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\mbox{}

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\begin{center}
{\bf A GRADED REPRESENTATION MODEL FOR MACDONALD'S POLYNOMIALS.}
\end{center}

\vfill

\begin{center}
{\it 
Adriano M. Garsia and Mark Haiman\\[1ex]
University of California, San Diego\\
Department of Mathematics, 0112\\
9500 Gilman Drive\\
La Jolla, CA 92093-0112\\[1ex]
Phone (Haiman): (619) 534-2704\\
E-mail: {\tt mhaiman@macaulay.ucsd.edu}\\[1ex]
Sept.\ 21, 1992}
\end{center}

\vfill

\begin{center}
{\sc Abstract}
\end{center}
\begin{quotation}
We define doubly graded $S_{n}$-modules $R_{\mu }$ for which we
conjecture that the multiplicities of irreducible representations in
various bi-degrees are given by the Macdonald coefficients $K_{\lambda
\mu}$.  Assuming one fundamental conjecture, the modules $R_{\mu }$
can be given several equivalent definitions, which we discuss.  We
prove the conjectures in various special cases.
\end{quotation}

\vfill

\newpage

\sect{1.  Introduction}

  Macdonald in (1, 2) introduced a remarkable new basis for the
ring of symmetric polynomials.  Macdonald's polynomials
$$
P_{\mu }(X;t,q)\eqno (1.1)
$$
are symmetric polynomials in the variables $X=\{x_{1},x_{2},\ldots
\}$, homogeneous of degree equal to the size of the indexing partition
$\mu $, with coefficients in the ring $\Bbb Q(t,q)$ of rational
functions of the two parameters $t$ and $q$.  Under suitable
specializations of these parameters, Macdonald's polynomials reduce to
various important bases of the symmetric function ring, including the
monomial and elementary symmetric functions, Schur functions,
Hall-Littlewood functions, and Jack functions.  Analogs of Macdonald's
polynomials for arbitrary root systems arise in connection with
$q$-analogs of the famous constant term identity conjectures of
Macdonald (3).

  The major outstanding problem on Macdonald's polynomials concerns
coefficients $\tilde{K}_{\lambda \mu }(t,q)$ expanding a certain
transformed variant of $P_{\mu }$ in terms of the Schur functions
$s_{\lambda }$.  As defined, $\tilde{K}_{\lambda \mu }(t,q)$ is a
rational function of $t$ and $q$; the problem is to show that it is in
fact a polynomial in $t$ and $q$, with non-negative integer
coefficients.  We refer to this assertion as the `Macdonald positive
$K$' conjecture, or {\it MPK}.

  In this note we describe certain doubly graded $S_{n}$ modules
$R_{\mu }$ for which we expect that $\tilde{K}_{\lambda \mu }(t,q)$ is
the generating function giving the multiplicities of the irreducible
representation $V_{\lambda }$ in the various bihomogeneous components,
which would prove {\it MPK}.  As we shall explain, the needed
properties and even aspects of the construction of $R_{\mu }$ remain
conjectural at present, but we have proven them (and with them {\it
MPK}) in the following instances:

  (1) For all $\mu $ a hook and all $\lambda $ (Stembridge (4) also
has a proof of {\it MPK} in this case).

  (2) For all $\mu $ a partition with two rows or two columns and all
$\lambda $.

  (3) For all $\lambda $ a hook and all $\mu $ (Macdonald (1) had already
proven {\it MPK} in this case).

  (4) By computer for all $\mu $ and $\lambda $ of size at most $7$
and for occasional cases of size $8$.

  Brief indications of the proof techniques for some of these cases
will be given in the last section of this note.

  We have also proven that the characters of our modules agree with
the specializations $\tilde{K}_{\lambda \mu }(t,0)$ and
$\tilde{K}_{\lambda \mu }(t,1)$ (whose values are known from (2)) and
with the symmetries $\tilde{K}_{\lambda \mu' }(t,q) =
\tilde{K}_{\lambda \mu }(q,t)$ and $\tilde{K}_{\lambda' \mu }(t,q) =
t^{n(\mu )}q^{n(\mu ')}\tilde{K}_{\lambda \mu }(t^{-1},q^{-1})$.  The
most interesting of these is the specialization
$$ 
\tilde{K}_{\lambda \mu }(t,0) = t^{n(\mu )}K_{\lambda \mu }(t^{-1}),
\eqno (1.1)
$$
where $K_{\lambda \mu }(t)$ denotes the $t$-Kostka coefficient.  Our
modules agree with (1.1) because their homogeneous components of
degree $(\cdot ,0)$ reduce to certain singly graded $S_{n}$ modules
known to model the coefficients $K_{\lambda \mu }(t)$.  These modules,
whose origins lie in the geometry of the flag variety, were first
given an elementary treatment in the work of De Concini and Procesi
(5), Kraft (6), and finally Garsia and Procesi (7).  The present
work was inspired primarily by the search for a `$t,q$-analog' of
constructions introduced by these authors.

\sect{2.  Definition and interpretation of $\tilde{K}_{\lambda \mu
}(t,q)$}

  We shall work from the start with transformed versions of
Macdonald's polynomials, as they are more convenient for us than the
original $P_{\mu }$.

\announce Theorem 1\states There exists a unique basis $H_{\mu
}(X;t,q)$ of the ring $\Lambda _{t,q}(X)$ of symmetric functions with
coefficients in $\Bbb Q(t,q)$ having the following properties:
\begin{itemize}
\item [(1)] $H_{\mu }((1-q)X; t,q) = \sum _{\lambda \geq \mu }
c_{\lambda \mu }(t,q)s_{\lambda }(X)$.
\item [(2)] $H_{\mu '}(X; t,q) = H_{\mu }(X; q,t)$.
\item [(3)] $\langle H_{\mu }(X;t,q),s_{(n)}(X) \rangle = 1$.
\end{itemize}

  In {\it (1)}, the notation $H_{\mu }((1-q)X; t,q)$ indicates a
`plethystic substitution,' defined by requiring the transformation
$f(X)\mapsto f((1-q)X)$ to be a ring homomorphism and putting
$p_{k}((1-q)X) = (1-q^{k})p_{k}(X)$ for the power sum $p_{k}(X) =
\sum_{i} x_{i}^{k}$.  The triangularity indicated by the sum is with
respect to the dominance order on partitions.  In {\it (2)}, $\mu '$
is the conjugate of the partition $\mu $; in {\it (3)}, $n$ is the
size of $\mu $ and $\langle \cdot ,\cdot \rangle$ is the usual inner
product of symmetric functions in which the Schur functions are
orthonormal.

  The theorem can be proved by defining $t^{n(\mu )}H_{\mu
}(X;t^{-1},q) = J_{\mu }(\frac{X}{1-t};t,q)$, where, as in (1),
$J_{\mu }(X;t,q) = h_{\mu }(t,q)P_{\mu }(X;t,q)$ for a certain
$(t,q)$-hook product $h_{\mu }(t,q)$.  Then the theorem follows by
straightforward calculations from Macdonald's existence and uniqueness
theorem for the basis $P_{\mu }$.

We now define the coefficients $\tilde{K}_{\lambda \mu }(t,q)$.

\announce Definition\defines $\tilde{K}_{\lambda \mu }(t,q)$ is the
coefficient of the Schur function $s_{\lambda }(X)$ in the expansion
$$
H_{\mu }(X;t,q) = \sum _{\lambda } \tilde{K}_{\lambda \mu }(t,q)
s_{\lambda }(X).\eqno (2.1)
$$

These coefficients are related to Macdonald's $K_{\lambda \mu }$ by 
$$ 
\tilde{K}_{\lambda \mu }(t,q) = t^{n(\mu )} K_{\lambda \mu
}(t^{-1},q).\eqno (2.2)
$$
It is known that if {\it MPK} holds, then $K_{\lambda \mu }$ has
degree $n(\mu )$ in $t$, so {\it MPK} for $\tilde{K}_{\lambda \mu }$
is equivalent to {\it MPK} for $K_{\lambda \mu }$.

Let 
$$ 
M = \bigoplus _{i,j\geq 0} (M)_{i,j} \eqno (2.3)
$$
be a doubly graded $S_{n}$ module, $(M)_{i,j}$ denoting its
homogeneous component of degree $(i,j)$.  It is useful to collect all
information about the characters of the $(M)_{i,j}$ into a {\it
Frobenius series}
$$ 
F_{M}(X;t,q) = \sum _{i,j} \psi ((M)_{i,j})t^{i}q^{j},\eqno (2.4)
$$
where $\psi $ is the usual `Frobenius map' assigning to an $S_{n}$
module $V$ the symmetric function $\psi (V) = \sum _{\lambda }
m_{\lambda }(V)s_{\lambda }(X)$, where $m_{\lambda }(V)$ is the
multiplicity of the irreducible $V_{\lambda }$ in $V$.

Trivially, {\it MPK} is equivalent to the assertion that $H_{\mu
}(X;t,q)$ is the Frobenius series of some finite dimensional doubly
graded $S_{n}$ module $R_{\mu }$.  Our contribution is to propose a
simple construction of the required $R_{\mu }$.

\sect{3.  Constructions of $R_{\mu }$}

In the following subsections we give three candidates for a
construction of $R_{\mu }$.  Assuming the fundamental `$n!$
conjecture' stated below, the three constructions yield the same
object.  Possible approaches to proving the $n!$ conjecture will be
treated in Section 6.

\sect{3.1  Construction from $\Delta _{\mu }$}

Throughout, $\mu $ will be a partition of $n$.  Let the coordinates of
the cells in the Ferrers diagram of $\mu $ be
$\{(p_{1},q_{1}),\ldots,(p_{n},q_{n}) \}$, where $p$ is the row
coordinate and $q$ is the column coordinate, indexed from zero, so
that the corner cell is $(0,0)$.  Put
$$
\Delta _{\mu }(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}) = \det
\left[\begin{array}{cccc}
x_{1}^{p_{1}}y_{1}^{q_{1}}&	x_{2}^{p_{1}}y_{2}^{q_{1}}&	\cdots
&	x_{n}^{p_{1}}y_{n}^{q_{1}}\\ 
x_{1}^{p_{2}}y_{1}^{q_{2}}&	x_{2}^{p_{2}}y_{2}^{q_{2}}&	\cdots
&	x_{n}^{p_{2}}y_{n}^{q_{2}}\\
\vdots&	\vdots &	&	\vdots \\
x_{1}^{p_{n}}y_{1}^{q_{n}}&	x_{2}^{p_{n}}y_{2}^{q_{n}}&	\cdots
&	x_{n}^{p_{n}}y_{n}^{q_{n}}
 \end{array}
\right]. \eqno (3.1)
$$
Since the bi-exponents $(p_{i},q_{i})$ are distinct, $\Delta _{\mu }$
is a non-zero $S_{n}$-alternating polynomial, homogeneous of degree
$n(\mu )$ in $x$ and $n(\mu ')$ in $y$.  Note that for $\mu =(1^{n})$
and $\mu =(n)$, $\Delta _{\mu }$ reduces to the Vandermonde
determinant in $x$ and in $y$, respectively.

Let $I_{\mu }\subseteq \Bbb Q[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}] =
\Bbb Q[X,Y]$ be the ideal of all polynomials
$p(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})$ with the property that the
corresponding differential operator $p(\frac{\partial }{\partial
x_{1}},\ldots,\frac{\partial }{\partial x_{n}},\frac{\partial
}{\partial y_{1}},\ldots,\frac{\partial }{\partial y_{n}})$
annihilates $\Delta_{\mu }$.  Clearly $I_{\mu }$ is an
$S_{n}$-invariant doubly homogeneous ideal.

\announce Definition\defines $R_{\mu }^{I}$ is the quotient ring $\Bbb
Q[X,Y]/I_{\mu }$, with its natural structure of doubly graded $S_{n}$
module.

Note that $R_{\mu }^{I}$ has additional structure: it is a doubly
graded 0-dimensional Gorenstein algebra, generated by its elements of
total degree $1$.

The ring $R_{\mu }^{I}$ is isomorphic as a doubly graded $S_{n}$
module to the space ${\cal H}_{\mu }$ of all partial derivatives of
all orders of $\Delta _{\mu }$, so we are free if we wish to regard
${\cal H}_{\mu }$ rather than $R_{\mu }^{I}$ as our model for the
Macdonald polynomial $H_{\mu }(X;t,q)$.  The part of the space ${\cal
H}_{\mu }$ homogeneous of degree zero in $y$ is spanned by all
derivatives of the {\it Garnir polynomials}
$$
\Delta _{T} = \Delta (T_{1})\Delta (T_{2})\cdots \Delta (T_{k}),
\eqno (3.2)
$$
where $T$ is a tableau of shape $\mu $ and $\Delta (T_{i})$ is the
Vandermonde determinant in those variables $x_{j}$ indexed by entries
in the $i$-th column of $T$.  By (8), this space is isomorphic to the
graded $S_{n}$ module studied in (7), showing that the Frobenius
series of $R_{\mu }^{I}$ agrees with $H_{\mu }(X;t,q)$ for $q=0$.

It is known from the theory of Macdonald polynomials that $H_{\mu
}(X;1,1)$ is the Frobenius characteristic of the regular
representation of $S_{n}$ ({\it i.e.}, $\tilde{K}_{\lambda \mu }(1,1)
= f_{\lambda }$).  In particular, our conjectures require that the
dimension of $R_{\mu }^{I}$ and ${\cal H}_{\mu }$ is $n!$.

\announce Conjecture\by ($n!$ conjecture)\states The dimension of the
space ${\cal H}_{\mu }$ of all partial derivatives of $\Delta _{\mu }$
is $n!$.

This conjecture will be discussed at length as we continue.  In
particular we will show that if it holds, then ${\cal H}_{\mu }$ affords
the regular representation of $S_{n}$.

\sect{3.2 Construction by grading an orbit}

Let $\alpha _{0},\alpha _{1},\ldots$ be a sequence of distinct
rational numbers, and let $\beta _{0},\beta _{1},\ldots$ be another
such sequence.  As before, let the coordinates of cells in the Ferrers
diagram of $\mu $ be $(p_{1},q_{1}),\ldots,(p_{n},q_{n})$, in some
order.  Consider the following vector in $\Bbb Q^{2n}$:
$$
a_{\mu } = (\alpha _{p_{1}},\ldots,a_{p_{n}},\beta
_{q_{1}},\ldots,\beta _{q_{n}}). \eqno (3.3)
$$
We let $S_{n}$ act on $\Bbb Q^{2n}$ by permuting the first $n$ and the
second $n$ coordinates simultaneously and identically.  We write
$[a_{\mu }]$ for the orbit of $a_{\mu }$ under this action.  Note that
$[a_{\mu }]$ does not depend on the order in which the Ferrers diagram
cells $(p_{i},q_{i})$ are numbered.  Note also that $[a_{\mu }]$ is a
regular orbit, {\it i.e.}, $a_{\mu }$ has trivial stabilizer.

We regard $\Bbb Q[X,Y]$ as the coordinate ring of $\Bbb Q^{2n}$ with
$x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ the coordinate functions.  Then
we have an ideal $J_{\mu }\subseteq \Bbb Q[X,Y]$ consisting of all
polynomials vanishing on $[a_{\mu }]$.  $\Bbb Q[X,Y]/J_{\mu }$ is the
coordinate ring of the finite set $[a_{\mu }]$; it has a natural
$S_{n}$ action and obviously affords the regular representation.

Since $J_{\mu }$ is not a homogeneous ideal, we construct a graded
$S_{n}$ module by replacing it with its {\it associated graded ideal}
$\gr\;J_{\mu }$, defined as the linear span of
the homogeneneous components of highest degree in elements of $J_{\mu }$.
One shows that $\gr\;J_{\mu }$ is a homogeneous ideal, it is
$S_{n}$ invariant, and $\Bbb Q[X,Y]/(\gr\;J_{\mu })$
affords the regular representation of $S_{n}$.

\announce Definition\defines $R_{\mu }^{II} = \Bbb Q[X,Y]/(\mbox{{\rm
gr}}\;J_{\mu })$.  

This definition is incomplete in two respects.  First, we do not know
that $\gr\;J_{\mu }$ is doubly homogeneous, so $R_{\mu
}^{II}$ is {\it a priori} only singly graded.  Second, the
construction might depend on the choice of the $\alpha _{i}$ and
$\beta _{i}$.  These defects disappear if we assume the $n!$
conjecture, by the following result.

\announce Proposition\states The ideal $I_{\mu }$ contains the ideal
$\gr\;J_{\mu }$; hence if the $n!$ conjecture holds then
$\gr\;J_{\mu } = I_{\mu }$, $R_{\mu }^{I} = R_{\mu
}^{II}$, and this ring affords the regular representation of $S_{n}$.

There is a way to produce a doubly homogeneous `associated graded
ideal' from $J_{\mu }$.  Namely, we may grade with respect to a
generalized degree in which the variables $x_{i}$ are assigned a
different weight from the variables $y_{i}$.  In particular, taking
the $X$ weight very large, we define an ideal $\gr _{Y}\gr
_{X}\;J_{\mu }$, and taking the $Y$ weight large we define $\gr
_{X}\gr _{Y}\;J_{\mu }$.  In general, different choices of weights
may yield different ideals, but in our situation we have the
following.

\announce Proposition\states The $n!$ conjecture is equivalent to $\gr
_{Y}\gr _{X}\;J_{\mu } = \gr _{X}\gr _{Y}\;J_{\mu }$, and if this
holds then all weighted associated graded ideals of $J_{\mu }$ are the
same.

\sect{3.3 Construction by bigrading an orbit}

Let $T_{\mu } = \Bbb Q[X,Y]/J_{\mu }$, the coordinate ring of the orbit
$[a_{\mu }]$.  We consider two filtrations of $T_{\mu }$ defined as follows.
$$
F^{X}_{d} = \{\hat{p}(X,Y)\in T_{\mu } \mid \mbox{{\rm
deg}}_{X}\;p(X,Y)\leq d \},
$$
$$\eqno (3.4) $$
$$
F^{Y}_{d} = \{\hat{p}(X,Y)\in T_{\mu } \mid \mbox{{\rm
deg}}_{Y}\;p(X,Y)\leq d \}.
$$
Here $p(X,Y)$ denotes a polynomial in $\Bbb Q[X,Y]$, $\mbox{{\rm
deg}}_{X}\;p(X,Y)$ is its degree in $X$, and similarly for $Y$, and
$\hat{p}(X,Y)$ is its image in $T_{\mu }$.  Clearly we have
$F^{(-)}_{0}\subseteq F^{(-)}_{1}\subseteq \cdots$ and
$F^{(-)}_{i}F^{(-)}_{j}\subseteq F^{(-)}_{i+j}$, as required for
filtrations.

Given the two filtrations of the ring $T_{\mu }$ we construct the
following doubly graded ring.
$$
\BiGr \;T_{\mu } = \bigoplus _{i,j} (F^{X}_{i}\cap
F^{Y}_{j})/(F^{X}_{i-1}\cap F^{Y}_{j}+F^{X}_{i}\cap
F^{Y}_{j-1}). \eqno (3.5)
$$
This has a well-defined ring structure, owing to the fact that the
$F^{(-)}_{d}$ are filtrations, and it is a doubly graded $S_{n}$
module because these filtrations are $S_{n}$ invariant.  The next
result makes precise the sense in which $\BiGr \;T_{\mu }$ is a
`bigrading' of $T_{\mu }$.

\announce Proposition\states As $S_{n}$ modules, $\BiGr \;T_{\mu }$
and $T_{\mu }$ are isomorphic.

We remark that this proposition is deeper than the corresponding
statement for a singly graded ring constructed from one filtration.
With more than two filtrations, such a statement would be false---{\it
i.e.}, there is no such thing as a `trigr.'

\announce Definition\defines $R_{\mu }^{III} = \BiGr \;T_{\mu }$.

The ring $R_{\mu }^{III}$ is always a doubly graded Gorenstein algebra
carrying the regular representation of $S_{n}$.  Its potential defect
is that it might not be generated by its elements of total degree 1,
{\it i.e.} it may not be the quotient of $\Bbb Q[X,Y]$ by some ideal.
Assuming the $n!$ conjecture corrects this flaw.

\announce Proposition\states $R_{\mu }^{III}$ is generated by its
elements of total degree 1 if and only if the $n!$ conjecture holds
for $\mu $, and in that case $R_{\mu }^{I}= R_{\mu }^{II}=R_{\mu
}^{III}$.

Considering only the part of $R_{\mu }^{III}$ homogeneous of degree
zero in $y$, its construction reduces to one given in (7) for the
module studied there.  Consequently, the Frobenius series of $R_{\mu
}^{III}$, like that of $R_{\mu }^{I}$, agrees with $H_{\mu }(X;t,q)$
for $q = 0$.

\sect{4.  Other conjectures}

Let $\mu $ and $\nu $ be partitions of $n$ obtained by deleting  different
corners from some partition of $n+1$.  Let $\hat{I}_{\nu }$ be the
image of the ideal $I_{\nu }$ in the ring $R_{\mu }^{I} = \Bbb
Q[X,Y]/I_{\mu }$.  Computational evidence supports the following
curious conjecture.

\announce Conjecture\by (Annihilator conjecture)\states With $\mu $
and $\nu $ as above, the ideal $\hat{I}_{\nu }$ is its own annihilator
in $R_{\mu }^{I}$.

The annihilator conjecture implies the $n!$ conjecture.  It does so
because assuming the annihilator conjecture, the dimension of $\Bbb
Q[X,Y]/(I_{\mu }+I_{\nu })$ is one half the dimension of $R_{\mu
}^{I}$ and also of $R_{\nu }^{I}$, so $\dim R_{\mu }^{I} = \dim R_{\nu
}^{I}$.  Since any partition can be reached from $(n)$ by steps of the form
$\mu \rightarrow \nu $ with $\mu $ and $\nu $ as in the conjecture,
every $R_{\mu }^{I}$ must have the same dimension, $n!$.

The annihilator conjecture also implies that the Frobenius series of
the modules $R_{\mu }$ agree with certain conjectures made by Butler,
and apparently noticed also by Lascoux, on the relationship between
$\tilde{K}_{\lambda \mu }$ and $\tilde{K}_{\lambda \nu }$ when $\mu $
and $\nu $ are as above.  (Originally, these conjectures were only for
the case when $\mu $ covers $\nu $ in dominance order.)

There is also computational evidence for an even stranger conjecture.

\announce Conjecture\states The lattice of ideals in $\Bbb Q[X,Y]$
generated by all the ideals $I_{\mu }$ is distributive.

Assuming both this conjecture and the annihilator conjecture, it is
possible to work out inductive recurrences for the Hilbert series of
the modules $R_{\mu }$.  These recurrences agree beautifully with what
one predicts supposing the Frobenius series of $R_{\mu }$ is in fact
$H_{\mu }(X;t,q)$.

\sect{5.  Triangularity and Tor modules}

Let $H'_{\mu }(X;t,q)$ denote the Frobenius series of the module
$R_{\mu }^{I}$ (or $R_{\mu }^{III}$, if desired).  Defining properties
{\it (2)} and {\it (3)} in Theorem 1 for $H_{\mu }(X;t,q)$ clearly
hold for $H'_{\mu }(X;t,q)$.  In principle, one might prove property
{\it (1)}, and thus $H'_{\mu } = H_{\mu }$, implying both the $n!$
conjecture and {\it MPK}, via the following interpretation of $H'_{\mu
}((1-q)X; t,q)$.

\announce Theorem 2\states Let $F_{i}(X;t,q)$ be the Frobenius series of
the module 
$$ 
\mbox{{\rm Tor}}_{i}^{\Bbb Q[X,Y]}(R_{\mu }^{I},\Bbb Q[X]) = 
\mbox{{\rm Tor}}_{i}^{\Bbb Q[Y]}(R_{\mu }^{I},\Bbb Q) \eqno (5.1)
$$
(which has a natural structure of doubly graded $S_{n}$ module).  Then
$$ 
H'_{\mu}((1-q)X; t,q) = \sum _{i} (-1)^{i}F_{i}(X;t,q).\eqno (5.2)
$$
In particular, if the modules (5.1) contain only irreducible components
$V_{\lambda }$ with $\lambda \geq \mu $, then $H'_{\mu }$ satisfies
property (1) of Theorem 1.

For those $\mu $ where we have carried out the computations, we find
that the modules (5.1) do have only the desired irreducible components.

\sect{6.  Toward proofs of the $n!$ conjecture and {\it MPK}}

Let $V$ be a finite set of points in $\Bbb Q^{N}$, $A = \Bbb
Q[x_{1},\ldots,x_{N}]$ the coordinate ring of $\Bbb Q^{N}$, and
$J(V)\subseteq A$ the ideal of polynomials vanishing on $V$.  We shall
say that $V$ is {\it Gorenstein} if the ring $A/(\gr\;J(V))$ is
Gorenstein.  Equivalently, if we embed $V$ in an affine hyperplane in
$(N+1)$-space and connect points of $V$ by lines to the origin, then
the local ring of this union of lines at the origin is Gorenstein.

If $V$ is Gorenstein, then the Hilbert function $h_{d} = \dim
(A/(\gr\;J(V)))_{d}$ is symmetric, meaning $h_{d} = h_{d_{0}-d}$,
where $d_{0}$ is the largest degree in which
$(A/(\gr\;J(V)))_{d_{0}}\not =0$.  We prove the following converse.

\announce Theorem 3\states Assume $V$ is an orbit of a finite linear
group action.  If $h_{0}+h_{1}+\cdots +h_{d}\geq
h_{d_{0}}+h_{d_{0}-1}+\cdots +h_{d_{0}-d}$ for all $0\leq d\leq
d_{0}$, then $V$ is Gorenstein.

This theorem has direct application to the $n!$ conjecture, by the
following result.

\announce Proposition\states The $n!$ conjecture holds for $\mu $ if
and only if $[a_{\mu }]$ is Gorenstein.

There is even some evidence for the following stronger statement.

\announce Conjecture\states Every regular orbit of $S_{n}$ acting on
$\Bbb Q^{2n}$ is Gorenstein.

(The corresponding statement for the action on $\Bbb Q^{n}$ is
classical; for the action on $\Bbb Q^{3n}$ it is false.)

  In the case that $\mu $ is a hook we are able to prove the $n!$
conjecture by means of the preceding Theorem and Proposition.  To
establish the hypothesis of the Theorem it suffices to exhibit for
each $d$ sufficiently many polynomials of degree at most $d$ which
define linearly independent functions on the set $V$.

  The proof technique leads to an explicit basis for $R_{\mu }$ in the
hook case, from which it is possible to analyze the restriction of the
character to $S_{n-1}$.  We obtain the following result.

\announce Proposition\states Let $H'_{r|s} = H'_{r|s}(X;t,q)$ be the
Frobenius series of $R_{\mu }$ for $\mu $ the hook $(s+1, 1^{r})$, and
let $\Gamma $ be the symmetric function operation corresponding under
the Frobenius map to restriction of an $S_{n}$ character to $S_{n-1}$.
Then
$$
\begin{array}{rcl}
\displaystyle \Gamma H'_{r|s}
&       =&	\displaystyle
                \frac{1-t^{r}}{1-t}H'_{r-1|s} + t^{r}H'_{r-1|0}H'_{0|s-1} +
                q\frac{1-q^{s}}{1-q}H'_{r|s-1} \\[3ex]
&	=&	\displaystyle
                t\frac{1-t^{r}}{1-t}H'_{r-1|s} + q^{s}H'_{r-1|0}H'_{0|s-1} +
                \frac{1-q^{s}}{1-q}H'_{r|s-1}.
\end{array} \eqno (6.1)
$$

The second equality in (6.1) gives a relation between $H'_{r-1|s}$,
$H'_{r|s-1}$, and $H'_{r-1|0}H'_{0|s-1}$ which determines $H'$ for any
hook in terms of products of $H'$ for a column by $H'$ for a row.  One
readily shows that the Macdonald polynomials $H_{\mu }(X;t,q)$ satisfy
the same relations and agree with $H'$ for $\mu $ a column or a row.
It follows that $H_{\mu }(X;t,q)=H'_{r|s}(X;t,q)$ when $\mu $ is a
hook, proving {\it MPK}.

Our approach for two-rowed $\mu = (s,r) $ ($s\geq r$) is somewhat
indirect.  We begin with the ring $R = \Bbb Q[u_{1},\ldots,u_{n}]/E$,
where $E$ is the ideal generated by all symmetric polynomials without
constant term.  $R$ is well-known to have dimension $n!$ as a vector
space.  In $R$ we consider the ideal $J =
(u_{1}^{s},\ldots,u_{n}^{s})$ and form the algebra
$$
S = \gr _{J}R = R/J\oplus J/J^{2}\oplus J^{2}/J^{3}\oplus \cdots,
\eqno (6.2)
$$
which can be given a ring structure in a standard way.
In $S$ we let $y_{i}$ stand for the element $u_{i}$ of $R/J$, and let
$x_{i}$ stand for the element $u_{i}^{s}$ of $J/J^{2}$.  Then we
prove:

\announce Proposition\states The ring $S$ is a graded Gorenstein ring,
generated by its elements $x_{i}$ and $y_{i}$.

It follows that $S = \Bbb Q[X,Y]/I$, where $I$ is the ideal of all
polynomials which annihilate a certain polynomial $\Delta $ when
applied as differential operators to $\Delta $.  One then easily shows
that in this case $\Delta =\Delta _{\mu }$, hence the $n!$ conjecture
holds, since $S$ has dimension $n!$.

Identification of the character in the case $\mu =(r,s)$ takes place
in two stages.  First, using Theorem 2 on Tor modules, we show that
the Frobenius series of $R_{\mu} $ is a linear combination of
Macdonald polynomials $H_{\nu }$ for two-rowed shapes $\nu $.  This is
achieved by showing that the restrictions of the Tor modules (5.1) to
$S_{3}$ contain no alternants.  Combining this with a recurrence for
the restriction to $S_{n-1}$ we obtain relations which determine the
Frobenius series.

Computational experiments suggest that the ring $S$ of (6.2) is
isomorphic to $R_{\mu }$ not only for two-rowed shapes but for any
`near rectangle' $(s^{k},r)$, $s\geq r$.  A crucial step in our proof
for two-rowed shapes involves induction on removal of corners, so a
different method will be required for near-rectangles.  For general
$\mu $ it is not the case that $R_{\mu }$ is isomorphic to an algebra
of the form (6.2).

\bigskip

{\small Both authors received support for this work from National
Science Foundation mathematical sciences research grants.}

\sect{References.}

\begin{itemize}

\item [{1.}]  Macdonald, I. G. (1988) {\sl Actes 20$^{e}$ S\'eminaire
Lotharingien}, Publ.\ I.R.M.A.\ Strasbourg 372/S--20, 131--171.

\item [{2.}]  Macdonald, I. G.  (Unpublished manuscript)
{\sl Symmetric Functions and Hall Polynomials, 2nd Edition}, Chapter
6.

\item [{3.}] Macdonald, I. G. (1982) {\sl SIAM J. Math.\ Anal.} {\bf
13}, no.~6, 988--1007.

\item [{4.}] Stembridge, J. R. (In press) {\sl Proc.\ Amer.\ Math.\
Soc.}

\item [{5.}] de Concini, C. \& Procesi, C. (1981) {\sl Invent.\
Math.} {\bf 64}, 203--230.

\item [{6.}] Kraft, H. (1981) {\sl Proc.\ 1980 Torun Conf., Poland},
Ast\'erisque {\bf 87--88}, 195--205.

\item [{7.}] Garsia, A. M. \& Procesi, C. (1992) {\sl Advances in
Math.} {\bf 94}, 82--138.

\item [{8.}] Bergeron, N. \& Garsia, A. M. (1992)
{\sl Contemporary Mathematics} {\bf 138}, 51--86.

\end{itemize}

\end{document}
