than Φ_{μ} and the module in 2.8 by Γ_{μ}^{y}. We shall then set

Φ_{μ}^{x}(x, y) = ∑_{σ∈S_{n}} ϵ(σ) σ ( e^{(y,b(T_{o}))} m_{T_{o}}(x)) ,  2.31

and let

Γ_{μ}^{x} = L{∂_{x}^{p}∂_{y}^{q} Φ_{μ}^{x} }  2.32

This given, the reader should have no difficulty obtaining the analogue of theorem 2.1 for Γ_{μ}^{x} and then translating the arguments that yielded 2.1 a) into arguments establishing 2.1 b). On the n! conjecture we would then deduce that the two modules m(Γ_{μ}^{y}) and m(Γ_{μ}^{x}), in spite of their apparently different definitions, must be identical. In fact, Theorem 2.2 and its analogue concerning m(Γ_{μ}^{x}) yield that they are then both equal to H_{μ}. These are the modules M_{μ}^{x} and M_{μ}^{y} we referred to in the introduction. In the next section we shall study their relation to the coordinate rings of the orbits studied in section 1 and as a by-product obtain that the equality m(Γ_{μ}^{x}) = m(Γ_{μ}^{y}) is in fact equivalent to the n!-conjecture.

3. Gradings and the n! conjecture.

We shall have to deal in this section with a number of related spaces, some of which we have already introduced. Before we prove anything it will be good to get across at least a general view of the relationships between these various ingredients. Let us recall that in Section 1 we have defined the ideal

J_{[ρ_{μ}]} = {p(x, y) ∈ Q[X, Y] : p(a, b) = 0 ∀ (a, b) ∈ [ρ_{μ}]_{S_{n}}}.

In words, J_{[ρ_{μ}]} is the ideal of polynomials in x, y which vanish at all points in the S_{n} orbit of ρ_{μ} = (a(T_{o}), b(T_{o})). We may therefore identify the quotient ring

R_{[ρ_{μ}]} = Q[X, Y]/J_{[ρ_{μ}]}

with the ring of rational-valued functions on the orbit in question.

Given an S_{n}-invariant choice of weights w assigning weight w_{x} to each variable x_{1}, …, x_{n} and w_{y} to each y_{1}, …, y_{n}, we have formed the associated graded ideal

gr_{w} J_{[ρ_{μ}]} = L{h_{w}(p) : p ∈ J_{[ρ_{μ}]}}

and graded ring

gr_{w} R_{[ρ_{μ}]} = Q[X, Y]/(gr_{w} J_{[ρ_{μ}]}),

where p → h_{w}(p) denotes the operation of extracting the w-homogeneous component of highest degree in the polynomial p(x, y).

For our present purpose, it will be useful to allow either or both components w_{x}, w_{y} of the weight w to be zero. Then, for example, taking w = (1, 0) makes the w-degree simply the x-degree, so that the operation gr_{w} is a grading with respect to x only. When this particular choice of w is in force, we write gr_{x} in place of gr_{w}. Similarly we write gr_{y} for gr_{w} when w = (0, 1).

Another important special grading occurs when we take both w_{x} and w_{y} positive, but w_{x} extremely large compared to w_{y}. If w_{x} is sufficiently large then the highest w-homogeneous component of a given polynomial will be the highest y-degree component of its highest x-degree component, so that the operation gr_{w} in this case reduces to gr_{y} gr_{x}, and we shall so denote it. In similar fashion we can realize gr_{x} gr_{y} as gr_{w} for a suitable w.

Extending the inner product defined in 1.1 to allow one of P or Q to be a formal power series, it is still well-defined, and in fact makes the formal power series ring into the dual space of the polynomial ring. The second part of Proposition 1.1 (see I.4) continues to hold: allowing power series solutions, J^{⊥} is the space of solutions to the differential equations p(∂_{x}, ∂_{y})f = 0, for all p(x, y) ∈ J, even if J is a non-homogeneous ideal. In our context J will always be S_{n}-invariant and Q[X, Y]/J finite-dimensional. Then we can prove that J^{⊥⊥} = J and that J^{⊥} is equivalent to Q[X, Y]/J as an S_{n} module, and indeed as a w-graded S_{n} module when J is w-homogeneous.

What we shall establish is that all the spaces

Γ_{μ}^{x}, Γ_{μ}^{y}, m(Γ_{μ}^{x}) and m(Γ_{μ}^{y})  3.1

have the form (gr_{w} J_{[ρ_{μ}]})^{⊥} for suitable choices of w. To achieve this we need to introduce another space which has been a missing piece from our considerations to this point. To this end, by analogy to 2.6–2.8 and 2.31–2.32, we set

Φ_{μ} = ∑_{σ∈S_{n}} ϵ(σ)σ(e^{(x,a(T_{o}))}e^{(y,b(T_{o}))})  3.2

and let

Γ_{μ} = L{∂_{x}^{p}∂_{y}^{q}Φ_{μ}}.  3.3

It then develops that Γ_{μ} = J_{[ρ_{μ}]}^{⊥}, and that the various spaces in 3.1 all occur as m_{w}(Γ_{μ}) for suitable weights w, where m_{w}(Γ_{μ}) = {m_{w}(f) : f ∈ Γ_{μ}}, and f → m_{w}(f) denotes the operation of extracting the w-homogeneous component of lowest degree from a formal power series f.

This given, our characterization of the spaces in 3.1 follows from the fact, which we can prove for any of the ideals J occuring in our picture, that m_{w}(J^{⊥}) = (gr_{w} J)^{⊥}.

Our first goal is now to prove the statements made above. Having completed that, and in particular having shown that m(Γ_{μ}^{x}) = (gr_{y} gr_{x} J_{[ρ_{μ}]})^{⊥} and m(Γ_{μ}^{y}) = (gr_{x} gr_{y} J_{[ρ_{μ}]})^{⊥}, we shall then turn to the question of deriving the n! conjecture on the assumption that gr_{y} gr_{x} J_{[ρ_{μ}]} = gr_{x} gr_{y} J_{[ρ_{μ}]}.

Proposition 3.1

Let J be an S_{n} invariant ideal in Q[X, Y], w-homogeneous for a weighted degree deg_{w}. Assume that Q[X, Y]/J is finite-dimensional. Then

(1) J^{⊥} = {f : p(∂_{x}, ∂_{y})f = 0 ∀ p(x, y) ∈ J}, J^{⊥} is a w-homogeneous space of formal power series, and in particular J^{⊥} is closed under differentiation;

(2) J^{⊥⊥} = J; and

(3) J^{⊥} and Q[X, Y]/J are equivalent as w-graded S_{n} modules.

Proof

The identity

J^{⊥} = {f : p(∂_{x}, ∂_{y})f = 0 ∀ p(x, y) ∈ J}

holds just as in Proposition 1.1, without essential change to the proof given there. Since whomogeneous polynomials and power series of different w-degrees are mutually orthogonal (indeed, distinct monomials are mutually orthogonal), it is clear that if f is in J^{⊥} so is each w-homogeneous component of f. That is what it means for J^{⊥} to be a w-homogeneous space.

For (2) we only have to prove J^{⊥⊥} ⊆ J since the reverse is obvious. Let x^{r_{1}}y^{s_{1}}, …, x^{r_{N}} y^{s_{N}} be a collection of monomials whose images modulo J form a basis of Q[X, Y]/J. For i = 1, …, N, let l_{i}(p) denote the coefficient of x^{r_{i}}y^{s_{i}} in the expansion of p on this basis, modulo J, for any polynomial p(x, y). Then there is a formal power series f_{i} such that the linear functional l_{i} is given by l_{i}(p) = ⟨f_{i}, p⟩ for all p. Namely, for monomials we have

⟨x^{p}y^{q}, x^{r}y^{s}⟩ = \{ \begin{matrix}p!q! & if (r, s) = (p, q), or \\ 0 & otherwise. \end{matrix}

From this it is plain that

f_{i} = ∑_{p,q} l_{i}(x^{p}y^{q})x^{p}y^{q}/p!q!

will do. Since by definition l_{i}(p) = 0 for p ∈ J we clearly have f_{i} ∈ J^{⊥}. Now if q is a polynomial in J^{⊥⊥} then ⟨f_{i}, q⟩ = 0 for all i and therefore q ≡ 0 modulo J, that is, q ∈ J.

For (3) note that the scalar product ⟨f, p⟩ is well-defined for f ∈ J^{⊥} and p ∈ Q[X, Y]/J, since if p′ and p are the same modulo J then p − p′ is orthogonal to f. This pairing makes J^{⊥} the dual space to Q[X, Y]/J, as we can easily see: if f is orthogonal to every p, then clearly f = 0, while if p ∈ Q[X, Y] is orthogonal to every f ∈ J^{⊥} then p ∈ J by (2). Likewise corresponding w-homogeneous components of J^{⊥} and Q[X, Y]/J are dually paired, and of course the pairing is S_{n} invariant. But a finite-dimensional rational representation of S_{n} is always equivalent to its dual, since they have the same character. Q.E.D.

The operation of grading an ideal is best understood in terms of a filtration of the corresponding quotient ring. We shall use this concept heavily later on to prove the main result of this section, but introduce it now because it will aid in proving our next proposition. Thus, given an ideal J in R = Q[X, Y] (or any polynomial ring), we define R_{J} = R/J and gr_{w} R_{J} = R/(gr_{w} J). Our object is to make a direct construction of gr_{w} R_{J} from R_{J}. To this end we define the w-filtration of R_{J} to be the sequence of subspaces

F_{≤0} ⊆ F_{≤1} ⊆ F_{≤2} ⊆ ⋯ ⊆ R_{J},  3.4

where

F_{≤k} = {p + J : p ∈ R, deg_{w}(p) ≤ k}  3.5

is the subspace spanned modulo J by all polynomials of w-degree at most k. To say that the spaces F_{≤k} form a filtration of the ring R_{J} means that they satisfy the evident property

F_{≤k}F_{≤l} ⊆ F_{≤k+l} for all k and l.  3.6

Owing to this property, there is a well-defined multiplication

(F_{≤k}/F_{≤k−1}) × (F_{≤l}/F_{≤l−1}) → F_{≤k+l}/F_{≤k+l−1},

which we use to make the a priori purely formal space

F_{≤0} ⊕ (F_{≤1}/F_{≤0}) ⊕ (F_{≤2}/F_{≤1}) ⊕ ⋯  3.7

into a graded ring. This construction is related to gr_{w} R_{J} by the following lemma.

Lemma 3.1

The graded ring defined in 3.7 is isomorphic in a natural way to the ring gr_{w} R_{J}, defined as R/(gr_{w} J).

Proof

Let p ∈ R be a w-homogeneous polynomial, say of degree k. Modulo gr_{w} J, p represents an element p̃ of gr_{w} R_{J}. Also, modulo J, p represents an element of F_{≤k} and therefore also an element q of F_{≤k}/F_{≤k−1}. Our procedure is to define the desired isomorphism α by the rule α(p̃) = q. Of course we must check that this is first well-defined, and then an isomorphism of graded rings.

To see that α is well-defined we have only to note that if p̃ = p̃′ then p − p′ belongs to gr_{w} J and is homogeneous of degree k, so we must have p − p′ = h_{w}(s) for some polynomial s ∈ J. Then decomposing s as h_{w}(s) + r, we have p − p′ + r ∈ J, where r has only terms of degree less than k. Now letting q′ denote the image of p′ in F_{≤k}/F_{≤k−1} we see that q − q′ ≡ −r ≡ 0.

That α is a ring homomorphism is immediate, since the multiplication in 3.7 is induced from that in R_{J}. It is also clear that α is surjective. It remains only to show that the kernel of α is zero. But if α(p̃) = 0, it means that the image of p modulo J belongs to F_{≤k−1}, that is, p = t + j for some j ∈ J and some t of w-degree less than k. But then p, being homogeneous of w-degree k, must be h_{w}(j). In other words p ∈ gr_{w}(J), so p̃ = 0. Q.E.D.

Proposition 3.2 Let J be an S_{n} invariant ideal in Q[X, Y], and assume Q[X, Y]/J is finite-dimesional. Then

(gr_{w} J)^{⊥} = m_{w}(J^{⊥}).  3.8

Moreover, Q[X, Y]/J and Q[X, Y]/(gr_{w} J) are equivalent as S_{n}-modules, and even as x- or y-graded S_{n}-modules in the event that J is x- or y-homogeneous to begin with.

Proof

One containment of the equality in 3.8 is easily derived. Namely, if p ∈ J and f ∈ J^{⊥}, we may write

p = h_{w}(p) + r, f = m_{w}(f) + R,

where r has only terms of degree less than deg_{w} h_{w}(p) and R likewise has only terms of degree greater than deg_{w} m_{w}(f). Then we have

p(∂_{x}, ∂_{y})f = h_{w}(p)(∂_{x}, ∂_{y})m_{w}(f) + S,

in which S has only terms of degree strictly greater than the degree of the first term. Since the left hand side is zero, so must be h_{w}(p)(∂_{x}, ∂_{y})m_{w}(f), which shows that m_{w}(J^{⊥}) ⊆ (gr_{w} J)^{⊥}.

The argument used to prove Proposition 2.2 shows, without essential changes, that dim m_{w}(J^{⊥}) = dim J^{⊥}, and even that these two spaces are equivalent as (possibly graded) S_{n} modules. In turn we have dim J^{⊥} = dim Q[X, Y]/J and dim (gr_{w} J)^{⊥} = dim Q[X, Y]/(gr_{w} J) by Proposition 3.1. Hence the equality in 3.8 follows once we prove the second part of the proposition, that Q[X, Y]/J and Q[X, Y]/(gr_{w} J) are equivalent modules.

But this is a completely general fact, as is clear from the previous Lemma and 3.7, since a filtered S_{n} module is always equivalent to the direct sum of the successive quotients in the filtration. For the additional assertions in the x- or y- graded case, we need only note that when J is already x- or y- homogeneous, so are the filtering spaces F_{≤k} in 3.4. Q.E.D.

Now we have the tools at hand to establish the relations between the various spaces reviewed at the outset.

Theorem 3.1 We have the equalities

\begin{aligned}a) Γ_{μ} & = J_{[ρ_{μ}]}^{⊥} \\ b) Γ_{μ}^{x} & = (gr_{x} J_{[ρ_{μ}]})^{⊥} \\ c) Γ_{μ}^{y} & = (gr_{y} J_{[ρ_{μ}]})^{⊥} \\ d) m(Γ_{μ}^{x}) & = (gr_{y} gr_{x} J_{[ρ_{μ}]})^{⊥} \\ e) m(Γ_{μ}^{y}) & = (gr_{x} gr_{y} J_{[ρ_{μ}]})^{⊥} \end{aligned}  3.9

In particular, all these spaces and the corresponding quotient rings are equivalent S_{n} modules, affording the left regular representation.

Proof

Our strategy here is to first prove (a). It then follows by Proposition 3.2 that each space on the right hand side will be m_{w}(Γ_{μ}) for the appropriate w, and that all these spaces have dimension n!. In Section 2 we have already shown that the spaces on the left hand side in (b)–(e) have dimension n!, so it will be enough to show that each is contained in m_{w}(Γ_{μ}) for its relevant w. This we can do by investigating the ‘kernels’ Φ_{μ}^{x}, Φ_{μ}^{y}, and Φ_{μ} that generate them.

To prove (a), we take note of the straightforward identity

p(∂_{x}, ∂_{y})e^{(x,a)}e^{(y,b)} = p(a, b)e^{(x,a)}e^{(y,b)},  3.10

which holds for any polynomial p(x, y). In particular, if we choose p so that p(a(T), b(T)) = 0 for every injective tableau T except T_{o}, while p(a(T_{o}), b(T_{o})) = 1 then we shall have

p(∂_{x}, ∂_{y})Φ_{μ} = e^{(x,a(T_{o}))}e^{(y,b(T_{o}))}.

By S_{n} invariance this shows that the collection

{e^{(x,a(T))}e^{(y,a(T))} : T ∈ IT_{μ}}  3.11

is contained in Γ_{μ}. Since its linear span is closed under differentiation and contains Φ_{μ}, this collection is evidently a basis of Γ_{μ}, and therefore dim Γ_{μ} = n!. From 3.10 we see that the collection in 3.11, and hence also the space Γ_{μ}, is contained in J_{[ρ_{μ}]}^{⊥}. Both spaces have the same dimension so they are equal.

Identity (d) is implied by (b), for given a power series f that is already x-homogeneous, its lowest customary total degree component m(f) is also its lowest y-degree component m_{y}(f), and consequently m(Γ_{μ}^{x}) is the same as m_{y}(Γ_{μ}^{x}). In the same way, exchanging x and y, identity (e) is implied by (c).

It remains to prove (b) and (c). Appealing to symmetry for (c), we only treat (b). All we need show is that Φ_{μ}^{x}, as defined by 2.31, belongs to the space m_{x}(Γ_{μ}), which is the content of the lemma below.

Lemma 3.2

Up to a scalar factor, Φ_{μ}^{x} is the x-homogeneous component of smallest x-degree in Φ_{μ}. Likewise Φ_{μ}^{y} is the y-homogeneous component of smallest y-degree. More precisely we have

\begin{aligned}a) m_{x}(Φ_{μ}) & = \frac{1}{p_{o}!}Δ_{T_{o}}(a(T_{o})) Φ_{μ}^{x} \\ b) m_{y}(Φ_{μ}) & = \frac{1}{q_{o}!}Δ_{T′_{o}}(b(T_{o})) Φ_{μ}^{y} \end{aligned}  3.12

Proof Expanding only the second exponential in 3.2 we may write

Φ_{μ} = ∑_{q} \frac{b^{q}(T_{o})}{q!} Φ_{q}(y)  3.13

where we have set

Φ_{q} = ∑_{σ∈S_{n}} ϵ(σ) σ( e^{(x,a(T_{o}))} y^{q} )  3.14

Note that if we rearrange the components of q by a permutation γ ∈ S_{n}, Φ_{q} changes according to the identity

Φ_{qγ} = ϵ(γ) ∑_{σ∈S_{n}} ϵ(σ) σ( e^{(x,a(γT_{o}))} y^{q} )  3.15

Thus, if γ is a permutation in the row group of T_{o}, 3.15 yields that

Φ_{qγ} = ϵ(γ) Φ_{q}  3.16

and therefore Φ_{q} must vanish if γ is a simple transpostion and q = qγ. So the only summands that survive in 3.13 are those corresponding to exponent sequences q whose components take different values as their index varies in a row of T_{o}. However, we can easily see that the minimum value of |q| for such an exponent sequence is obtained when q = q_{o}α where q_{o} is the exponent sequence that gives

m_{T′_{o}}(y) = y^{q_{o}}  3.17

and α is any element of the row group of T_{o}. Since Φ_{q} is y-homogeneous of y-degree |q|, we see from 3.13 that the homogeneous component of smallest y-degree in Φ_{μ} must be given by the expression

m_{y}(Φ_{μ}) = \frac{1}{q_{o}!} ∑_{α∈P (T_{o})} b^{q_{o}α}(T_{o}) Φ_{q_{o}α} .

On the other hand, since α is in the row group of T_{o} we can use 3.16 and obtain the factorization

m_{y}(Φ_{μ}) = \frac{1}{q_{o}!} Φ_{q_{o}} ∑_{α∈P (T_{o})} ϵ(α) b^{q_{o}α}(T_{o}) .

Recalling the definition 1.18 of a Garnir polynomial, we finally obtain that

m_{y}(Φ_{μ}) = \frac{1}{q_{o}!} Φ_{q_{o}}Δ_{T′_{o}}(b(T_{o})) .

However, for q = q_{o}, Φ_{q_{o}} reduces to Φ_{μ}^{x} by Lemma 2.1, equation 2.10, which establishes 3.12 (b). The proof of 3.12 (a) is analogous. Q.E.D.

We come now to the main technical device available to us for establishing the n! conjecture in any particular case.

Proposition 3.3

Let G be a finite group acting linearly on Q^{m}, and let x_{1}, …, x_{m} be the coordinate functions on Q^{m}, so that G also acts on the polynomial ring R = Q[x_{1}, …, x_{m}]. Let ρ be a point in Q^{m}, let [ρ] denote its G-orbit, and let J_{[ρ]} be the ideal in R consisting of polynomials that vanish at all points of [ρ]. Finally let w be a G-invariant system of weights making deg_{w} x_{i} strictly positive for each variable x_{i}, and let the Hilbert series of gr_{w} R_{J_{[ρ]}} = R/(gr_{w} J_{[ρ]}) be

H(q) = ∑_{k=0}^{m_{o}} h_{k}q^{k},

where h_{m_{o}} ≠ 0, that is, m_{o} is the largest degree k for which H_{k}(gr_{w} R_{J_{[ρ]}}) does not reduce to zero. Then the following conditions are equivalent:

a) h_{0} + h_{1} + ⋯ h_{k} ≥ h_{m_{o}−k} + h_{m_{o}−k+1} + ⋯ + h_{m_{o}} for all 0 ≤ k ≤ m_{0};
b) h_{k} = h_{m_{o}−k} for all 0 ≤ k ≤ m_{0};
c) gr_{w} R_{J_{[ρ]}} is a Gorenstein graded algebra; and
d) the space (gr_{w} J_{[ρ]})^{⊥} is the linear span
L{∂_{x}^{p}Δ}

of all derivatives of a single polynomial Δ. (We describe this by saying that (gr_{w} J_{[ρ]})^{⊥} is a cone, and Δ is its summit.)

Proof

The equivalence of (c) and (d) is well-known (see [3]) as is the implication (c) ⇒ (b). Alternatively, one can obtain (d) ⇒ (b) without recourse to (c) by using the flip operator of 1.14 for Δ.

So the only new feature, and indeed the only one that depends upon the set [ρ] being a Gorbit, is the implication (a) ⇒ (c). Given (a), we have a fortiori h_{m_{o}} = h_{0} = 1. Thus up to a scalar multiple gr_{w} R_{J_{[ρ]}} contains a unique homogeneous element δ of w-degree m_{o}, and correspondingly (gr_{w} J_{[ρ]})^{⊥} contains a unique w-homogeneous polynomial Δ of the same degree. The 1-dimensional space L{δ} affords a 1-dimensional representation of G, with character ϵ, let us say.

Since [ρ] is a G-orbit, the representation of G afforded by R_{J_{[ρ]}} and gr_{w} R_{J_{[ρ]}} is equivalent to a quotient of the regular representation, and thus contains the character ϵ with multiplicity one. In particular this character cannot occur at all in any degree less than m_{o}; in terms of the filtration 3.4 of R_{J_{[ρ]}} this means that ϵ does not occur in F_{≤m_{o}−1}. Let us define on R_{J_{[ρ]}} a scalar product ⟨, ⟩_{ϵ} by the rule

∑_{σ∈G} ϵ(σ^{−1})σ(pq) = ⟨p, q⟩_{ϵ}δ.  3.18

This definition makes sense because the element of R_{J_{[ρ]}} represented by δ is non-zero (otherwise δ, being homogeneous, would belong to gr_{w} J_{[ρ]}), and its scalar multiples are the only elements transforming under G according to the character ϵ.

The scalar product in 3.18 is non-degenerate, as we can see by the following argument. Suppose p were such that ⟨p, q⟩_{ϵ} = 0 for every q. Unless p vanishes at every point of [ρ], we can choose q so that pq = 1 at one point γρ and pq = 0 at all other points of [ρ]. But if the sum in 3.18 were zero for this function pq, then by construction it would also have to be zero for α(pq), for every α ∈ G, and hence it would be zero with any function f in place of pq, that is, the scalar product ⟨, ⟩_{ϵ} would be identically zero. And this is clearly not the case, for ⟨1, δ⟩_{ϵ} = |G|.

As we have already noted, F_{≤m_{o}−1} does not contain the character ϵ, which implies by the filtration property 3.6 that F_{≤k} and F_{≤m_{o}−1−k} are orthogonal to each other for the scalar product in 3.18. However the inequalities in (a) mean precisely that dim F_{≤k} + dim F_{≤m_{o}−1−k} ≥ dim R_{J_{[ρ]}}. Therefore these inequalities must be equalities, and we must have

(F_{≤k})^{⊥} = F_{≤m_{o}−1−k}  3.19

for each k. Then the scalar product ⟨, ⟩_{ϵ} induces a non-degenerate pairing of F_{≤k}/F_{≤k−1} with F_{≤m_{o}−k}/F_{≤m_{o}−1−k}, which by 3.7 and Lemma 3.1 is simply the multiplication pairing H_{k}(gr_{w} R_{J_{[ρ]}})× H_{m_{o}−k}(gr_{w} R_{J_{[ρ]}}) → H_{m_{o}}(gr_{w} R_{J_{[ρ]}}). By definition, the non-degeneracy of this pairing means that (gr_{w} R_{J_{[ρ]}}) is Gorenstein. Q.E.D.

For an alternative elementary proof of this proposition, not using the theory of Gorenstein rings, we refer the reader once again to [9].

We are now ready to proceed in earnest toward our main result. Throughout the rest of this section, we fix the partition μ and the S_{n} orbit [ρ_{μ}] of ρ_{μ} = (a(T_{o}), b(T_{o})), and set

R = Q[X, Y], J = J_{[ρ_{μ}]}, R_{J} = R/J.

Associated with gr_{x} R_{J} and gr_{y} R_{J} are two filtrations

F_{≤0}^{x} ⊆ F_{≤1}^{x} ⊆ ⋯ ⊆ F_{≤n(μ)}^{x} = R_{J}  3.20

and

F_{≤0}^{y} ⊆ F_{≤1}^{y} ⊆ ⋯ ⊆ F_{≤n(μ′)}^{x} = R_{J},  3.21

where F_{≤k}^{x} consists of all elements in R_{J} representable by a polynomial p(x, y) of degree at most k in x; and similarly for y. Note that the termination of these filtrations at degrees n(μ) and n(μ′), respectively, follows from Theorem 2.2, together with the identifications 3.9 (b) and (c) of Γ_{μ}^{x} and Γ_{μ}^{y} as (gr_{x} J)^{⊥} and (gr_{y} J)^{⊥}.

The x-graded ring gr_{x} R_{J} comes equipped with two possible y-filtrations, either of which may be used to produce from it a doubly graded ring. The first is the normal y-filtration of gr_{x} R_{J} = R/(gr_{x} J) defined by 3.5, which is the filtration associated to gr_{y} gr_{x} R_{J}. We may denote this filtration

G_{≤0}^{y} ⊆ G_{≤1}^{y} ⊆ ⋯ ⊆ gr_{x} R_{J}.  3.22

The second and more remarkable filtration is induced by the filtration F^{y} of R_{J}. Specifically, for each k the k-th graded component of gr_{x} R_{J}, namely F_{≤k}^{x}/F_{≤k−1}^{x}, is itself filtered by its subspaces (F_{≤k}^{x} ∩ F_{≤l}^{y})/(F_{≤k−1}^{x} ∩ F_{≤l}^{y}), and we therefore set

G_{≤l}^{′y} = ⨁_{k=0}^{n(μ)}(F_{≤k}^{x} ∩ F_{≤l}^{y})/(F_{≤k−1}^{x} ∩ F_{≤l}^{y})  3.23

Of course the filtration property 3.6 for F^{y} immediately implies the same for G^{′y}, and so we obtain a doubly graded ring

\begin{aligned}bigr R_{J} & = ⨁_{k=0}^{n(μ)} ⨁_{l=0}^{n(μ′)} H_{k}^{x}(G_{≤l}^{′y}/G_{≤l−1}^{′y}) \\ & = ⨁_{k=0}^{n(μ)} ⨁_{l=0}^{n(μ′)}(F_{≤k}^{x} ∩ F_{≤l}^{y})/(F_{≤k−1}^{x} ∩ F_{≤l}^{y} + F_{≤k}^{x} ∩ F_{≤l−1}^{y}). \end{aligned}  3.24

From the second equality in 3.24 we see that in the construction of bigr R_{J}, x and y actually play perfectly symmetrical roles.

Proposition 3.4 The ring bigr R_{J} defined in 3.24 is a Gorenstein graded algebra.

Proof

We are going to imitate the proof of Proposition 3.3, with some refinements to take account of the double grading. To begin, on R_{J} we define a scalar product ⟨, ⟩ϵ just as in 3.18 by

∑_{σ∈S_{n}} ϵ(σ)σ(pq) = ⟨p, q⟩_{ϵ}Δ_{μ}.  3.25

In the present case, of course, ϵ is the sign character of S_{n}, and we know that the bihomogeneous polynomial Δ_{μ} represents a non-zero element of R_{J}, since otherwise it would belong to gr_{w} J_{μ} for any w, contrary to Theorem 1.1. Then 3.25 is well-defined since every S_{n}-alternating element of R_{J} must be a scalar multiple of Δ_{μ}.

From the equivalence of gr_{x} R_{J} with Γ_{μ}^{x} as x-graded S_{n}-modules, we see that the sole instance of the alternating representation in gr_{x} R_{J} occurs in the highest x-degree, n(μ), which is to say that F_{≤n(μ)−1}^{x} contains no alternating elements. It follows that the spaces F_{≤k}^{x} and F_{≤n(μ)−1−k}^{x} are mutually orthogonal for the scalar product ⟨, ⟩_{ϵ}. Now we already know that each space F_{≤k}^{x} has dimension

dim F_{≤k}^{x} = ∑_{k′≤k} dim H_{k}(Γ_{μ}^{x})

and further, as a consequence of Theorem 2.2, that

dim H_{k}(Γ_{μ}^{x}) = \binom{n}{μ′} ∏_{i} \frac{1 − q^{μ′_{i}}}{1 − q} |_{q^{k}} ,

where the vertical bar denotes extracting the coefficient of q^{k}. In particular,

dim H_{k}(Γ_{μ}^{x}) = dim H_{n(μ)−k}(Γ_{μ}^{x})

and therefore dim F_{≤k}^{x} +dim F_{≤n(μ)−1−k}^{x} = n! = dim R_{J}. Hence, with respect to our scalar product ⟨, ⟩_{ϵ}, we have for all k

(F_{≤k}^{x})^{⊥} = F_{≤n(μ)−1−k}^{x}.  3.26

By analogous reasoning with y in place of x we also have for all l

(F_{≤l}^{y})^{⊥} = F_{≤n(μ′)−1−l}^{y}  3.27

Given these orthogonalities, we see that our scalar product induces for each k and l a welldefined pairing of the space

H_{k,l}(bigr R_{J}) = (F_{≤k}^{x} ∩ F_{≤l}^{y})/(F_{≤k−1}^{x} ∩ F_{≤l}^{y} + F_{≤k}^{x} ∩ F_{≤l−1}^{y})

with

H_{h,j}(bigr R_{J}) = (F_{≤h}^{x} ∩ F_{≤j}^{y})/(F_{≤h−1}^{x} ∩ F_{≤j}^{y} + F_{≤h}^{x} ∩ F_{≤j−1}^{y}),

where we have set h = n(μ) − k and j = n(μ′) − l. Moreover this pairing is clearly the same as that given by multiplication

H_{k,l}(bigr R_{J}) × H_{h,k}(bigr R_{J}) → H_{n(μ),n(μ′)}(bigr R_{J}),

followed by extracting the coefficient of Δ_{μ}. We are to show that the pairing is non-degenerate, which is to say, that the orthogonal complement of F_{≤k}^{x} ∩F_{≤l}^{y} in F_{≤h}^{x} ∩F_{≤j}^{y} is (F_{≤h−1}^{x} ∩F_{≤j}^{y} +F_{≤h}^{x} ∩F_{≤j−1}^{y}). Now the orthogonal complement in question is

F_{≤h}^{x} ∩ F_{≤j}^{y} ∩ (F_{≤k}^{x} ∩ F_{≤l}^{y})^{⊥} = F_{≤h}^{x} ∩ F_{≤j}^{y} ∩ (F_{h−1}^{x} + F_{j−1}^{y}),

by 3.26–3.27, so we are reduced to proving

F_{≤h}^{x} ∩ F_{≤j}^{y} ∩ (F_{h−1}^{x} + F_{j−1}^{y}) = (F_{≤h−1}^{x} ∩ F_{≤j}^{y} + F_{≤h}^{x} ∩ F_{≤j−1}^{y}).  3.28

But 3.28 is an instance of a completely general and easy to prove identity holding for any subspaces of a vector space:

if X_{1} ⊆ X and Y_{1} ⊆ Y, then X ∩ Y ∩ (X_{1} + Y_{1}) = X_{1} ∩ Y + X ∩ Y_{1}.

Q.E.D.

Remark 3.1

One might at first suppose that Proposition 3.4 combined with the ‘cone’ condition (d) of Proposition 3.3 could be used to prove the n! conjecture outright. It is quite interesting to explore just what goes wrong with this supposition. The difficulty is that bigr R_{J} need not be a quotient of R = Q[X, Y] at all. The reason for this is that while every element a of F_{≤k}^{x} ∩ F_{≤l}^{y} is representable modulo J by a polynomial p(x, y) of x-degree at most k, and also by a polynomial p′(x, y) of y-degree at most l, it may not be possible to represent it so simultaneously by the same polynomial. In that case, a can represent an element of H_{k,l}(bigr R_{J}) which is not the image of any polynomial in R.

In fact, this situation actually occurs: in place of a Ferrers diagram μ, we may take any diagram D consisting of n lattice squares in the plane, and construct [ρ_{D}], J_{[ρ_{D}]}, and bigr R/J_{[ρ_{D}]} all by exact analogy to what we have done for partitions. When D is not a partition the ring bigr R_{J} usually does fail to be generated by X and Y. Nevertheless it is a very interesting ring in its own right, which we expect not to depend on the parameters α and β introduced into the construction by 1.6. Assuming this and the C = H̃ conjecture, it is appropriate to think of the Frobenius characteristic of bigr R/J_{[ρ_{D}]} as a kind of Macdonald polynomial H̃_{D}(x; q, t) indexed by an arbitrary diagram D.

At present we can prove very little about H̃_{D}, although it does possess the pleasant property that when D = D_{1} ⊕ D_{2}, meaning that D_{1} and D_{2} occupy disjoint sets of rows and columns, and D is their union, then H̃_{D} = H̃_{D_{1}} H̃_{D_{2}}. There is also reason to suspect further relations, especially when D is a skew-Ferrers diagram. In [10], we managed to define the polynomials H̃_{D} for all tworowed skew diagrams based solely on their expected properties, without having to prove anything about the underlying rings. Everything done there was however necessarily motivated by hidden considerations involving bigr R/J_{[ρ_{D}]}.

Returning from this digression, we arrive at our main theorem.

Theorem 3.2

The condition m(Γ_{μ}^{x}) = m(Γ_{μ}^{y}) is equivalent to the n! conjecture for the partition μ. Indeed, both spaces are then equal to H_{μ}, and moreover for J = J_{[ρ_{μ}]}, we have

bigr R_{J} = gr_{x} gr_{y} R_{J} = gr_{y} gr_{x} R_{J}.  3.29

Proof

It is actually 3.29 that we are going to prove. For then the ring gr_{y} gr_{x} R_{J} is Gorenstein, and by Proposition 3.3 the space (gr_{y} gr_{x} J)^{⊥} is a cone, necessarily generated by its element Δ of bidegree (n(μ), n(μ′)). However by Theorem 1.1, this element Δ must be a scalar multiple of Δ_{μ}, which shows that H_{μ} = (gr_{y} gr_{x} J)^{⊥} = m(Γ_{μ}^{x}) = m(Γ_{μ}^{y}).

What we must show is that in gr_{x} R_{J} the two filtrations G^{y} and G^{′y} given by 3.22 and 3.23 are the same. Now an element of H_{k}^{x}(G_{≤l}^{y}) is by definition represented modulo gr_{x} J by a polynomial p(x, y) homogeneous of bidegree k, l. Hence it also belongs to H_{k}^{x}(G_{≤l}^{′y}), that is, we always have

G_{≤l}^{y} ⊆ G_{≤l}^{′y}.  3.30

On the assumption that gr_{x} gr_{y} R_{J} = gr_{y} gr_{x} R_{J}, the dimension of G_{≤l}^{y} is equal to

∑_{l′≤l} ∑_{k} dim H_{k,l′}(gr_{x} gr_{y} R_{J}) = ∑_{l′≤l} dim H_{l′}^{y}(gr_{y} R_{J}) = dim F_{≤l}^{y}.  3.31

On the other hand G_{l}^{′y} is the direct sum of the successive quotients in the filtration of the space F_{≤l}^{y} by its intersections with the spaces F_{≤k}^{x}. Therefore we always have dim G_{l}^{′y} = dim F_{≤l}^{y}, and this with 3.30 and 3.31 establishes the result. Q.E.D.

4. Our bigraded modules H_{μ} under Raising Operators.

There is an elementary approach to proving the n! conjecture which can be successfully used in various special cases. More importantly, this approach leads to the discovery of some truly remarkable properties of the modules H_{μ}. The basic idea can be best understood by working out some examples. Note that Theorem 1.5 guarantees that the 0 y-degree portion of H_{μ} has dimension \binom{n}{μ}. In the particular case that μ = (1^{n−2}, 2) this dimension is n!/2. Since flip (see 1.14) yields a sign-twisted S_{n}-module isomorphism of the 0 y-degree portion into the n(μ′) y-degree portion, we see that for all these partitions (and their conjugates) the n! conjecture must necessarily hold true. Moreover the case of μ = 1^{n} (single columns) or μ = (n) (single rows) is classical since Δ_{μ} then reduces to the Vandermonde determinant in the x′s and the y′s respectively. In particular, for n = 4 we are only left with μ = (2, 2). We shall start by dealing with this case.

Expanding with respect to the last column we get

Δ_{2,2} = det ( \begin{matrix}1 & 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} & x_{4} \\ y_{1} & y_{2} & y_{3} & y_{4} \\ x_{1}y_{1} & x_{2}y_{2} & x_{3}y_{3} & x_{4}y_{4} \end{matrix} ) = Φ_{oo} + x_{4} Φ_{1o} + y_{4} Φ_{o1} + x_{4}y_{4}Φ_{11} .  4.1

So we may write

∂_{x_{4}}Δ_{2,2} = Φ_{1o} + y_{4}Φ_{11} , ∂_{y_{4}}Δ_{2,2} = Φ_{o1} + x_{4}Φ_{11} , ∂_{x_{4}}∂_{y_{4}}Δ_{2,2} = Φ_{11} ,  4.2

and note that

Φ_{11} = Δ_{1,2} = det ( \begin{matrix}1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{matrix} )  4.3

If P is a polynomial in x_{1}, .., x_{n}; y_{1}, .., y_{n}, the differential operator P (∂_{x_{1}}, .., ∂_{x_{n}}; ∂_{y_{1}}, .., ∂_{y_{n}}) here and after will be denoted by P (∂). This given, let B be a collection of monomials in x_{1}, x_{2}, x_{3}; y_{1}, y_{2}, y_{3} chosen so that {b(∂)Δ_{1,2}}_{b∈B} is a basis for H_{21}. We claim that the polynomials in the union

B_{2,2} = {b(∂)Δ_{2,2}}_{b∈B} + {b(∂)∂_{x_{4}}Δ_{2,2}}_{b∈B} + {b(∂)∂_{y_{4}}Δ_{2,2}}_{b∈B} + {b(∂)∂_{x_{4}}∂_{y_{4}}Δ_{2,2}}_{b∈B} 4.4

are linearly independent. Note that since we know in full generality that dimB_{μ} ≤ n!, we can thus conclude that B_{2,2} is a basis for H_{2,2} and that the conjecture is true for μ = (2, 2). To this end, let there be b_{oo}, b_{1o}, b_{o1}, b_{11} in the linear span of B giving

b_{oo}(∂)Δ_{2,2} + b_{1o}(∂)∂_{x_{4}}Δ_{2,2} + b_{o1}(∂)∂_{y_{4}}Δ_{2,2} + b_{11}(∂)∂_{x_{4}}∂_{y_{4}}Δ_{2,2} = 0 .

Using 4.1, 4.2 and 4.3 and extracting the coefficients of 1, x_{4}, y_{4}, x_{4}y_{4} we derive that we must have the simultaneous equations

b_{oo}(∂)Φ_{oo} + b_{1o}(∂)Φ_{1o} + b_{o1}(∂)Φ_{o1} + b_{11}(∂)Δ_{1,2} = 0
b_{oo}(∂)Φ_{1o}  + b_{o1}(∂)Δ_{1,2} = 0
b_{oo}(∂)Φ_{o1} + b_{1o}(∂)Δ_{1,2} = 0
b_{oo}(∂)Δ_{1,2} = 0

Note now that if b_{oo} = ∑_{b∈B} c_{b} b then the last equation may be written as

∑_{b∈B} c_{b} b(∂)Δ_{1,2} = 0 ,

but this contraddicts the choice of B unless the coefficients c_{b} are all equal to zero. Substituting b_{oo} = 0 in the third equation reduces it to b_{1o}(∂)Δ_{1,2} = 0 which again implies that the coefficients of b_{1o} must all vanish as well. Substituting b_{oo} = 0 in the second equation yields the same for the coefficients of b_{o1}. Finally, setting b_{oo} = b_{1o} = b_{o1} = 0 in the first equation forces the vanishing of the coefficients of b_{11}. This proves that B_{2,2} is an independent set as asserted. We should note that our argument here establishes a bit more that the validity of the n! conjecture for μ = (2, 2).

Let us recall that if θ is the Frobenius image of an S_{n} character χ then the partial derivative ∂_{p_{1}}θ (*) yields the Frobenius image of the restriction of χ to S_{n−1}. In particular, if C_{μ}(x; q, t) (as in section 1.) denotes the Frobenius characteristic of H_{μ} then the polynomial

G_{μ}(q, t) = ∂_{p_{1}}^{n} C_{μ}(x; q, t) = ∑_{λ⊢n} f_{λ}C_{λμ}(q, t)  4.5

gives the bigraded Hilbert series of H_{μ}. Of course under the C = H̃ conjecture this Hilbert series is given by the polynomial

F̃_{μ}(q, t) = ∂_{p_{1}}^{n} H̃_{μ}(x; q, t) = ∑_{λ⊢n} f_{λ}K̃_{λμ}(q, t) .  4.6

(*) Here p_{1} denotes the first power symmetric polynomial and the symbol ∂_{p_{1}}θ is to mean the partial derivative of θ as a polynomial in the power symmetric functions.

We should also keep in mind (see [9]) that the “duality” result for Macdonald polynomials in our notation becomes

ωH̃_{μ}(x; 1/q, 1/t)t^{n(μ)}q^{n(μ′)} = H̃_{μ}(x; q, t)  4.7

where ω is the involution which sends S_{λ} into S_{λ′}. Thus 4.7 implies that we must also have

F̃_{μ}(1/q, 1/t)t^{n(μ)}q^{n(μ′)} = F̃_{μ}(q, t) .  4.8

The analogous identities

ωC_{μ}(x; 1/q, 1/t)t^{n(μ)}q^{n(μ′)} = C_{μ}(x; q, t)  4.9
G_{μ}(1/q, 1/t)t^{n(μ)}q^{n(μ′)} = G_{μ}(q, t) .  4.10

follow from the fact that flip is a sign-twisting, degree-complementing S_{n}-module automorphism of H_{μ}.

It is easy to see that if β(q, t) gives the bidegree distribution of the monomials in B then qtβ(1/q, 1/t) must give the Hilbert series of H_{1,2}. Thus from 4.10 we get that β = G_{1,2}(t, q). From our construction 4.4 of the basis B_{2,2} we can then immediately derive that the Hilbert series of H_{2,2} must be given by formula

G_{2,2}(q, t) = β(1/q, 1/t)(1 + 1/t + 1/q + 1/tq)q^{2}t^{2} = (1 + t + q + tq)G_{1,2}(q, t) .

A slightly more refined argument which takes account of the action of S_{3} on the basis B_{2,2}, yields that we must also have

∂_{p_{1}}C_{2,2}(x; q, t) = (1 + t + q + tq)C_{1,2}(x; q, t) .

It is not difficult to see that the argument we have given here can be generalized to the case of arbitrary rectangular partitions μ = r^{s} and obtain that

Theorem 4.1 If the n! conjecture holds for the partition (r − 1, r^{s−1}) then it holds for r^{s}, Moreover,

∂_{p_{1}}C_{r^{s}}(x; q, t) = B_{r^{s}}(q, t) C_{r−1,r^{s−1}}(x; q, t)  4.11
G_{r^{s}}(q, t) = B_{r^{s}}(q, t) G_{r−1,r^{s−1}}(q, t)  4.12

Continuing with our examples we see that for n = 5 we need to deal with the partitions (1, 2, 2), (1, 1, 3), (3, 2). In view of the identities

C_{μ′}(x; q, t) = C_{μ}(x; t, q)  H̃_{μ′}(x; q, t) = H̃_{μ}(x; t, q) ,  4.13

proving the n! conjecture for μ yields it also for μ′. So we are left with (1, 2, 2) and (1, 1, 3). We start with (1, 2, 2). Expanding again with respect to the last column we get

Δ_{1,2,2} = det ( \begin{matrix}1 & 1 & 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ y_{1} & y_{2} & y_{3} & y_{4} & y_{5} \\ x_{1}y_{1} & x_{2}y_{2} & x_{3}y_{3} & x_{4}y_{4} & x_{5}y_{5} \\ x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & x_{4}^{2} & x_{5}^{2} \end{matrix} ) = Φ_{oo} + x_{5} Φ_{1o} + y_{5} Φ_{o1} + x_{5}y_{5}Φ_{11} + x_{5}^{2}Φ_{2o} .

4.14

So we may write

Δ_{1,2,2}  = Φ_{oo} + y_{5}Φ_{o1} + x_{5}Φ_{1o} + x_{5}y_{5}Δ_{1,1,2} + x_{5}^{2}Δ_{2,2}
∂_{y_{5}}Δ_{1,2,2} = Φ_{o1} + x_{5}Δ_{1,1,2}
∂_{x_{5}}Δ_{1,2,2} =  Φ_{1o} + y_{5}Δ_{1,1,2} + 2x_{5}Δ_{2,2}
∂_{x_{5}}∂_{y_{5}}Δ_{1,2,2} =  Δ_{1,1,2}
∂_{x_{5}}^{2}Δ_{1,2,2} =  2Δ_{2,2}

4.15

Proceeding as we did for μ = (2, 2) we are to construct 5 collections B_{oo}, B_{1o}, B_{o1}, B_{11}, B_{20} of polynomials in x_{1}, x_{2}, x_{3}, x_{4}; y_{1}, y_{2}, y_{3}, y_{4} such that the polynomials in the union

\begin{aligned}B_{1,2,2} = \{b(∂)Δ_{1,2,2}\}_{b∈B_{oo}} + \{b(∂)∂_{y_{4}} & Δ_{1,2,2}\}_{b∈B_{o1}} + \{b(∂)∂_{x_{4}}Δ_{1,2,2}\}_{b∈B_{1o}}+ \\ & + \{b(∂)∂_{x_{4}}∂_{y_{4}}Δ_{1,2,2}\}_{b∈B_{11}}\{b(∂)∂_{x_{4}}^{2}Δ_{1,2,2}\}_{b∈B_{2o}} \end{aligned}

are independent. If we succeed in choosing them so that B_{1,2,2} has altogether 120 elements we will have proved the n! conjecture for μ = (1, 2, 2). In this case we shall not venture a guess but rather determine the B_{ij} from the equations that would have to hold if there was a dependence between the elements of B_{1,2,2}. So let b_{ij} be in the linear span of B_{ij} and assume that we have

b_{oo}(∂)Δ_{1,2,2} + b_{o1}(∂)∂_{y_{5}}Δ_{1,2,2} + b_{1o}(∂)∂_{x_{5}}Δ_{1,2,2} + b_{11}(∂)∂_{x_{5}}∂_{y_{5}}Δ_{1,2,2} + b_{2o}(∂)∂_{x_{5}}^{2}Δ_{1,2,2} = 0

Using the expansions in 4.12 and 4.13 and equating coefficients of 1, x_{5}, y_{5}, x_{5}y_{5}, x_{5}^{2} we get the system of equations

b_{oo}(∂)Φ_{oo} + b_{o1}(∂)Φ_{o1} + b_{1o}(∂)Φ_{1o} + b_{11}(∂)Δ_{1,1,2} + 2b_{2o}(∂)Δ_{2,2} = 0
b_{oo}(∂)Φ_{o1} + b_{1o}(∂)Δ_{1,1,2} = 0
b_{oo}(∂)Φ_{1o} + b_{o1}(∂)Δ_{1,1,2} + 2b_{1o}(∂)Δ_{2,2} = 0
b_{oo}(∂)Δ_{1,1,2}  = 0
b_{oo}(∂)Δ_{2,2} = 0

4.16

Recall that in section 1 (see 1.12) we denoted by I_{μ} the ideal of polynomials which kill Δ_{μ}. With this notation the last two equations yield us that b_{oo} ∈ I_{1,1,2} ∩ I_{2,2} and since (I_{1,1,2} ∩ I_{2,2})^{⊥} = H_{1,1,2} +H_{2,2}, we see that we can choose B_{oo} to be any bihomogeenous basis of H_{1,1,2} +H_{2,2}. Indeed, if we do so any linear combination b_{oo} of elements of B_{oo} which satisfies the last two equations in 4.13 would have to be orthogonal to itself and therefore must have all its coefficients equal to zero. We shall visualize this choice by writing

B_{oo} = ∨  4.17

This done the second equation reduces to b_{1o}(∂)Δ_{1,1,2} = 0 which gives b_{1o} ∈ I_{1,1,2}. Thus using the same imagery we can set

B_{1o} =  4.18

This reduces the third equation to b_{o1}(∂)Δ_{1,1,2} = 0. Thus we can set

B_{o1} =  4.19

Substituting b_{oo} = b_{1o} = b_{o1} = 0 in the first equation reduces it to b_{11}(∂)Δ_{1,1,2} + b_{2o}(∂)Δ_{2,2} = 0. Here we have two possible choices. We take

B_{2o} =  4.20

Next, we want b_{11}(∂)Δ_{1,1,2} ∈ H_{2,2} to imply b_{11} = 0. We can assure this if b_{11}(∂)Δ_{1,1,2} lies in the orthogonal complement of H_{1,1,2} ∩ H_{2,2} in H_{1,1,2}. In other words we want flip_{1,1,2}b_{11} (*) to lie in I_{2,2} ∩ H_{1,1,2}. So our final choice is

B_{11} = flip_{1,1,2}^{−1}[( ∧ )^{⊥} ∧ ]

These choices assure that B_{1,1,2} is an independent set. Let us now count how many elements it has. Now, 4.18, 4.19 and 4.20 yield 3 × 24 elements and 4.17 yields 2 × 24 − dim H_{1,1,2} ∩ H_{2,2} elements. Since H_{1,2,2} ≤ 120 the independence of B_{1,2,2} yields the inequality

5 × 24 − dim ∧ + dim ( ∧ )^{⊥} ∧ ≤ 120

or better

dim ( ∧ )^{⊥} ∧ ≤ dim ∧  4.21

In particular, B_{1,2,2} would be a basis for H_{1,2,2} and the n! conjecture would then be established for Δ_{1,2,2} if we prove that equality holds in 4.21. Amazingly it develops that this equality holds true in the strongest possible sense. More precisely, we can prove that

flip_{1,1,2} ∧ = ( ∧ )^{⊥} ∧ .  4.22

Let us postpone the proof of this identity and proceed to examine all its implications. To begin with it shows that we can take B_{11} to be any bihomogeneous basis of H_{1,1,2} ∩ H_{2,2}. This done we can visualise the Hilbert series of H_{1,2,2} by writing

G_{1,2,2} =  +  − ∧ + q + t + t^{2} + tq ∨ .

To be precise, the left hand side of this relation is G_{1,2,2}(1/q, 1/t)t^{4}q^{2}, however we can write it that way because of 4.10. It will be convenient to extend the definition of flip_{μ} to act on a Frobenius characteristic H(x; q, t) and on a Hilbert series F (q, t) by setting

flip_{μ} H(x; q, t) = ωH(x; 1/q, 1/t) t^{n(μ)}q^{n(μ′)} , flip_{μ} F (q, t) = F (1/q, 1/t) t^{n(μ)}q^{n(μ′)}  4.23

(*) Here and after flip_{μ} will denote flipping with respect to Δ_{μ}.

This given, the more refined argument which takes account of the action of S_{4} on the basis B_{1,2,2} yields the relation

∂_{p_{1}}C_{1,2,2}(x; q, t) = (1 + t + q)C_{1,1,2}(x; q, t) + (1 + t^{2})C_{2,2}(x; q, t) + (tq − 1) ϕ  4.24

where for convenience we have denoted by ϕ the Frobenius characteristic of ∧ . Here again the left-hand side should have been flip_{1,2,2}∂_{p_{1}}C_{1,2,2}, but 4.9 makes it right the way it is.

Note that if we follow the other alternative and choose B_{11} = instead of 4.20, then we must take

B_{2o} = flip_{2,2}^{−1} [( ∧ )^{⊥} ∧ ]

However, the analogue of 4.22

flip_{2,2} ∧ = ( ∧ )^{⊥} ∧ .  4.25

yields 4.24 back again. Note that formulas 4.22 and 4.25 give us the decompositions

C_{1,1,2}(x; q, t) = ϕ + flip_{1,1,2} ϕ ,  C_{2,2}(x; q, t) = ϕ + flip_{2,2} ϕ ,  4.26

from which we derive that

ϕ = \frac{q C_{1,1,2} − t C_{2,2}}{q − t}  4.27

It goes without saying that all these relations can be verified by computer. In particular in this manner we can establish 4.22 and 4.25 and also check that the C = H̃ conjecture is in fact true for (1, 2, 2), (1, 1, 2) and (2, 2). Nevertheless it is more illuminating to verify these identities using representation theory. Recalling that

Δ_{2,2} = det ( \begin{matrix}1 & 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} & x_{4} \\ y_{1} & y_{2} & y_{3} & y_{4} \\ x_{1}y_{1} & x_{2}y_{2} & x_{3}y_{3} & x_{4}y_{4} \end{matrix} ) , Δ_{1,1,2} = det ( \begin{matrix}1 & 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} & x_{4} \\ y_{1} & y_{2} & y_{3} & y_{4} \\ x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & x_{4}^{2} \end{matrix} ) ,

we get

∂_{x_{4}}∂_{y_{4}}Δ_{2,2} = \frac{1}{2} ∂_{x_{4}}^{2}Δ_{1,1,2} = det ( \begin{matrix}1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{matrix} )

Now (by Theorem 1.8) the action of S_{4} on this element generates a 3-dimensional irreducible representation with character χ^{1,1,2} and weight tq which is shared by H_{1,1,2} and H_{2,2}. Similarly, we see that its partial derivatives

∂_{x_{3}}det ( \begin{matrix}1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{matrix} ) = det ( \begin{matrix}1 & 1 \\ y_{1} & y_{2} \end{matrix} ) , ∂_{y_{3}}det ( \begin{matrix}1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{matrix} ) = det ( \begin{matrix}1 & 1 \\ x_{1} & x_{2} \end{matrix} ) ,

generate common 3-dimensional representations with character χ^{1,3} and weights q and t respectively. Similar calculations yields that

(∂_{x_{3}}∂_{y_{4}} + ∂_{x_{4}}∂_{y_{3}})Δ_{1,1,2} = 2∂_{y_{3}}∂_{y_{4}}Δ_{2,2} = 2det ( \begin{matrix}1 & 1 \\ x_{1} & x_{2} \end{matrix} ) det ( \begin{matrix}1 & 1 \\ x_{3} & x_{4} \end{matrix} ) ,

and this generates a common 2 dimensional representation with character χ^{2,2} and weight t^{2}. Including the trivial, we have thus identified a submodule of H_{1,1,2} ∩ H_{2,2} with bigraded Frobenius chracteristic

Π = S_{4} + (t + q) S_{1,3} + tq S_{1,1,2} + t^{2} S_{2,2} .

Using 4.9 and the definition in 4.23 we then immediately derive that H_{1,1,2} and H_{2,2} contain submodules with respective Frobenius characteristics

flip_{1,1,2} Π = t^{3}q S_{1^{4}} + (t^{2}q + t^{3}) S_{1,1,2} + t^{2} S_{1,3} + tq S_{2,2} ,
flip_{2,2} Π = t^{2}q^{2}S_{1^{4}} + (tq^{2} + t^{2}q) S_{1,1,2} + tq S_{1,3} + q^{2} S_{2,2} . 4.28

Since Π and flip_{1,1,2} Π have no common terms and each accounts for 12 dimensions we conclude that together they must give the Frobenius characteristic of H_{1,1,2}. A similar reasoning applies to Π and flip_{2,2} Π and we can write

C_{1,1,2} = Π + flip_{1,1,2} Π , C_{2,2} = Π + flip_{2,2} Π ,

Comparing the expressions in 4.28, we see that the possibility still remains that H_{1,1,2} and H_{2,2} may have in common an irreducible submodule with character χ^{1,1,2} and weight t^{2}q. However, this submodule is generated in H_{2,2} by the action of S_{4} on the polynomial

P = ( \begin{matrix}1 & 1 \\ ∂_{y_{1}} & ∂_{y_{2}} \end{matrix} ) Δ_{2,2}

If this polynomial were to belong to H_{1,1,2} it would be killed by any element that kills Δ_{1,1,2}. In particular we should have ∂_{x_{4}}∂_{y_{4}}P = 0. But we see that

∂_{x_{4}}∂_{y_{4}}P = ( \begin{matrix}1 & 1 \\ ∂_{y_{1}} & ∂_{y_{2}} \end{matrix} ) det ( \begin{matrix}1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{matrix} ) = x_{2} − x_{1} ≠ 0 .

This completes the proof of 4.22 and 4.25 and shows that Π is none other than the Frobenius characteristic ϕ of the intersection of H_{112} and H_{2,2}.

Remark 4.1

It is interesting to see what 4.24 reduces to when we express its right hand side entirely in terms of ϕ. To this end we use the identities in 4.26 to replace C_{1,1,2} and C_{2,2} and obtain

∂_{p_{1}} C_{1,2,2} = (1 + t + q)(ϕ + t^{3}qω ↓ ϕ) + (1 + t^{2})(ϕ + t^{2}q^{2}ω ↓ ϕ) + (tq − 1)ϕ ,

where for convenience we have let ↓ denote the operator on rational functions of t, q which replaces t by 1/t and q by 1/q. Collecting terms in ϕ and ω ↓ ϕ gives

\begin{aligned}∂_{p_{1}} C_{1,2,2} & = (1 + t + q + tq + t^{2}) ϕ + (t^{4}q^{2} + t^{3}q^{2} + t^{4}q + t^{3}q + t^{2}q^{2})ω ↓ ϕ \\ & = (1 + t + q + tq + t^{2}) ϕ + t^{4}q^{2}(1 + 1/t + 1/q + 1/tq + 1/t^{2})ω ↓ ϕ \end{aligned} ,

and this may be rewritten in the very suggestive form

∂_{p_{1}} C_{1,2,2} = B_{1,2,2}(q, t) ϕ + flip_{1,2,2}( B_{1,2,2}(q, t) ϕ ) .  4.29

What we have discovered in this particular example appears to hold true in full generality. We are thus lead to a number of results and conjectures of various strengths.

Proposition 4.1

Let λ be a two-corner partition of n+1 and let μ and ν be the partitions obtained by removing one of the corners. Assume the n! conjecture holds for μ and ν. Then

dim (H_{μ} ∩ H_{ν})^{⊥} ∩ H_{μ} ≤ dim H_{μ} ∩ H_{ν} .  4.30

Moreover, if equality holds here then the n! conjecture holds for λ as well.

This result can be established by following very closely what we did for the partition 1, 2, 2.

Let μ and ν be two partitions of the same number n. Following A. Young it is customary to say that μ is obtained from ν by a “Raising Operator” if μ can be obtained by lifting a number of cells of ν from lower to upper rows.(*) A raising operator will be called minimal if it lifts a single cell up a single row or to the right a single column. It can be shown (see [17]) that the dominance partial order is the transitive closure of the relation μ = Rν with R a minimal raising operator. L. Butler [4] observed that the Macdonald coefficients K̃_{λμ}(q, t) change in a remarkably simple way when μ is obtained from ν by a minimal raising operator. Her observation which was originally made from tables computed by hand by Macdonald [17] (up to n ≤ 6) is now confirmed by more extensive tables obtained by computer (up to n ≤ 8). Butler noticed that for any such pair of partitions μ, ν and any λ there are two polynomials ϕ_{λ} and θ_{λ} with non-negative integer coefficients such that

\begin{matrix}K̃_{λμ}(q, t) = ϕ_{λ} + θ_{λ} \\ K̃_{λν}(q, t) = ϕ_{λ} + θ_{λ} T_{ν}/T_{μ}, \end{matrix}  4.31

where for convenience we have set T_{μ} = q^{n(μ′)}t^{n(μ)}, T_{ν} = q^{n(ν′)}t^{n(ν)}. This has come to be referred to as the Butler conjecture. To explore the full implications of this conjecture let us set for a moment ψ_{λ} = θ_{λ}/T_{μ} and write these two identities in the more symmetric form

\begin{matrix}K̃_{λμ}(q, t) = ϕ_{λ} + ψ_{λ} T_{μ} \\ K̃_{λν}(q, t) = ϕ_{λ} + ψ_{λ} T_{ν}, \end{matrix}

(*) This definition requires that diagrams of partitions be drawn by the English convention Equivalently we may write

ϕ_{λ} = \frac{T_{ν} K̃_{λμ} − T_{μ} K̃_{λν}}{T_{ν} − T_{μ}} ,  ψ_{λ} = \frac{K̃_{λμ} − K̃_{λν}}{T_{μ} − T_{ν}} .  4.32

Now the identity in 4.7 (written for μ and ν) gives

K̃_{λμ}(1/q, 1/t) T_{μ} = K̃_{λ′μ}(q, t) ,  K̃_{λν}(1/q, 1/t) T_{ν} = K̃_{λ′ν}(q, t) .

These relations show that ψ_{λ} is superfluous here. In fact, combinining them with 4.31 we can easily derive that

ψ_{λ}(q, t) = ϕ_{λ′}(1/q, 1/t) .

Setting ϕ(x; q, t) = ∑_{λ} ϕ_{λ}(q, t) S_{λ}(x) the identites in 4.31 become

H̃_{μ}(x; q, t) = ϕ(x; q, t) + flip_{μ} ϕ(x; q, t) , H̃_{ν}(x; q, t) = ϕ(x; q, t) + flip_{ν} ϕ(x; q, t) .  4.33

Now we see that our findings in the case of the pair (1, 1, 2), (2, 2) reveal that the phenomenon observed by Butler may be the result of a beautiful mechanism which we venture to state as follows.

Conjecture 4.1

If μ = Rν with R a minimal raising operator then the identities in 4.33 hold true with ϕ(x; q, t) the bigraded Frobenius characteristic of H_{μ} ∩ H_{ν}

On the validity of our C = H̃ conjecture, this property of the Macdonald polynomials would be an immediate consequence of

Conjecture 4.2 If μ = Rν with R a minimal raising operator then

H_{μ} = H_{μ} ∩ H_{ν} ⊕ flip_{μ} H_{μ} ∩ H_{ν} ,
H_{ν} = H_{μ} ∩ H_{ν} ⊕ flip_{ν} H_{ν} ∩ H_{ν} .

4.34

Remark 4.2 There are even stronger forms of this conjecture. Indeed, we expect it to hold true that

H_{μ} ∩ H_{ν} = flip_{μ} H_{μ} ∩ I_{ν} = flip_{ν} H_{ν} ∩ I_{μ}  4.35

which would imply 4.34. Our evidence suggests that all these equalities hold true whenever μ = Rν with R any raising operator, not necessarily minimal, which lifts a single cell. Remark 4.3

We should note that the equalities

dim (H_{μ} ∩ H_{ν})^{⊥} ∩ H_{μ} = dim H_{μ} ∩ H_{ν} = dim (H_{μ} ∩ H_{ν})^{⊥} ∩ H_{ν} .  4.36

for all pairs μ = Rν with R minimal do imply the n! conjecture. This is because every partition of n can be reached from μ = 1^{n} by a sequence of minimal raisings. At the start 4.36 would give dim H_{1^{n}} ∩ H_{ν} = n!/2 which recursively would yield

dim H_{μ} ∩ H_{ν} = n!/2  4.37

for all pairs μ = Rν with R minimal. For this reason we have come to refer to 4.36 as the n!/2 conjecture.

Remark 4.4

Let ρ be any two-corner partition and μ and ν be obtained by removing one of its corner cells. Given 4.36, the same mechanism we applied to the partition (1, 2, 2) yields the n! conjecture for ρ whenever it is known to be true for both μ and ν. By taking account of the action of S_{n−1} on the resulting basis of H_{ρ} we can obtain a general result relating the Frobenius characteristics of H_{ρ}, H_{μ} and H_{ν} which specializes to 4.29 when ρ = (1, 2, 2).

To state it we need some notation. Let ρ be the partition which has L_{2} rows of length L_{1} and A_{2} rows of length L_{1} + A_{1}. To visualize our identities, we shall act as if ρ is the Ferrers diagram of the partition obtained by setting L_{1} = A_{2} = 4, L_{2} = 3 and A_{1} = 5. Pictorially, this is setting

ρ = .

Now let μ be the partition obtained by removing the upper corner cell from ρ and ν be obtained by removing the lower corner cell. Following the same conventions we adopted in [10] we shall represent a Frobenius characteristic by the Ferrers diagram of the partition that indexes the corresponing module. This given, the Frobenius characteristics of H_{ρ}, H_{μ} and H_{ν} are depicted as follows:

C_{ρ}(x; q, t) =  , C_{μ}(x; q, t) =  , C_{ν}(x; q, t) =  .

Letting ϕ, U and V denote the Frobenius characteristics of H_{μ} ∩ H_{ν}, (H_{μ} ∩ H_{ν})^{⊥} ∩ H_{μ} and (H_{μ} ∩ H_{ν})^{⊥} ∩ H_{ν} respectively and assuming that the identities in 4.34 hold true even in this more general situation, we get

= ϕ + flip_{μ}ϕ ,  = ϕ + flip_{ν}ϕ ,  4.38

from which we derive that

ϕ = \frac{T_{ν}  − T_{μ}}{T_{μ} − T_{ν}}  = \frac{t^{L_{2}}  − q^{A_{1}}}{t^{L_{2}} − q^{A_{1}}}  .  4.39

The mechanism of proof we have illustrated in the case L_{1} = L_{2} = A_{1} = 1 and A_{2} = 2 can be extended to the general case and (upon the validity of 4.38) it yields the identity

∂_{p_{1}}  = B_{ρ}(q, t) ϕ + flip_{ρ} ( B_{ρ}(q, t) ϕ ) .  4.40

This given, we can eliminate ϕ from this relation by means of 4.39 and obtain

∂_{p_{1}}  = B_{ρ} \frac{T_{ν}  − T_{μ}}{T_{μ} − T_{ν}}  + T_{ρ} ↓B_{ρ} \frac{T_{μ}ω ↓  − T_{ν}ω ↓}{T_{ν} − T_{μ}}

Using 4.9 we can collect terms and rewrite this on the form

∂_{p_{1}}  = \frac{P}{t^{L_{2}} − q^{A_{1}}}  + \frac{Q}{t^{L_{2}} − q^{A_{1}}}  4.41

with

P = B_{ρ}t^{L_{2}}− ↓B_{ρ}q^{L_{1}−1}t^{A_{2}+L_{2}−1}  and  Q = −B_{ρ}q^{A_{1}}+ ↓B_{ρ}t^{A_{2}−1}q^{A_{1}+L_{1}−1}

Since with our choice of parameters we can write

B_{ρ}(q, t) = [L_{1}]_{q}[L_{2}]_{t} + [L_{1} + A_{1}]_{q}[A_{2}]_{t}

a simple calculation yields that

P = (t^{L_{2}+A_{2}} − q^{A_{1}}) [L_{1}]_{q}[L_{2}]_{t}  and  Q = (t^{L_{2}} − q^{L_{1}+A_{1}}) [A_{1}]_{q}[A_{2}]_{t}

Now it was shown in [10] (see 4.40 there) that the Macdonald-Stanley Pieri rules yield the identity in 4.41 with precisely the same P and Q when

= H̃_{ρ}(x; q, t) ,  = H̃_{μ}(x; q, t) ,  = H̃_{ν}(x; q, t) .

This not only supports the C = H̃ conjecture and the validity of the identities in 4.34 (even when R is not minimal) but it reveals the combinatorial mechanism which underlies the mysterious intricacy of the coefficients appearing in this particular instance of the Pieri rules.

5. Hooks.

In this section we shall show how the various ingredients we have put together allow us to prove the C = H̃ conjecture for μ a hook. To guide us and to help our understanding of the proof it is best to begin with a close study of the symmetric function H̃_{μ}(x; q, t) and the polynomial F̃_{μ}(q, t) = ∂_{p_{1}}^{n} H̃_{μ}(x; q, t) for μ a hook. To this end it will be convenient to use a notation that is consistent with that introduced in [10] so we may freely refer to [10] for the proof of some of the auxiliary identities. To begin with we need to imbed the collection of hook indexed H̃_{μ} ′s into a larger family which includes H̃ ′s indexed by broken hooks.(*) More precisely we set

\{ \begin{matrix}a) & H̃[l, 0, a] = H̃_{1^{l}}(x; q, t) H̃_{a}(x; q, t) \\ b) & H̃[l, 1, a] = H̃_{1^{a}b+1}(x; q, t) \end{matrix} .  5.1

(*) Diagrams which consist of the disjoint union of a row and a column.

Since [l, 1, a] represents a hook with leg l and arm a we may visualize the symbol [l, 0, a] as the diagram obtained by removing the corner cell from [l, 1, a].

Remark 5.1

We should note at this point that the symmetric function H̃[l, 0, a], for l + a + 1 = n, is the Frobenius characteristic of a module which yields a bigraded version of the regular representation of S_{n−1}. The easiest way to obtain such a module is to induce from S_{[1,...,l]} × S_{[l+1,...,n−1]} to S_{n−1} the tensor product of the S_{[1,...,l]}-Harmonics in x_{1}, …, x_{l} by the S_{[l+1,...,n−1]}- Harmonics in y_{l+1}, …, y_{n−1}. In the present context, this representation arises in the following manner. Set for a moment

Δ_{[l,0,a]}(x; y) = ∏_{1≤i<j≤l}(x_{j} − x_{i}) ∏_{l+1≤i<j≤n−1}(y_{j} − y_{i}) ,  5.2

and let

\begin{matrix}b^{ϵ,η}(x, y) = ∂_{x_{1}}^{ϵ_{1}} ⋯ ∂_{x_{l}}^{ϵ_{l}}∂_{y_{l+1}}^{η_{1}} ⋯ ∂_{y_{n−1}}^{η_{n−1−l}}y_{l+1} ⋯ & y_{n−1}Δ_{[l,0,a]}(x; y) . \\ ( with 0 ≤ ϵ_{i}, η_{i} ≤ i − 1 ) \end{matrix}

Now it is easy to see that the lexicographically first monomial in b^{ϵ,η}(x, y) is

y_{l+1} ⋯ y_{n−1} ∏_{i=1}^{l} x_{i}^{i−1−ϵ_{i}} ∏_{j=1}^{n−1−l} y_{l+j}^{j−1−η_{j}} .

Thus the b^{ϵ,η} ′s form an independent set of polynomials. Now for a given decomposition of the indices {1, 2, …, n − 1} into two disjoint subsets S = {i_{1} < ⋯ < i_{l}}, T = ^{c}S = {j_{1} < ⋯, j_{a}} let b_{S}^{ϵ,η}(x, y) denote the polynomial obtained by substituting in b^{ϵ,η}(x, y) x_{s} by x_{i_{s}} and y_{j} by y_{j_{t}}. It is not difficult to see that not only the b_{S}^{ϵ,η}(x, y), for fixed S, are independent but the whole collection

B_{[l,o,a]} = \{ b_{S}^{ϵ,η}(x, y) \}_{{S=\{1≤i_{1}<···<i_{l}≤n−1\} \atop 0≤ϵ_{i},η_{i}≤i−1}}  5.3

forms a basis for the module obtained by letting S_{n−1} act diagonally on the linear span of the b^{ϵ,η} ′s. Moreover, this action induces a representation with bigraded Frobenius Chacteristic precisely given by the product

q^{a}H̃[l, 0, a] = q^{a}H̃_{1^{l}}(x; q, t) H̃_{a}(x; q, t) .

It will be good to keep this observation in mind as we continue our proof of the C = H̃ conjecture for hooks.

Note next that using the theory of Macdonald polynomial we can derive the following basic identities.

Proposition 5.1

\begin{matrix}a) H̃_{[l,0,a]} = \frac{t^{l} − 1}{t^{l} − q^{a}} H̃_{[l−1,1,a]} + \frac{1 − q^{a}}{t^{l} − q^{a}} H̃_{[l,1,a−1]} , \\ b) ∂_{p_{1}} H̃_{[l,1,a]} = [l]_{t} \frac{t^{l+1} − q^{a}}{t^{l} − q^{a}} H̃_{[l−1,1,a]} + [a]_{q} \frac{t^{l} − q^{a+1}}{t^{l} − q^{a}} H̃_{[l,1,a−1]} . \end{matrix}  5.4

The proof of a) is a straightforward translation in the present notation of one of the Stanley-Macdonald Pieri rules ([20],[17]). As for 5.4 b), we need only observe that differentiation by p_{1} is Cauchy-dual to multiplication by e_{1}. This given, we may obtain it as another consequence of the same Pieri rules. Details can be found in [9].

Simple manipulations yield that 5.4 (a) & (b) are equivalent to the following two different ways of expressing ∂_{p_{1}} H̃_{[l,1,a]}.

Proposition 5.2

\begin{matrix}a) ∂_{p_{1}} H̃_{[l,1,a]} = [l]_{t} H̃_{[l−1,1,a]} + t^{l} H̃_{[l,0,a]} + q[a]_{q} H̃_{[l,1,a−1]} , \\ b) ∂_{p_{1}} H̃_{[l,1,a]} = t[l]_{t} H̃_{[l−1,1,a]} + q^{a} H̃_{[l,0,a]} + [a]_{q} H̃_{[l,1,a−1]} . \end{matrix}  5.5

On the validity of the C = H̃ conjectures the Hilbert series of our module H_{1^{l},a+1} should be given by the expression

F̃_{1^{l},a+1}(q, t) = ∂_{p_{1}}^{n} H̃_{[l,1,a]} ,  5.6

Note that the original theory [17] only defines this to be a rational function of q and t. However, we can easily show here that it is a polynomial. To this end note that F̃_{1^{l},a+1}(q, t) satisfies either of the following two recursions.

Proposition 5.3

\begin{matrix}a) F̃_{[l,1,a]} = [l]_{t} F̃_{[l−1,1,a]} + \binom{l + a}{a}t^{l} [l]_{t}! [a]_{q}! + q[a]_{q} F̃_{[l,1,a−1]} , \\ b) F̃_{[l,1,a]} = t[l]_{t} F̃_{[l−1,1,a]} + \binom{l + a}{a}q^{a} [l]_{t}! [a]_{q}! + [a]_{q} F̃_{[l,1,a−1]} . \end{matrix}  5.7

Proof A simple use of Leibnitz formula gives

∂_{p_{1}}^{n−1} H̃_{[l,0,a]} = ∂_{p_{1}}^{n−1} H̃_{1^{l}}(x; q, t) H̃_{a}(x; q, t) = \binom{l + a}{a} [l]_{t}! [a]_{q}!

Thus 5.7 a) and b) are obtained by applying ∂_{p_{1}}^{n−1} to both sides of 5.5 a) and b).

This yields

Proposition 5.4

F̃_{[l,1,a]} is a polynomial in q, t whose component of highest degree is bihomogeneous of degree \binom{l+1}{2} in t and degree \binom{a+1}{2} in q. Moreover, it is bisymmetric in t and q, that is we have

t^{\binom{l+1}{2}}q^{\binom{a+1}{2}} F̃_{1^{l},a+1}(1/q, 1/t) = F̃_{1^{l},a+1}(q, t)  5.8

Proof

The polynomiality follows from the fact that F̃_{[l,1,a]} may be recursively computed by using either of 5.7 a) or b) with the initial condition F̃_{1^{l},a+1}(q, t)|_{l=a=0} = 1. On the other hand we see from either recursion that each step increases the degree in t by l and the degree in q by a, this establishes the degree assertion. This given, 5.8 follows immediately from the fact that replacing t by 1/t and q by 1/q in 5.7 and multiplying both equations by t^{\binom{l+1}{2}}q^{\binom{a+1}{2}} amounts to switching the right-hand sides of 5.7 a) and b). Thus the left-hand side of 5.8 satisfies the same recursions and initial conditions as the right-hand side and thus they must necessarily be identical.

A corollary of the identity in 5.8 is that the specialization F̃_{1^{l},a+1}(t, t) has symmetric coefficient sequence. Our plan is to derive the n! conjecture for μ = (1^{l}, a + 1) by means of Proposition 3.3. More precisely, we shall construct a basis B_{1^{l},a+1} for the module R_{[ρ]} (with ρ = ρ_{1^{l},a+1}) with the property that

∑_{b∈B_{1^{l},a+1}} t^{degree(b)} = F̃_{1^{l},a+1}(t, t) .  5.9

The symmetry of F̃_{1^{l},a+1}(t, t) then guarantees the validity of condition (a) in Proposition 3.3, it follows that H_{[ρ]} is a cone, necessarily with summit Δ_{μ}.

The recursions in 5.7 have a natural combinatorial interpretation which we shall use as a guide to the construction of our basis B_{1^{l},a+1}. For n = l + a + 1 let us say that π is a labelled hook if π is a filling of the shape [l, 1, a] by the entries σ_{1}, σ_{2}, …, σ_{n} of a permutation σ ∈ S_{n} by placing σ_{1}, σ_{2}, …, σ_{a} from right to left in its arm and σ_{a+1}, σ_{a+2}, …, σ_{n} from bottom to top in its leg. For instance the figure below gives the labelled hook corresponding to the filling of [3, 1, 5] by 143679258.

π = 8 5 2 9 7 6 3 4 1

Setting A(π) = σ_{1}, σ_{2}, …, σ_{a} and L(π) = σ_{a}, σ_{a+1}, …, σ_{n} we can easily show that we have

Proposition 5.5

F̃_{1^{l},a+1}(q, t) = ∑_{π} t^{inv(L(π))}q^{inv(A(π))},  5.10

where π runs over all labelled hooks of shape [l, 1, a] and “inv” denotes the number of inversions in a word.

Proof

Let l + a + 1 = n and let, for a moment, Σ_{1^{l},a+1}(q, t) denote the right hand side of 5.10. Since the initial condition Σ_{[l−1,1,a]}|_{a=l=0} = 1 may be taken as the definition of our sum for these parameter values, it is sufficient to show that Σ_{1^{l},a+1} satisfies one of the recursions in 5.7. We can derive the first recursion by grouping summands according to the position of n. To this end note that if n is in the leg of π it will produce an inversion in the word L(π) = σ_{a} ⋯ σ_{n} for each label that lies above it in π. A moment’s reflection should reveal that the terms where there are 0 ≤ i ≤ l − 1 labels above n must add up to t^{i} Σ_{1^{l−1},a+1}(q, t). So the contribution of all the terms where n is in the leg is [l]_{t}Σ_{1^{l−1},a+1}(q, t). On the other hand, if n is in the arm of π it will produce a number 1 ≤ i ≤ a inversions in the word A(π) according to the number of labels that are to the left of it in π. Thus all of these terms will add up to q[a]_{q}Σ_{1^{l},a}(q, t). Note next that we can construct each labelled hook with n in the corner cell by first choosing a labels out of 1, …, n − 1 and place them in the arm in one of the a! possible ways. Then place n in the corner cell and finally place the remaining labels in the leg in one of the l! possible ways. Let π be the resulting labelled hook. By our construction n will be the last entry in A(π) and the first entry in L(π) thus n will contribute no inversions to A(π) and l inversions to L(π). This given we see that the sum of t^{inv(L(π))}q^{inv(A(π))} over all these terms must be none other than

\binom{l + a}{a}t^{l} [l]_{t}![a]_{q}! .

Putting all this together we see that we must have

Σ_{[l,1,a]} = [l]_{t} Σ_{[l−1,1,a]} + \binom{l + a}{a}t^{l} [l]_{t}! [a]_{q}! + q[a]_{q} Σ_{[l,1,a−1]}

as desired.

We should note that a similar reasoning based on the position of 1 yields that Σ_{[l,1,a]} satisfies the second recursion as well.

We shall next carry out the construction of our module R_{[ρ_{μ}]} by the process outlined in Section 1, specializing the “injective tableaux of shape μ” we used there to the labelled hooks of shape [l, 1, a]. Having chosen the parameters α_{1}, α_{2}, …, α_{l+1}; β_{1}, β_{2}, …, β_{a+1} to label the rows and columns of [l, 1, a], we shall total order the elements of the corresponding orbit according to the lexicographic order of the labeling permutation σ = σ_{1}, σ_{2}, …, σ_{n}. For instance, for a = l = 1 the labeled hooks in this order are

3 2 1 ,  2 3 1 ,  3 1 2 ,  1 3 2 ,  2 1 3 ,  1 2 3 ,

and the corresponding orbit points in this total order are

\begin{matrix}(α_{1}, α_{1}, α_{2}; β_{2}, β_{1}, & β_{1}) , (α_{1}, α_{2}, α_{1}; β_{2}, β_{1}, β_{1}) , (α_{1}, α_{1}, α_{2}; β_{1}, β_{2}, β_{1}) , \\ (α_{2}, α_{1}, α_{1}; β_{1}, β_{2}, β_{1}) , (α_{1}, α_{2}, α_{1}; β_{1}, β_{1}, β_{2}) , (α_{2}, α_{1}, α_{1}; β_{1}, β_{1}, β_{2}) . \end{matrix}

The idea in our construction of the basis B_{1^{l},a+1} is to produce, for each labelled hook π a polynomial ϕ_{π}(x; y) such that

ϕ_{π}(a_{π′}; b_{π′}) = \{ \begin{matrix}1 & if π′ = π \\ 0 & if π′ <_{T} π \end{matrix} ,  5.11

where (a_{π′}; b_{π′}) is the orbit point corresponding to the labeled hook π′ and π′ <_{T} π means that π′ precedes π in the given total order. This given, we let

B_{1^{l},a+1} = { ϕ_{π}(x; y) }_{π}  5.12

Clearly, the equalities in 5.11 assure that B_{1^{l},a+1} is a basis. To satisfy the additional condition 5.9 we shall construct ϕ_{π}(x; y) so that its component of highest degree is bihomogeneous of degree inv(L(π)) in t and of degree inv(A(π)) in q. To better get across how we do this we shall use a jargon that helps visualize the successive steps of the construction. Note that the linear factor x_{i} − α_{j} is different from zero at (a_{π}, b_{π}) if and only if i is not in the j^{th} row of π. We shall express this property by saying that x_{i} − α_{j} kicks the label “i” out of the j^{th} row of [l, 1, a]. We shall put together our polynomial ϕ_{π}(x; y) as the product of linear factors which together kick all the labels into the exact position they occupy in π. By this we mean that (a_{π}, b_{π}) will be the lexicographically first orbit point at which none of the factors vanish. We describe this construction in general terms first then make it precise by carrying it out in a special case.

Let i_{o} be the label that occupies the corner cell of π and let L_{<i_{o}} and A_{<i_{o}} respectively denote the sets of labels less than i_{o} that fall in the leg and the arm of π. Similarly let L_{>i_{o}} and A_{>i_{o}} denote the sets of labels greater than i_{o} that fall in the leg and the arm of π. Our first step is to kick every label in L_{<i_{o}} out of the first row and every label in A_{>i_{o}} out of the first column. This can be achieved by means of the polynomial

ψ(x; y) = ∏_{i∈L_{<io}} (x_{i} − α_{1}) ∏_{i∈A_{>io}} (y_{i} − β_{1}) .  5.13

It is easy to see that if (a_{π′}, b_{π′}) is the lexicographically first orbit element at which ψ doesn’t vanish then all the labels in L_{<i_{o}} ∪ L_{>i_{o}} fall in increasing order up the leg of π′ and all the labels in A_{<i_{o}} ∪ A_{>i_{o}} fall in the arm of π′ in increasing order from right to left. To force the labels in L_{<i_{o}} ∪ L_{>i_{o}} precisely into the cells they occupy in π, we need to throw in a few additional kicks. We shall illustrate how this is done in a special case The reader should have no difficulty seing that how this last step is to be carried out in the general case. Note that if

π = 2 5 6 4 1 3 7

Then ψ = (x_{2} −α_{1})(x_{5} −α_{1})(y_{7} −β_{1}) and the lexicographically first labelled hook at which ψ doesn’t vanish is

π′ = 5 2 6 7 4 3 1

Now we can kick 5 out of the third row by means of x_{5} − α_{3} and kick 7 into the 5^{th} column by means of (y_{7} − β_{2})(y_{7} − β_{3})(y_{7} − β_{4}). We see that the lexicographically first labelled hook at which the polynomial

ψ′(x; y) = (x_{5} − α_{3})(y_{7} − β_{2})(y_{7} − β_{3})(y_{7} − β_{4})ψ

doesn’t vanish is

π′′ = 2 5 6 4 3 1 7

Now to force 3 into its desired position we need only kick it out of the 3^{rd} column. This makes π the lexicographically first labelled hook at which the polynomial

ψ′′(x; y) = (y_{3} − β_{3})ψ′′

doesn’t vanish. So we may set

ϕ_{π}(x; y) = \frac{ψ′′(x; y)}{ψ′′(a_{π}; b_{π})} .

A moment’s reflection should reveal that, in the general case, a label i, in the arm of π that lies in the j^{th} column can be forced there by kicking out of the j^{th} column all the labels that are greater than i and are to the right of i in π. But that means that the contribution of the label i to the y-degree of ϕ_{π}(x; y) is equal to the number of inversions that i causes in the word A(π). A similar reasoning applies to the labels in the leg of π. Since the x and y degrees of the polynomial ψ in 5.13 are respectively equal to the number of inversions that i_{o} makes in L(π) and A(π) we see that the homogeneous component of highest degree in ϕ_{π}(x; y) will be bihomogeneous of of bidegree (inv(L(π)), inv(A(π))). This assures that our basis B_{1^{l},a+1} satisfies 5.9 and via Proposition 1.4 proves the n! conjecture for all hook shapes.

Our next and final task is to verify that the Frobenius characteristic of the module H_{1^{l},a+1} is indeed given by the polynomial H̃_{1^{l},a+1}(x; q, t). There are several ways to do this, we shall follow a path that appears most amenable to extensions to more general shapes. Our plan is to show that the Frobenius characterstic C_{1^{l},a+1}(x; q, t) satisfies the recursions in 5.5.To be precise and to conform with the notation in 5.5 we shall set C_{[l,1,a]} = C_{1^{l},a+1}(x; q, t) (for all choices of l, a ≥ 0) and show that we must have

\begin{matrix}a) ∂_{p_{1}} C_{[l,1,a]} = [l]_{t} C_{[l−1,1,a]} + t^{l} H̃_{[l,0,a]} + q[a]_{q} C_{[l,1,a−1]} , \\ b) ∂_{p_{1}} C_{[l,1,a]} = t[l]_{t} C_{[l−1,1,a]} + q^{a} H̃_{[l,0,a]} + [a]_{q} C_{[l,1,a−1]} . \end{matrix}  5.14

This given, subtracting 5.14 b) from 5.14 a) and 5.5 b) from 5.5 a) we can write the difference of the resulting equations in the form

(1 − t^{l})( C_{[l−1,1,a]} − H̃_{[l−1,1,a]}) = (q^{a} − 1)( C_{[l,1,a−1]} − H̃_{[l,1,a−1]}) .  5.15

Since we trivially have the equality H̃_{[l−1,1,a]} = C_{[l−1,1,a]} for a = 0 (*) the desired equality H̃_{[l−1,1,a]} = C_{[l−1,1,a]} immediately follows for each n = l + a + 1 by induction on a.

For n = l + 1 + a, the equalities in 5.14 may be obtained by a suitable decomposition of the action of S_{n−1} on H_{1^{l},a+1}. Having proved the n! conjecture in this case, we know that H_{1^{l},a+1} affords the left regular representation of S_{n}. Thus, its restriction to S_{n−1} must be decomposable into a direct sum of n copies of the left regular representation of S_{n−1}. The equations in 5.14 result from two different ways of carrying out such a decomposition. We shall prove them by studying the action of S_{n−1} on some special bases of H_{1^{l},a+1}. To this end we shall pick once and for all two bases B_{1^{l−1},a+1} and B_{1^{l},a} for H_{1^{l−1},a+1} and H_{1^{l},a} respectively, then let

\begin{aligned}B_{1^{l},a+1} = ∑_{i=1}^{l} \{f_{i}(∂)∂_{x_{n}}^{i−1} & Δ_{1^{l},a+1} \}_{f_{i}∈B_{1^{l−1},a+1}} + \\ & \{ m_{S}^{ϵ,η}(∂)∂_{x_{i_{1}}} ⋯ ∂_{x_{i_{l}}} Δ_{1^{l},a+1} \}_{{S=\{1≤i_{1}<···<i_{l}≤n−1\} \atop 0≤ϵ_{i},η_{i}≤i−1}} + \\ & + ∑_{j=1}^{a} \{g_{j}(∂)∂_{y_{n}}^{j}Δ_{1^{l},a+1} \}_{g_{j}∈B_{1^{l−1},a+1}} , \end{aligned} 5.16

(*) in this case Δ_{μ} reduces to the Vandermonde determinant in the x_{i} ′s where, for a given decomposition of the indices {1, …, n − 1} into two disjoint subsets,

S = {i_{1} < ⋯ < i_{l}} and T = ^{c}S = {j_{1} < ⋯ < j_{a}}

we let

m_{S}^{ϵ,η}(x, y) = x_{i_{1}}^{ϵ_{1}} ⋯ x_{i_{l}}^{ϵ_{l}} y_{j_{1}}^{η_{1}} ⋯ y_{j_{a}}^{η_{a}} .  5.17

We claim that B_{1^{l},a+1} is a basis for H_{1^{l},a+1} and the action of S_{n−1} on this basis induces precisely the decomposition in 5.14 a). The proof of this is best understood by carrying it out on a special case. We shall work with the case l = 3, a = 4. We start by showing that B_{1^{3}5} is a basis. So let f_{1}, f_{2}, f_{3} be elements in the linear span of B_{1^{2}5} and let g_{1}, g_{2}, g_{3}, g_{4} be in the linear span of B_{1^{3}4}. Note that our construction requires that these are polynomials in x_{1}, …, x_{7}; y_{1}, …, y_{7}. Furthermore, let h_{i_{1},i_{2},i_{3}}(x, y) be an element in the linear span of the monomials in 5.17 for fixed S = {1 ≤ i_{1} < i_{2} < i_{3} ≤ 7}. We claim that the equality

f_{1}(∂)Δ_{1^{3}5} + f_{2}(∂)∂_{x_{8}}Δ_{1^{3}5} + f_{3}(∂)∂_{x_{8}}^{2}Δ_{1^{3}5} +

+ ∑_{S={1≤i_{1}<i_{2}<i_{3}≤7}} h_{i_{1},i_{2},i_{3}}(∂)∂_{x_{i_{1}}} ∂_{x_{i_{2}}} ∂_{x_{i_{3}}} Δ_{1^{3}5} +

+ g_{1}(∂)∂_{y_{8}}Δ_{1^{3}5} + g_{2}(∂)∂_{y_{8}}^{2}Δ_{1^{3}5} + g_{3}(∂)∂_{y_{8}}^{3}Δ_{1^{3}5} + g_{4}(∂)∂_{y_{8}}^{4}Δ_{1^{3}5} = 0

5.18

implies that f_{1}, f_{2}, f_{3}, g_{1}, g_{2}, g_{3}, g_{4} and all h_{i_{1},i_{2},i_{3}} are equal to zero. To start with note that from the definition I.3 we have that

Δ_{1^{3}5}(x; y) = det ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} \\ x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & x_{4}^{2} & x_{5}^{2} & x_{6}^{2} & x_{7}^{2} & x_{8}^{2} \\ x_{1}^{3} & x_{2}^{3} & x_{3}^{3} & x_{4}^{3} & x_{5}^{3} & x_{6}^{3} & x_{7}^{3} & x_{8}^{3} \\ y_{1} & y_{2} & y_{3} & y_{4} & y_{5} & y_{6} & y_{7} & y_{8} \\ y_{1}^{2} & y_{2}^{2} & y_{3}^{2} & y_{4}^{2} & y_{5}^{2} & y_{6}^{2} & y_{7}^{2} & y_{8}^{2} \\ y_{1}^{3} & y_{2}^{3} & y_{3}^{3} & y_{4}^{3} & y_{5}^{3} & y_{6}^{3} & y_{7}^{3} & y_{8}^{3} \\ y_{1}^{4} & y_{2}^{4} & y_{3}^{4} & y_{4}^{4} & y_{5}^{4} & y_{6}^{4} & y_{7}^{4} & y_{8}^{4} \end{matrix} )  5.19

and we see that are no terms in x_{8}^{4}, x_{8}y_{8} or y_{8}^{5} in the expansion of Δ_{1^{3}5}. This implies that we have the equalities

∂_{x_{8}}^{4}Δ_{1^{3}5} = 0  ,  ∂_{x_{8}}∂_{y_{8}}Δ_{1^{3}5} = 0  and  ∂_{y_{8}}^{5}Δ_{1^{3}5} = 0  5.20

Moreover, since for i_{1} < i_{2} < i_{3} < 8, the polynomial

(x_{i_{1}} − α_{1})(x_{i_{2}} − α_{1})(x_{i_{3}} − α_{1})(x_{8} − α_{1})  5.21

vanishes identically on the orbit of ρ_{1^{3}5} we deduce that the monomial x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{8} lies in the ideal gr J_{[ρ_{135}]} and therefore we must also have

∂_{x_{i_{1}}} ∂_{x_{i_{2}}} ∂_{x_{i_{3}}} ∂_{x_{8}}Δ_{1^{3}5} = 0 .  5.22

This given, we see that differentiating 5.18 by ∂_{x_{8}}^{3} and using 5.20 and 5.22 we get that

0 = f_{1}(∂)∂_{x_{8}}^{3}Δ_{1^{3}5} = 6f_{1}(∂)Δ_{1^{2}5} .

However, this cannot happen for a harmonic f_{1} in the linear span of B_{1^{2}5} without it being identically zero. Replacing f_{1} by zero in 5.18 and differentiating by ∂_{x_{8}}^{2} now gives

0 = f_{2}(∂)∂_{x_{8}}^{3}Δ_{1^{3}5} = 6f_{2}(∂)Δ_{1^{2}5} ,

which for the same reason yields that f_{2} must be a zero linear combination of the elements of B_{1^{2}5}. Finally, replacing f_{1}&f_{2} by zero and differentiating by ∂_{x_{8}}^{2} 5.18 gives that f_{3} = 0.

Next we must see what becomes of 5.18 if we replace f_{1}, f_{2}, f_{3} by zero and set y_{8} = 0. To this end observe that

∂_{x_{i_{1}}} ∂_{x_{i_{2}}} ∂_{x_{i_{3}}} Δ_{1^{3}5} |_{y_{8}=0} = y_{j_{1}}y_{j_{2}}y_{j_{3}}y_{j_{4}}det ( \begin{matrix}1 & 1 & 1 \\ x_{i_{1}} & x_{i_{2}} & x_{i_{3}} \\ x_{i_{1}}^{2} & x_{i_{2}}^{2} & x_{i_{3}}^{2} \end{matrix} ) det ( \begin{matrix}1 & 1 & 1 & 1 \\ y_{j_{1}} & y_{j_{2}} & y_{j_{3}} & y_{j_{4}} \\ y_{j_{1}}^{2} & y_{j_{2}}^{2} & y_{j_{3}}^{2} & y_{j_{4}}^{2} \\ y_{j_{1}}^{3} & y_{j_{2}}^{3} & y_{j_{3}}^{3} & y_{j_{4}}^{3} \end{matrix} ) ,

where {j_{1}, j_{2}, j_{3}, j_{4}} is the complement of {i_{1}, i_{2}, , i_{3}} in {1, 2, 3, …, 7}. Furthermore it is easy to check that by contrast none of the determinants

g_{1}(∂)∂_{y_{8}}Δ_{1^{3}5} |_{y_{8}=0} , g_{2}(∂)∂_{y_{8}}^{2}Δ_{1^{3}5} |_{y_{8}=0} , g_{3}(∂)∂_{y_{8}}^{3}Δ_{1^{3}5} |_{y_{8}=0} , g_{4}(∂)∂_{y_{8}}^{4}Δ_{1^{3}5} |_{y_{8}=0}

have in their expansions monomials with any of the products y_{j_{1}}y_{j_{2}}y_{j_{3}}y_{j_{4}} as a factor. These observations allow us to conclude that the relation

∑_{S={1≤i_{1}<i_{2}<i_{3}≤7}} h_{i_{1},i_{2},i_{3}}(∂)∂_{x_{i_{1}}} ∂_{x_{i_{2}}} ∂_{x_{i_{3}}} Δ_{1^{3}5}|_{y_{8}=0} +

+ g_{1}(∂)∂_{y_{8}}Δ_{1^{3}5}|_{y_{8}=0} + g_{2}(∂)∂_{y_{8}}^{2}Δ_{1^{3}5|_{y_{8}=0}} + g_{3}(∂)∂_{y_{8}}^{3}Δ_{1^{3}5}|_{y_{8}=0} + g_{4}(∂)∂_{y_{8}}^{4}Δ_{1^{3}5}|_{y_{8}=0} = 0

can hold true only if each of the individual summands in the first sum vanishes separately. Note that the term

h_{i_{1},i_{2},i_{3}}(∂)∂_{x_{i_{1}}} ∂_{x_{i_{2}}} ∂_{x_{i_{3}}} Δ_{1^{3}5}|_{y_{8}=0}

is, by construction, a linear combination of the polynomials

b_{{i_{1},i_{2},i_{3}}}^{ϵ,η}(x_{i_{1}}, x_{i_{2}}, x_{i_{3}}; y_{j_{1}}, y_{j_{2}}, y_{j_{3}}, y_{j_{4}}) (with 0 ≤ ϵ_{i}, η_{i} ≤ i − 1 )

which by our Remark 5.1 form an independent set. This gives that each of the h_{i_{1},i_{2},i_{3}} must be the zero linear combination of the monomials in 5.17. We can now set f_{1} = f_{2} = f_{3} = 0 and h_{i_{1},i_{2},i_{3}} = 0 for all subsets {i_{1}, i_{2}, i_{3}} of {1, 2, …, 7} in 5.18 and reduce it to

g_{1}(∂)∂_{y_{8}}Δ_{1^{3}5} + g_{2}(∂)∂_{y_{8}}^{2}Δ_{1^{3}5} + g_{3}(∂)∂_{y_{8}}^{3}Δ_{1^{3}5} + g_{4}(∂)∂_{y_{8}}^{4}Δ_{1^{3}5} = 0

Differentiating by ∂_{y_{8}}^{3} and using the last of 5.20 we derive that

0 = g_{1}(∂)∂_{y_{8}}^{4}Δ_{1^{3}5} = 24g_{1}(∂)Δ_{1^{3}4}

which forces g_{1} = 0 since a non trivial linear combination harmonics in B_{1^{3}4} cannot kill Δ_{1^{3}4}. We should easily see that the process which yielded the vanishing of f_{2} and f_{3} can be repeated here to deliver in the same manner the vanishing of g_{2}, g_{3} and g_{4}. This shows that the collection B_{1^{3}5} defined in 5.16 is an independent set of harmonics in the linear span of derivatives of Δ_{1^{3}5} and since this collection has exactly 8! elements it must be a basis of our module H_{1^{3}5}. The reader should have no difficulty seeing that the argument we carried out for l = 3 and a = 4 can be extended to the general case.

We observe next that each of the summands in the definition 5.16 of B_{1^{l},a+1} is a basis of a subspace of H_{1^{l},a+1} that is invariant under the action of S_{n−1}. However it is not difficult to derive that the bigraded Frobenius characteristic of the action of S_{n−1} on the linear span of the basis {f_{i}(∂)∂_{x_{n}}^{i−1}Δ_{1^{l},a+1} }_{f_{i}∈B_{1^{l−1},a+1}} is given by the symmetric polynonial

t^{\binom{l}{2}−i+1}q^{\binom{a}{2}}C_{[l−1,1,a]}(x; 1/q, 1/t) .

Similarly we derive that the bigraded Frobenius characteristic of the action of S_{n−1} on the linear span of the basis {g_{j}(∂)∂_{y_{n}}^{j}Δ_{1^{l},a+1} }_{g_{j}∈B_{1^{l},a}} is

t^{\binom{l}{2}}q^{\binom{a}{2}−j}C_{[l,1,a−1]}(x; 1/q, 1/t) .

Finally, using Remark 5.1, we obtain that the bigraded Frobenius characteristic of the action of S_{n−1} on the linear span of \{ m_{S}^{ϵ,η}(∂)∂_{x_{i_{1}}} ⋯ ∂_{x_{i_{l}}} Δ_{1^{l},a+1} \}_{{S=\{1≤i_{1}<···<i_{l}≤n−1\} \atop 0≤ϵ_{i},η_{i}≤i−1}} is

q^{a} H̃_{1^{l}}(x; q, t)H̃_{a}(x; q, t) .

Putting all this together, gives the recursion

\begin{aligned}∂_{p_{1}}C_{[l,1,a]}(x; q, t) = ∑_{i=1}^{l} & t^{\binom{l}{2}−i+1}q^{\binom{a}{2}}C_{[l−1,1,a]}(x; 1/q, 1/t) + \\ & + q^{a} H̃_{1^{l}}(x; q, t)H̃_{a}(x; q, t) + \\ & + ∑_{j=1}^{a} t^{\binom{l}{2}}q^{\binom{a}{2}−j}C_{[l,1,a−1]}(x; 1/q, 1/t) \end{aligned} .

5.23

Since we do have the identities

\begin{aligned}∂_{p_{1}}C_{[l,1,a]}(x; q, t) & = t^{\binom{l}{2}}q^{\binom{a}{2}} ∂_{p_{1}}C_{[l,1,a]}(x; 1/q, 1/t) \\ C_{[l−1,1,a]}(x; q, t) & = t^{\binom{l−1}{2}}q^{\binom{a}{2}} ∂_{p_{1}}C_{[l−1,1,a]}(x; 1/q, 1/t) \\ C_{[l,1,a−1]}(x; q, t) & = t^{\binom{l}{2}}q^{\binom{a−1}{2}} ∂_{p_{1}}C_{[l,1,a−1]}(x; 1/q, 1/t) \\ H̃_{1^{l}}(x; q, t)H̃_{a}(x; q, t) & = t^{\binom{l−1}{2}}q^{\binom{a−1}{2}}H̃_{1^{l}}(x; 1/q, 1/t)H̃_{a}(x; 1/q, 1/t) \end{aligned} ,

it is easy to see that 5.23 is just another way of writing the identity in 5.14 a).

To establish 5.14 b) we can proceed in two ways. We can follow the argument which gave 5.14 a) reversing the roles of x and y. Or, (which is basically equivalent) note that it follows from the fact that C_{[l,1,a]}(x; t, q) = C_{[a,1,l]}(x; q, t). In other words 5.14 b) follows from 5.14 a) by interchanging arm with leg and q with t. This completes our proof of the C = H̃ conjecture for hooks.

Remark 5.2

The above proof of C = H̃ for hooks also establishes the n! conjecture for hooks, theoretically making the ‘kicking’ proof superfluous. It can be shown, however, that on the validity of the n! conjecture, the kicking method which we followed in the hook case is bound to work in general. Nevertheless, the complexity of the polynomials that accomplish the desired kicking increases considerably as we pass to more general shapes. For instance, as we pass from hooks to extended hooks (three-corner shapes obtained by adding a corner square to a hook) we can no longer achieve the needed kicking by a product of linear factors. E. Reiner [18] discovered that in this case some of the kicking has to be achieved by means of polynomials which can be recognized as skew flag Schur functions. This additional ingredient permitted Reiner to establish the n! conjecture for all extended hooks. In view of the formidable complexity of Reiner’s proof it is apparent that some new ingredients will have to be added in order to extend this method of proof to general shapes. We should also mention that completely different approaches to the n! conjecture have been developed by E. Allen [1] and Dunkl & Hanlon [5]. Allen’s aproach, which uses Rota’s biletter algebras yielded a combinatorial interpretation of the K̃_{λμ}(q, t) for μ a Hook. An equivalent interpretation had been also been previously obtained by Stembridge [22]. The work of Dunkl & Hanlon uses Dunkl operators and although falls short of proving the n! conjecture it does reduce it to remarkable new conjectures.

Remark 5.3

We should mention that the spaces H_{μ} are all contained in the space DH_{n}[X; Y] of harmonics corresponding to the diagonal action of S_{n}. It can be shown that DH_{n}[X; Y] consists of all solutions P (x; y) of the differential equations

∑_{i=1}^{n} ∂_{x_{i}}^{h}∂_{y_{i}}^{k} P (x; y) = 0 . ( h + k ≥ 1 )

Our discovery of the relation between the Frobenius characteristics of the H_{μ} ′s and the Macdonald polynomials naturally brought particular attention to the space DH_{n}[X; Y] itself. Computer data revealed remarkable properties of DH_{n}[X; Y] which have been translated by several observers (see [15]) into a collection of outstanding conjectures. An algebraic geometrical study of this space led M. Haiman to conjecture an explicit expansion of the bigraded Frobenius characteristic of DH_{n}[X; Y] in terms of our polynomials H̃_{μ}[x; q, t]. The consequences of this expansion in the theory of symmetric functions and its combinatorial implications are given in [11].

REFERENCES

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[3] D. Bernstein and A. Iarrobino, A Non-unimodal Graded Gorenstein Artin Algebra in Codimension Five, Communications in Algebra, 20, No. 8 (1992) 2323-2336.

[4] L. Butler, Personal communication.

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[6] Y. M. Chen, A. M. Garsia and J. Remmel, Algorithms for Plethysm, Contemporary Math. #34. Combinatorics and Algebra, Curtis Greene ed. (1984) pp. 109-153

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[10] A. M. Garsia and M. Haiman, Factorizations of Pieri rules for Macdonald polynomials, To appear in the Proceedings of the IV^{th} Colloquium on Algebraic Combinatorics

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[13] A. Garsia and M. Wachs, Combinatorial Aspects of Skew Representations of the Symmetric Group, J. Comb. Theory, A V. 50 (1989) 47-81.

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[17] I. G. Macdonald, A new class of symmetric functions, Actes du 20^{e} Séminaire Lotharingien, Publ. I.R.M.A. Strasbourg, (1988) 131-171. (To appear also in the the new edition of [17] above)

[18] E. Reiner, A proof of the n! conjecture for extended hooks, to appear in the Journal of Comb. Theory Series A. See also Some applications of the theory of orbit harmonics, Doctoral dissertation UCSD (1993)

[19] W. Specht, Die irreduciblen Darstellungen der Symmetrischen Gruppen, Math. Zeit. 39 (1935) 696-711.

[20] R.Stanley, Some combinatorial properties of Jack Symmetric functions, Adv. Math. 77 (1989) 76-117.

[21] R. Steinberg, Differential equations invariant under finite reflection groups, A. M. S. Transactions 112 (1964) 392-400

[22] J. Stembridge, Some particular entries of the two-parameter Kostka matrix, Proc. Amer. Math. Soc. 121 (1994), 367–373.

[23] A. Young, On quantitative substitutional analysis (sixth paper), The collected papers of A. Young, University of Toronto Press (1977) pp. 434-435.