%  Hirzebruch    modifie  2 juin 98  
\magnification 1200 \overfullrule 0mm
\def\S{{\cal S}}
\def\F{{\cal F}}
\def\M{{\cal M}}
\def\O{{\cal O}}

\def\rh{\hookrightarrow }
\def\s{\scriptstyle }
\def\Sym{ {\sl {\cal S}ym} }
\def\puce{\vskip 5mm\noindent {$\scriptstyle\bullet$}\kern 8pt}
\def\d{\partial}
\def\e{\epsilon}
\def\car{$\chi_y$ characteristic }
\def\T{\bar{T}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf About the "$y$" in the $\chi_y$-characteristic of Hirzebruch}
\smallskip\centerline{\it Alain Lascoux}
\vskip 5mm
{\it What concerns Macdonald polynomials is a joint work 
with Jennifer Morse and Luc Lapointe }
\vskip 5mm

Geometers want to compute the dimensions of spaces of cohomology 
$H^i(V\otimes\Omega^j)$, where $V$ is a vector bundle and $\Omega^j$ the sheaf
of $j$-differential forms.  In his study of Riemann-Roch theorem, Hirzebruch introduced a generating function
  $$ \sum_{i,j} (-1)^i\, y^j\, dim(H^i(V\otimes\Omega^j)) $$
that he called $\chi_y${\it -genus} or $\chi_y${\it -characteristic}.

I shall only speak of the case of a flag variety and shall enumerate 
many missed opportunities of giving to the extra parameter "$y$"
the status it deserves.

\puce {\bf Relative flag variety} \quad 
Let $V$ be a vector bundle of rank $n$ over a complex variety  $\M$ and 
$\F(V) \mapsto \M$ the associated relative flag variety, the fibers of which
are the usual flag variety of complete flags of vector spaces 
$$V_1 \rh V_2 \rh \cdots \rh V_n=C^n$$
of successive ranks $1,2,\ldots \, n$.

The variety $\F(V)$ comes equipped with {\it tautological line bundles}
$L_1,\ldots,\, L_n$. Their Chern classes generate the cohomology ring 
$H(\F(V))$ (wih coefficients in $H(\M)$), and their classes generate the
Grothendieck ring $K(\F(V))$ of classes of vector bundles (with coefficient
in $K(\M)$). We shall denote the first Chern classes, as well as the classes
of the line bundles by the variables $x_1,\ldots,\, x_n$. Thus the two rings
are the quotient of the ring of polynomials in $x_1,\ldots,\, x_n$,
modulo some ideal of symmetric functions. We can skip describing the ideal
by considering $H(\F(V))$ and  $K(\F(V))$ as free modules over the ring of 
symmetric polynomials $\Sym$ in $x_1,\ldots,\, x_n$. Taking more general 
symmetric functions as coefficients (we shall need exponentials), we still 
have  to operate only on polynomial distinguished elements of the
free modules.

When $\M$ is a point, symmetric functions have to be specialized
to their constant term in the case of  the cohomology ring, 
while they have to be
specialized to their value in $x_1=1, \ldots,\, x_n=1$ in the
case of the Grothendieck ring. 


However, the two Gysin morphisms from $H(\F(v))$ to $H(\M)$, 
and from $K(\F(V))$ to $K(\M)$ are different. For the cohomology,
it is 
$$  \d_\omega :\ \ H(\F(v)) \ni f(x_1,\ldots,\, x_n) \longrightarrow 
\sum_{\mu\in \S(n)} \bigl( {f\over \prod_{i<j} x_i- x_j}
\bigr)^\mu   $$
while it is 
$$ \pi_\omega :\ \ K(\F(v)) \ni f(x_1,\ldots,\, x_n) \longrightarrow 
\sum_{\mu\in \S(n)} \bigl( {f\over \prod_{i<j} 1- x_j/x_i}
\bigr)^\mu  $$
for the Grothendieck ring, denoting by $f^\mu$ the image of a function 
$f(x_1,\ldots,\, x_n)$ under the permutation $\mu$ of the $x_i$.

The two rings are connected by the Chern morphism $x\mapsto e^x$.
However, one summation involves dividing by the Vandermonde in the
$x_i$, while the other uses the Vandermonde in the exponentials $e^{x_i}$.
 To get a commutative diagram where the vertical arrows are the two Gysin
morphisms,  
 one must introduce the correcting factor of proportionality  between
the two summations 
$$ Todd:=   \prod_{i<j} (x_i- x_j) \, /\, (1-e^{x_j-x_i}) \ , $$


\setbox2=\hbox{$\longrightarrow H(\F) 
 \buildrel {\s Todd}\over { \longrightarrow}  $}
\setbox3=\hbox{$
 \buildrel {\s Chern}\over {\hbox to 20 mm{ \rightarrowfill}}    $}


$$ \matrix{ \ K(\F) & \box2 & H(\F) \cr\cr
            \pi_\omega\Big\downarrow & & \Big\downarrow\d_\omega \cr\cr
    K(\M) & \box3 & H(\M) \cr\cr }$$
 


Knowing that the class of the relative cotangent bundle in $K(\F(v))$ is
$\sum_{i<j} x_j/x_i$, one recognizes into the factor of proportionality between the two above summations the {\it Todd class} of the 
relative cotangent bundle  $\Omega_{\F/\M}$ and in the commutative diagram,
Riemann-Roch theorem for a relative flag manifold.  


The weighted sum of all the classes of the  exterior powers 
of $\Omega_{\F/\M}$ is $$ \prod_{i<j} (1 + y x_j/x_i) \ .$$ 
 Thus, Hirzezbruch's characteristic $\chi_y$  is the morphism
 $$ K(\F(v)) \ni f(x_1,\ldots,\, x_n) \rightarrow 
\sum_{\mu\in \S(n)} \bigl(f\, \prod_{i<j}{x_i+yx_j\over x_i -x_j}  
\bigr)^\mu \ .  $$

It is $K(\M)$-linear, and thus is determined by its values on monomials
(it would be sufficient, since $K(\F(V))$ is a free module of rank 
$n!$, to give only $n!$ values, but one would have then to use interpolation
technics to compute the image of a general polynomial). 

\puce {\bf Hall-Littlewood polynomials}  \quad 
The above summation, when the function $f$ is a {\it dominant} monomial, 
taking the factor $\prod_{i<j}{x_i-y x_j\over x_i -x_j}$
 instead of  $\prod_{i<j}{x_i+yx_j\over x_i -x_j}$, has been used by Littlewood
to define {\it Hall-Littlewood polynomials } (cf. [Li], [Ma]).
 Denote it by $\chi$.

However, the $\chi(x^I)$, $I\in N^n$, are not linearly independent over 
$C(t)$.  A basis is given by the Hall-Littlewood polynomials 
$Q_I:=\chi(x^I)$, $I$ {\it dominant}
(i.e. $I=[i_1,i_2,\ldots,\, i_n],\, i_1\geq i_2\geq \cdots \geq i_n\geq 0$;
one also says that $I$ is a {\it partition}. ). 

In the case when the monomial is not dominant, one must "straighten" it 
into a sum of dominant monomials. This amounts using the following relations,
where $\star$ denotes a function not depending on $x_i, x_{i+1}$ 
  $$ \chi\, \bigl(\star\,  (x_i^{a+b} x_{i+1}^b + x_i^a x_{i+1}^{a+b})
    \, (yx_i-x_{i+1}) \bigr) = 0  \eqno (\clubsuit) $$
(which come from the fact that the image of a function symmetrical in
$x_i,x_{i+1}$, under the morphism  $\sum (-1)^{\ell(\mu)} \mu$, is zero). 

Now, from the tables of these polynomials, one reads dimensions of 
cohomology spaces or the image of any element in the Grothendieck ring
$K(\F(V))$ under the \car of Hirzebruch. 
  
 For example, to get the cohomology of the sheaf of 
 differential forms of any order on the Grasmannian $G_{3,2}(C^5)$, twisted by the fourth power of
the line bundle giving the Pl\"ucker embedding, one must look at
$Q_{44400}(x_1,\ldots,\, x_5)$. The precise decoding is given in the 
appendix.  

\puce {\bf Associativity} \quad Flags of vector spaces can be split into pieces. This induces  factorisations of the \car. Another way to formulate this
splitting is to state 
that there is an associative law on Hall-Littlewood polynomials indexed
by  integral vectors (i.e. the set of all $\chi(x^I)$, $I\in N^\star$), 
 which is just the concatenation of the indexing vectors.
Any  Hall-Littlewood polynomial can be decomposed (uniquely) into
the basis of standard Hall-Littlewood polynomials indexed by dominant vectors
 by successive applications of $(\clubsuit)$, that is to say, according
to the following commutation rules 
$$ y[\star, a+1,b,\star] + y[\star,b+1,a,\star] - [\star,a,b+1,\star]
 - [\star, b, a+1,\star] = 0 \eqno (\clubsuit)   $$

The special case $y=0$ has many applications. In fact, it is the translation,
at the level of the Grothendieck ring, 
of Bott's theorem for the cohomology of homogeneous bundles on a relative 
grassmannian  $G_{q,r}(V)$. This variety has a {\it tautological quotient}
$Q$ of rank $q$ and a {\it tautological kernel} $R$ of rank $r$. Two
dominant weights $I$, $J$ determine two irreducible representations of
the linear groups $Gl(q)$ and $Gl(r)$ and two homogenous bundles 
$S_I(Q)$ and $S_J(R)$, the classes of which in $K(G_{q,r})$ are two Schur
functions. Bott's theorem is that the cohomology of the tensor product 
$S_I(Q) \otimes S_J(R)$ is either null, or concentrated in only one degree,
and in that case  is equal to an irreducible representation of $Gl(q+r)$.
The class of this representation is the Schur function of index the concatenation of $I$ and $J$.  In the case that the index is not dominant,
 recall that the rule for straightening Schur functions is
$$S_{\star,\, a ,b ,\,\star} = - S_{\star, \,  b-1,a+1,\, \star } 
\eqno(\diamond) $$

\puce {\bf Yang-Baxter equation}\quad 
Geometry however indicates that the description of the cohomology or the
Grothendieck ring of $\F(V)$ is a pure algebraic consequence of the case
of a $P^1$-fibration (i.e. the case where $V$ is of rank 2). 

Indeed, calling $x_i$ and $x_{i+1}$ the fisrt Chern classes, or the classes
of the two tautological line bundles of $\F(V)$,
 the Gysin morphism for the cohomology
is {\it Newton's divided difference} 
$$ \d_i : \ f\mapsto { f(x_i,x_{i+1}) - f(x_{i+1},x_i) \over x_i - x_{i+1} }$$
and the Gysin morphism for the Grothendieck ring is 
$$ \pi_i : \ f\mapsto { x_i f(x_i,x_{i+1}) - x_{i+1} f(x_{i+1},x_i) 
\over x_i - x_{i+1} } \ .$$

Now these elementary operators satisfy the so-called {\it braid relations}
(due to Moore, and Coxeter for more general groups):
$$ \d_i \d_{i+1}\d_i = \d_{i+1} \d_i \d_{i+1} \ {\rm and} \
  \d_i \d_j=\d_j\d_i \ , \ |i-j|>1 $$
 
$$ \pi_i \pi_{i+1}\pi_i = \pi_{i+1} \pi_i \pi_{i+1} \ {\rm and} \
  \pi_i \pi_j=\pi_j\pi_i \ , \ |i-j|>1 $$

This allows to define by products a divided difference $\d_\mu$
and an isobaric divided difference $\pi_\mu$ for any $\mu \in \S_n$.
In particular, the two Gysin morphisms for a relative flag variety  
 correspond to the maximal 
permutation $\omega:= [n\ldots 1]$. For any {\it reduced } decomposition
of $\omega$ one gets a factorisation of $\d_\omega$ and of $\pi_\omega$.
Thus, for $n=4$  
$$ \d_{4321}= \d_1 \d_2 \d_1 \d_3 \d_2 \d_1= \cdots =
 \d_3 \d_2\d_1 \d_3 \d_2 \d_3 $$
 
$$ \pi_{4321}= \pi_1 \pi_2 \pi_1 \pi_3 \pi_2 \pi_1= \cdots =
  \pi_3\pi_2\pi_1 \pi_3\pi_2\pi_3 $$

Given that success, one  defines the $i$-th elementary characteristic to be
 $$\chi_i : \ f\mapsto 
{ (tx_i -x_{i+1})f(x_i,x_{i+1}) - 
(tx_{i+1}-x_i) f(x_{i+1},x_i) \over x_i - x_{i+1} }$$ 

However 
$$ \chi_1 \chi_2 \chi_1 \neq \chi_2 \chi_1 \chi_2 \ .$$
It can be corrected by introducing some shifts 
$$ \chi_1 \, (\chi_2 -{t\over 1+t} )\, \chi_1 =
 \chi_2 \, (\chi_1  -{t\over 1+t} )\,  \chi_2 \ ,$$
and this product happens to be a factorisation of the characteristic for $n=3$
(and with now, the factor $\prod_{i<j} (tx_i -x_j)$ instead of 
$\prod_{i<j} (x_i -yx_j)$; we shall keep on using the notation $\chi$,
nothwithstanding the change of parameter). 

More generally, looking at the action of the operators on a specific basis
of the ring of polynomials (as a free module over $\Sym$ ) one can check that
$\chi$ factorizes into simple factors $\chi_i - {t+\cdots +t^k\over 
1+t+\cdots t^k }$ which involve shifts by the $t$-integers 
$[k]:= 1+t+\cdots +t^{k-1}$. 

For example, for $n=4$, one has the following decomposition of $\chi$ :
$$ \chi =  \chi_1 \, (\chi_2 -{t\over 1+t} )\, \chi_1 \, (\chi_3 -{t+t^2
\over 1+t+t^2})\,  (\chi_2 -{t\over 1+t} )\, \chi_1  \ ,$$
the general law being not difficult to infer for the special reduced 
decompositions of the same type. But we must have recourse to  the more
 more general Yang-Baxter equation (cf. [Ya]) to explain 
the preceeding decomposition  ($a,b,c$ being any distinct constants)

$$ (\chi_1 - {ta-b\over a-b})    (\chi_2 -{ta-c\over a-c})
  (\chi_1 - {tb-c\over b-c}) = 
(\chi_2 - {tb-c\over b-c})    (\chi_1 - {ta-c\over a-c})
  (\chi_2 - {ta-b\over a-b})  $$

These relations (for any pair $i,i+1$), together with the trivial 
commutations\par\noindent  
$(\chi_i +a) (\chi_j+b) = (\chi_j+b) (\chi_i +a)$,
$|i-j|>1$, allow to define $n!$ Yang-Baxter elements indexed by permutations
in $\S_n$, taking arbitrary
reduced decompositions of these permutations.  One can use
these elements to recover the classes of Schubert varieties in the
cohomology ring, or the classes of their structure sheaves in the
Grothendieck ring  (cf. [LLT]). 

For the \car of Hirzebruch, one has to use the special parameters
$a=1,b=t,c=t^2, \ldots \, t^{n-1}$. In other words, \it 
Hirzebruch's caracteristic for a relative flag manifold is the specialization
of the Yang-Baxter element corresponding to the maximal permutation
$[n\ldots 1]$, taking the spectral parameters $1,t,\ldots,\, t^{n-1}$.  \rm
 
This is not yet the way that we shall choose to compute in flag manifolds.

\puce{\bf Iwahori-Hecke algebra} \quad 
Defining $T_i:= \chi_i -1$, one sees that these last elements satisfy the 
braid relations
$$ T_i T_{i+1} T_i = T_{i+1} T_i T_{i+1} \ {\rm and} \
  T_i T_j= T_j T_i \ , \ |i-j|>1 $$

However, contrary to the two degenerate cases $\d_i^2=0$, $\pi_i^2=\pi_i$,
they satisfy the {\it Hecke relation} 
$$   (T_i+1) (T_i -t) = 0  \ .   $$

More general operators on the ring of polynomials representing the 
{\it Iwahori-Hecke algebra} can be found in [LS1]. The algebra generated
by the $T_i$ and $x_j$ is usually called the {\it degenerate affine
Hecke algebra}. With Sch\"utzenberger [LS2], we had called it the 
{\it Alg\`ebre des diff\'erences divis\'ees}  since the
relations between the $T_i$ and $x_j$ are no other that the 
(Leibniz) commutations between the $\d_i $ and $x_j$. 

This still is not the way that we favor to compute with the \car,
 we shall rather have recourse to eigenfunctions.

\puce{\bf Eigenfunctions}\quad 
All the operators that we have used up to now commute with the multiplication
with symmetric functions. A priori, their action on Hall-Littlewood polynomials
presents no interest since these polynomials are symmetric.
 However, given a symmetric function, one can 
introduce a disymmetry, so that symmetrizing it afterwards provides identities. 

Instead of using $\chi$ on the ring $\Sym$, let us take the composite $\nabla$: 
$$ \Sym \ni f \mapsto f(0,x_2,x_3,\ldots ,\, x_n)
\mapsto \chi (f(0,x_2,x_3,\ldots ,\, x_n)) $$ 

It is an easy computation  that for a
dominant $I=[i_1,\ldots,\, i_\ell,\, 0,\ldots,\, 0]$, 
$i_\ell >0$, 
$$ \nabla( Q_I) = (1+t+\cdots + t^{n-\ell(I)})\, Q_I$$
We see that the eigenspaces of $\nabla$ are not one-dimensional, and thus
we cannot characterize  Hall-Littlewood polynomials this way.  

\puce{\bf Macdonald operators} \quad 
Macdonald eliminated multiplicities by introducing an extra parameter 
$q$ and by  considering the operator 
$$ \Sym \ni f \mapsto f(qx_1,x_2,x_3,\ldots ,\, x_n)
\mapsto \chi\, (f(qx_1,x_2,x_3,\ldots ,\, x_n)) \ . $$ 

However, taking into account that $\chi$ operates on a function which is
symmetrical in $x_2,\ldots,\, x_n$, one eliminates from an appropriate
 factorisation of $\chi$  the part which acts trivially, so that
Macdonald's operator is  $\nabla^q$: 
$$ {\Sym} \ni f \mapsto f(qx_1,x_2,x_3,\ldots ,\, x_n)
\buildrel {\nabla^q}\over {\longrightarrow} \d_{n-1}\cdots \d_1
\bigl( \prod_{i>1}(tx_1-x_i) \, f(qx_1,x_2,x_3,\ldots ,\, x_n)  \bigr) \ . $$
Let us notice that the operator 
$ \d_{n-1}\cdots \d_1$ is the Gysin morphism for the cohomology of a projective
space ${\cal P(V)}$, $rank(V)=n$. 


For any integer $k$ :$ 1\leq k\leq n$, one can define an 
operator similar to $\nabla^q$ : 
$$ {\Sym} \ni f \mapsto \chi(f(qx_1,\ldots,\, qx_k, \ x_{k+1},
\ldots ,\, x_n)) $$ 
but it can be deduced from $\nabla^q$ and brings no new information. 

Now one can check that eigenspaces of $\nabla^q$ 
 are 1-dimensional. We shall give in the next section 
a more precise statement by computing 
the explicit action of $\nabla^q$ on Schur functions.
 By specialization, 
we shall thus be able to handle Hall-Littlewood polynomials and the \car. 


\puce{\bf Action of Macdonald operators on Schur functions} \quad 
 

We must first adapt notations to $\lambda$-rings. A symmetric function
$f$ of $x_1,x_2,\ldots, \, x_n$ will be noted $f(X)$, with 
$X= x_1+x_2+\cdots +x_n$. The two parameters $t$ and $q$ will be considered
as classes of line bundles.  Now an element like 
$X^{tq} := X {1-t\over 1-q}$ makes sense as a formal power series in $q$.
The generating function $\lambda_{-1} (X^{tq})$ for all the exterior powers
is by definition equal to
$$\lambda_{-1}( X^{tq}) = \prod_i { 1- tx_i\over \prod_{j\geq 0 } 1-q^j x_i}$$

Now the first step in Macdonald's operator is replacing $x_1$ by $qx_1$,
which in a $\lambda$-ring must be written
 $$X \mapsto X -x_1 +qx_1 $$
However, if we start from a function of $X^{tq}$, we get  
$$X^{tq} \mapsto (X-x_1 +qx_1){t-1\over q-1} = X^{tq} +x_1(t-1) \, $$
that one can as well write
$$(X^{tq}-X) +x_1+\cdots +x_n \mapsto (X^{tq}-X) + tx_1 +x_2+\cdots +x_n \ . $$
Let us write $X^{tq}-X =Y$ to indicate that this element will be invariant
under all the operators we shall use. Our first step is
$$ Y +x_1+\cdots +x_n \mapsto Y + tx_1 +x_2+\cdots +x_n \ . $$


Now the second step consists into a multiplication by the
factor $\prod_{i>1} (tx_1 -x_i)$.  One can interchange the operations, the first step being now  
$$ f(X^{tq}) =f(Y+x_1+\cdots + x_n) \mapsto 
f(Y+ x_1+\cdots +x_n) \prod_{i>1}(x_1-x_i)$$
followed by the operation $x_1 \mapsto tx_1$.

However, if one starts from a Schur function $f$, then its product by the
Vandermonde $\prod_{i<j}(x_i -x_j)$ is, according to Jacobi, a determinant
of powers in the variables $x_1,x_2,\ldots,\, x_n$, each of them 
being in different columns. We still have $x_1$ separated from the
other variables in 
 the case of a Schur function of $Y+X$ multiplied by the factor 
$\prod_{i>1} (x_1 -x_i)$.  We are in fact reduced to operate on a determinant
where all the columns are invariant, except the first one whose entries
are polynomials in $x_1$ with constant coefficients.  The only ingredient
which is finally needed was known to Newton, that is, the image of
$1/(1-tx_1)$ under the sequence of divided differences 
 $\d_1$, $\d_2, \ldots , \, \d_{n-1}$
is equal to 
$$  t^{n-1}/(1-tx_1)\cdots (1-tx_n) $$
(one can also say that the Gysin push-down 
 of a power of the first Chern class of
the tautological line bundle on ${\cal P(V)}$ is equal to a {\it Segre} class
of $V$, i.e. a {\it complete function} in the $x_i$).

To be more precise, denote by $\Xi_j$ the operator which transforms 
a determinant of classes of symmetric powers $S^i(A)$ by changing in the
$j$-th column each entry  $S^i(A)$ into $S^i(A +X(t-1))$. Then

\noindent{\bf Theorem}  The action of Macdonald's operator $\nabla^q$
on a Schur function of $X^{tq}$ coincides with the operator
$$ t^{n-1} \Xi_1 +t^{n-2} \Xi_2 +\cdots + t^0 \Xi_n   $$

For example, for $n=3$
$$S_{632}(X^{tq}) = det \left| \matrix{
S^6(X^{tq}) & S^7(X^{tq}) & S^8(X^{tq})\cr
S^2(X^{tq}) & S^3(X^{tq}) & S^4(X^{tq})\cr
S^0(X^{tq}) & S^1(X^{tq}) & S^2(X^{tq})\cr
}\right|$$
Its image under $\nabla^q$ is equal to the following sum of three determinants
taking into account that $X^{tq} + X(t-1) = q X^{tq}$, and writing $S^i$ for
$S^i(X^{tq})$: 

$$\kern -1cm  t^2\left|\matrix{
q^6S^6 & S^7 & S^8\cr
q^2S^2 & S^3 & S^4\cr
q^0S^0 & S^1 & S^2\cr}\right|  +
t\left|\matrix{
S^6 & q^7S^7 & S^8\cr
S^2 & q^3S^3 & S^4\cr
S^0 & q^1S^1 & S^2\cr}\right|  +
\left|\matrix{
S^6 & S^7 & q^8 S^8\cr
S^2 & S^3 &q^4 S^4\cr
S^0 & S^1 & q^2 S^2\cr}\right|  $$


\puce {\bf Determinantal expression of a Macdonald polynomial} 

If one takes  the diagonals of all the  determinants in the sum 
expressing the image
of a Schur function, one obtains that the image of $S_I(X^{tq})$ under Macdonald's operator is 
$$ t^{n-1} (q^{i_1} S^{i_1}) S^{i_2} S^{i_3}\cdots + 
 t^{n-2}  S^{i_1} (q^{i_2} S^{i_2}) S^{i_3}\cdots + 
 t^{n-3}  S^{i_1} S^{i_2} (q^{i_3} S^{i_3})\cdots  + \cdots $$
i.e. that the leading term (with respect to the so-called {\it natural order
} on partitions) of $\nabla^q(S_I(X^{tq})$ is
$$ ( t^{n-1} q^{i_1} + t^{n-2} q^{i_2} + \cdots + t^0 q^{i_n}) S_I(X^{tq}) \ .$$  
Therefore, puting 
$$ [|I|] :=  t^{n-1} q^{i_1} + t^{n-2} q^{i_2} + \cdots + t^0 q^{i_n} \ ,
\ I\in N^n \ ,$$
one has that there exists a family of symmetric functions indexed by
partitions, which are eigenvalues of $\nabla^q$ with eigenvalues the 
$[|I|]$, and are obtained by a triangular elimination from the 
$S_I^{tq}$. 

 
The {\it Macdonald polynomial} $M_I$, $I\in N^n$ dominant, 
is defined to be the eigenfunction of $\nabla^q$ with eigenvalue
$[|I|]$,  satisfying a certain
normalisation that we shall ignore here.  Many properties of 
Macdonald's polynomials have been obtained during these last ten years, 
 but no satisfactory explicit expression of them.
Using the structure of
$\lambda$-ring of the Grothendieck ring, we propose a determinantal expression.

 Indeed, the elimination process induced from the order on partitions 
produces a determinant of the type 
$$ \left|\matrix{ \star &\star &\star &\star\cr
 0 &\star &\star &\star\cr 0&0&\star &\star\cr   S &S&S&S \cr
  } \right| $$
that is, a triangular matrix of linear combinations of eigenvalues $\star$,
apart from the last row which is constituted of Schur functions.

\setbox2=\hbox{$\matrix{ {\s (5,0,0)}\cr {\s (4,1,0)}\cr{\s (3,2,0)}\cr
{\s (3,1,1)}\cr} $  }
For example, for $\lambda=(2,2,1)$, one has 
$$ M_{(2,2,1)} =  
\left|
\matrix{ 
[5, 0, 0]&-[0, 5, 0]& 0& [0, 0, 5]& 0\cr 0& 
[4, 1, 0]&-[1, 4, 0]&-[0, 4, 1]& [1, 0, 4]\cr 0& 0& [3, 
2, 0]&-[3, 0, 2]&-[2, 0, 3]\cr 0& 0& 0& [3, 1, 1]&-[1, 3
, 1]\cr S_{5}& S_{4, 1}& S_{3, 2}& S_{3, 1, 1}& S_{2, 2, 
1} }  \right| \raise 2mm \box2  $$ 

Let us explain how to write the formal determinant before interpreting it.
The bottom row is the list of all Schur functions whose index is smaller or
equal to the partition $I$.  In the column of $S_J$, one writes the
(ordered) expansion of the Jacobi-Trudi 
determinant in the complete functions $S^k$ 
expressing  $S_J$, puting on the same row all the vectors equal up to permutation. For example, the column of $S_{3,1,1}$ corresponds to the
ordered expansion of 
$S_{3,1,1}= \left|\matrix{S^3 & S^4 & S^5\cr S^0 &S^1 &S^2\cr 0 &S^0 &S^1}
\right|$ which is 

$$ \matrix{ & & -[041] &\rightarrow &[005] \cr
    &\nearrow &        &            &    &\searrow \cr
[311] &      &       &       &     &   & 0\cr
 &\searrow &        &            &    &\nearrow \cr
 & & -[302] &\rightarrow &0 \cr
}  $$

In the rightmost column, one does not take the top element $S_I$. Let us
remark that these ordered expansions are also used to compute 
Kostant's partition function.

\smallskip
A bigger example, with multiplicities,  for $M_{2,2,2}$ :
$$ \left|   \matrix{
[6, 0, 0]&-[0, 6, 0]& 0& [0, 0, 6]& 0& 0& 0\cr
0& [5, 1, 0]&-[1, 5, 0]&-[0, 5, 1]& 0& [1, 0, 5]& 0\cr 
0& 0& [4, 2, 0]&-[4, 0, 2]&-[2, 4, 0]& 0&-[0, 2, 4]\cr 0
& 0& 0& [4, 1, 1]& 0&-[1, 4, 1]& [1, 1, 4]\cr 0& 0& 0& 
0& [3, 3, 0]&-[3, 0, 3]& [0, 3, 3]\cr 0& 0& 0& 0& 0& 
[3, 2, 1]&-[2, 1, 3]-[1, 3, 2]\cr S_{6}& S_{5, 1}& S_{4, 2
}& S_{4, 1, 1}& S_{3, 3}& S_{3, 2, 1}& S_{2, 2, 2}  }
\right| $$

\smallskip
How to interpret the determinant meant to express the Macdonald polynomial
$M_I$?  Each entry $[J]$ must be simply replaced by the difference 
 $[|J|] -[|I|]$, and $S_J$ must be interpreted as the Schur function
$S_J( X^{tq})$. 

For example, 
 
$$ M_{2,2} =  
\left| \matrix{  [4, 0]&-[0,4]&  0\cr 0&[3,1]&-[1,3] \cr S_{4}& S_{3, 1}& S_{2, 2} } \right| = 
\left| \matrix{
[|2,2|]-[|4, 0|]&[|0,4|]-[|2,2|]& 
0\cr 0&[|2,2|]-[|3,1|]&[|1,3|]-[|2, 2|]
\cr 
S_{4}(X^{tq})& S_{3, 1}(X^{tq})& S_{2, 2}(X^{tq}) }
\right| $$

and more explicitely 
$$ M_{2,2} = 
\left|  \matrix{
(1-q^2)(q^2t-1)& (1-q^2)(t-q^2)& 
0\cr 0& q(1-q)(qt-1)& q(q-1)(t-1) \cr
S_{4}(X^{tq})& S_{3, 1}(X^{tq})& S_{2, 2}(X^{tq})  } \right| $$

\puce {\bf Associativity again} \quad 
As for Hall-Littlewood polynomials, one wants to retain only the 
associative algebra underlying the indices of a choosen basis of symmetric
polynomials, at this stage, Macdonald polynomials (that we shall 
index by the conjugate increasing partitions and note $N_J$, in this section).


Let $I$ be an increasing vector: $I=[i_1,\ldots,\, i_n]$, $0\leq i_1\leq
\cdots \leq i_n$.  We have to interpret $I$ as a sequence of operators
such that the image of $1=N_0$ under 
$B_{i_1}$ is $N_{i_1}$, the image of $N_{i_1}$
under $B_{i_2}$ is $N_{i_1,i_2}$, \& c. ,
the image of $N_{i_1,i_2,\ldots, i_{n-1}}$ under $B_{i_n}$
is $N_I$.

It happens that there are algebraic operators on the ring of polynomials
satisfying such properties, they have been defined and baptized 
{\it creation operators} by [LL] and [KN]. 

These operators were originally defined as symmetric functions in some
distinguished elements of the affine Hecke algebra due to Cherednik [Ch].
However, as Macdonald's operators, they can be factorised, one of the
factor being the Euler-Poincar\'e characteristic.  The degree  raising step
 just consists into multiplication by a factor $x_1\cdots x_k$. 
One needs moreover to combine the  operation $x_1 \mapsto qx_1$ 
with circular permutations of variables. Thus let 
$$ \zeta_k : [x_1,\ldots,\, x_k,\, x_{k+1},\ldots,\, x_n]
\mapsto [qx_k,\,  x_1, \ldots,\, x_{k-1},\, x_{k+1},\ldots,\, x_n]$$
and define, for every positive integer $k$ the operator $B_k$ 
$$ \Sym \ni f \buildrel {B_k}\over {\longrightarrow}
\chi\Bigl( (x_1\cdots x_k) (1-t\zeta_k)
\cdots (1-t^k\zeta_k) f \Bigr) \ .$$

The same type of computations as for Macdonald's operators give

\noindent {\bf Theorem} \ Let $k$ be  a positive integer, $I$ a partition in
$N^k$, $J$ its (increasing) conjugate. Then

1) $B_k( S_I(X^{tq}))$ is the determinant obtained from the one expressing
$S_I(X^{tq})$ by replacing in each entry $S^i(X^{tq})$ in column $h$, 
$h=1,\ldots,k$, by $(1-t^{k-h}q^{i+1}) S^(X^{tq})$.

2) $ B_k(N_J)= N_{J,k} $, i.e. the image is the Macdonald polynomial indexed
by the concatenation of $J$ and $k$.

For example, still noting $S^i$ instead of $S^i(X^{tq})$, one has 

$$ B_3( S_{421}) = B_3 \left( 
\matrix{ S^4 & S^5 &S^6\cr S^1 &S^2 &S^3\cr S^{-1} &S^0 &S^1\cr   }
\right)=
\left|\matrix{ (1-t^2q^5)S^5 & (1-tq^6)S^6 & (1-q^7)S^7\cr
(1-t^2q^2)S^2 & (1-tq^3)S^3 & (1-q^4)S^4\cr
(1-t^2q^0)S^0 & (1-tq^1)S^1 & (1-q^2)S^2\cr}
  \right| $$




\puce {\bf Back to the \car : q=0} \quad 
In the preceeding determinants, we have to put $q=0$
(after dividing in each row by the minimum power of q)
 to recover the case of interest for the \car. 

For example,
$$ M_{221} = 
\left|\matrix{ 
[5, 0, 0]&-[0, 5, 0]& 0& [0, 0, 5]& 0\cr 0& 
[4, 1, 0]&-[1, 4, 0]&-[0, 4, 1]& [1, 0, 4]\cr 0& 0& [3, 
2, 0]&-[3, 0, 2]&-[2, 0, 3]\cr0& 0& 0& [3, 1, 1]&-[1, 3
, 1]\cr S_{5}& S_{4, 1}& S_{3, 2}& S_{3, 1, 1}& S_{2, 2,1} }
\right| $$ 
specializes to 
$$ \left|\matrix{ 
-t-1& t^{2}+1& 0& -t-t^{2}& 0\cr 0& -1& 1& t^{2
}& -t\cr 0& 0& -1& t&t \cr 0& 0& 0& -t& t^{2}\cr S_{
5}& S_{4, 1}& S_{3, 2}& S_{3, 1, 1}& S_{2, 2, 1} }\right|
=
\matrix{ (t^2+t)S_{2, 2, 1} +t^{2}{(t+1)}^{2}S_{3, 2}+\cr 
+t^{2}(t+1)S_{3,1,1}+t^{5}(t+1)S_5 +t^{3}{(t+1)}^{2}S_{4, 1}  }$$

The specialized determinant can be written {\it \`a vue}: the powers of 
$t$ just record the positions of the minimum components in the vectors
$[J]$ and $[I]$.  For example, $[8,1,8,8,1,8]$ would give $t+t^4$ 
(counting the positions of the 1's from the right, starting from 0).

However, the preceeding determinant is not equal to the Hall-Littlewood function  $Q_{221}$, because there is a parasitic factor ( 
$=t^2+t$), which however is easily eliminated, knowing that 
$Q_I= S_I  + \cdots $.

\smallskip 
Of course, instead of "crystalizing" the flag variety at $q=0$,
it would be better to explain why geometry needs the extra parameter $q$.

\smallskip
This, I cannot explain. 

\puce {\bf Appendix : reading tables} \ Let $G_{q,r}(V)$ be a Grassmann
variety, $Q$ its tautological quotient vector bundle, $R$ its tautological
kernel. The relative cotangent bundle is isomorphic to $R\otimes Q^\vee$,
and $\O(1) :=\wedge^q(Q)$ is the line bundle associated to the Pl\"ucker
 embedding. Cauchy formula allows to decompose the exterior powers of the
cotangent bundle:
$$ \wedge^j( R\otimes Q^\vee) \simeq \oplus_{J: |J|=j}\  S_J(R)
\otimes S_{J'}(Q^\vee)  \ ,$$
where $J'$ denotes the partition conjugate to $J$. 

Bott's theorem gives the cohomology of each individual module 
$S_J(R) \otimes S_{J'}(Q^\vee) \otimes O(k)$~: it is null or an irreducible
representation of $Gl(q+r)$.  Computing in the Grothendieck ring of the 
base variety $\M$, simultaneously for all partitions $J$, one loses no
information. Now, with the same notations as for the flag manifold,
one has the following equality in the Grothendieck ring of $G_{q,r}(V)$ :
$$\sum_j y^j [ \wedge^j( R\otimes Q^\vee)] = 
\prod_{ 1\leq i\leq q, q+1\leq h\leq q+r} (1 +y {x_h\over x_i}) $$
and $[\O(k)] = (x_1\cdots x_q)^k$. Therefore, 
$$ \sum_{i,j} y^j H^i(\Omega^j\otimes \O(k)) = 
\sum_{i,J} y^j H^i(S_J(R) \otimes S_{J'}(Q^\vee) \otimes \O(k)) =
\chi( (x_1\cdots x_q)^k)= Q_{k^q} $$



Because the Hall-Littlewood functions occuring in this problem are special,
one has in fact a direct algorithm to obtain them
(in general, the expansion of Hall-Littlewood polynomials
 in the basis of Schur functions can  be obtained
by putting $t=0$ in the  determinants expressing Macdonald's polynomials). 
It amounts to enumerate
all possible words $w$ of length $q+r$ in the two letters alphabet $\{ 0,k\}$
satisfying the condition that there exists no $i$ such that
 $w_i=0, w_{i+k}=k$ (otherwise the corresponding Schur determinant would
have two identical rows).

For example, if $q=r=k=3$, then the tables give 
$$Q_{333}= S_{333} -y S_{3321} +y^2 S_{3222} + y^2 S_{33111} - y^4 S_{22221}
 -y^4 S_{321111} +y^5 S_{222111} \ . $$

The coefficient of $(-y)^4$ is  $S_{22221}+S_{321111}$, which means that
the space of sections of differential forms of order 4, twisted by
$\O(3)$, is a representation of $Gl(6)$ which breaks into two irreducible
components indexed by the partitions $22221$ and $321111$. 


 However the above expansion exactly corresponds to the easier enumeration of 
all words with three "0" and three "3" which do not possess any factor of
length 4 of the type $[0,\star,\star,3]$, that is to the set 
$$ [3,3,3,0,0,0], [[3,3,0,3,0,0], [3,0,3,3,0,0], [3,0,3,0,0,3],
[0,3,3,0,3,0], [0,3,3,0,0,3] \ .$$

To obtain easily such sets of words, one can have recourse to the 
non commutative technics of automata theory. 




  
\vskip 5mm
\puce {\bf Bibliography}

\vskip 3mm\parindent 0mm

{\bf [At]} \ M.F. Atiyah, {\it Collected Works}

{\bf [Ch]}\ I. Cherednik, {\it Non symmetric Macdonald polynomials},
Intern. Math. Res. Notices {\bf 10} (1995) 483--515. 

{\bf [DKLLST]}\ G. Duchamp, D. Krob, A. Lascoux, B. Leclerc, T. 
Scharf et J.Y. Thibon, {\it Euler-Poincar\'e characteristics and polynomial 
representations of Iwahori-Hecke algebras}, 
  Pub RIMS  (Kyoto), 1995, pp. 179-201.

{\bf [Hi]}\  F. Hirzebruch,  {\it Collected Works}.

{\bf [HBJ]}\  F. Hirzebruch, T. Berger, R. Jung, {\it Manifolds and modular
forms}, Aspects of Mathematics, Vieweg (1992).

{\bf [KN]}\ A. Kirillov, M. Noumi, {\it Affine Hecke algebras and  raising
 operators for Macdonald polynomials },
to appear in Duke Math. 

{\bf [LL]}\ L. Lapointe, L. Vinet, {\it A short proof of the integrality of the
Macdonald $(q,t)$-Kostka coefficients}, {\bf 91} (1998) 205--214. 

{\bf [LLT]}\  A. Lascoux, B. Leclerc et J.Y. Thibon,
{\it Flag Varieties and the Yang-Baxter Equation},
Letters in Math. Phys. {\bf 40} (1997) 75--90.

{\bf [LS1]}\ A. Lascoux, M.P. Sch\"utzenberger,
 Symmetry and Flag manifolds, 
{\it in Invariant Theory, Springer L.N.} {\bf 996} (1983) 118-144.

{\bf [LS2]} \ A. Lascoux, M.P. Sch\"utzenberger,  {\it Alg\`ebre des
diff\'erences divis\'ees}, Discrete Maths {\bf  99} (1992) 165-179.


{\bf [Li]}\ D.E. Littlewood, {\it On certain symmetric functions},
Proc. London M.S. {\bf 43} (1961) 485--498. 

{\bf [Ma]}\ I.G. Macdonald, {\it Symmetric functions and Hall Polynomials},
$2^{nd}$ ed., Oxford Math. Monograph, Clarendon, Oxford (1995).

{\bf [Sn]}\ D.M. Snow, {\it Cohomology of twisted holomorphic forms
on Grassmann manifolds and quadric hypersurfaces}, 
Math. Ann. {\bf 276} (1986) 159--176.

{\bf [Ya]} \ C.N. Yang, {\it Some exact results for the many-body problem
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Phys. Rev. Lett {\bf 19} (1967) 1312--1315. 

\bye




  

