Jean-Christophe Aval
email: aval@math.u-bordeaux.fr

On certain spaces of lattice diagram determinants

. We recall that ML denotes the space of all partial derivatives of ∆L. In this paper, we want to study the space M k i,j (X, Y ) which is defined as the sum of ML spaces where the lattice diagrams L are obtained by removing k cells from a given partition, these cells being in the "shadow" of a given cell (i, j) in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space M k i,j (X, Y ), that we conjecture to be optimal. This dimension is a multiple of n! and thus we obtain a generalization of the n! conjecture. Moreover, these upper bounds associated to nice properties of some special symmetric differential operators (the "shift" operators) allow us to construct explicit bases in the case of one set of variables, i.e. for the subspace M k i,j (X) consisting of elements of 0 Y -degree.

1 Introduction Definition 1.1 A lattice diagram is a finite subset of N × N. For µ 1 ≥ µ 2 ≥ • • • ≥ µ k > 0, we say that µ = (µ 1 , µ 2 , . . . , µ k ) is a partition of n if |µ| = µ 1 + • • •+ µ k equals n.
We associate to a partition µ its Ferrers diagram {(i, j) : 0 ≤ i ≤ k -1, 0 ≤ j ≤ µ i+1 -1} and we shall use the symbol µ for both the partition and its Ferrers diagram.

Most definitions and conventions we use are similar to [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF]. For example, given the partition (4, 2, 1), its partition diagram is 2,0 1,0 1,1 0,0 0,1 0,2 0,3 . It consists of the lattice cells {(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (0, 2), (0, 3)}.

Let now X = X m = {x 1 , x 2 , . . . , x m } and Y = Y m = {y 1 , y 2 , . . . , y m } be two sets of m variables and Q[X] = Q[x 1 , x 2 , . . . , x m ] and Q[X, Y ] = Q[x 1 , x 2 , . . . , x m , y 1 , y 2 , . . . , y m ] denote respectively the rings of polynomials in m and 2m variables with rational coefficients. Since we have to deal with polynomials in Q[X] or Q[X, Y ], we shall denote by Z a subalphabet of (X, Y ) and by Q[Z] the corresponding ring of polynomials. Definition 1.2 Given a lattice diagram L = {(p 1 , q 1 ), (p 2 , q 2 ), . . . , (p n , q n )} with n cells we define the lattice determinant ∆ L (X, Y ) = det x pj i y qj i 1≤i,j≤n .

(1.1)

The

polynomial ∆ L (X, Y ) ∈ Q[X n , Y n ] = Q[X, Y ]
(with m = n, the number of cells in the diagram L) is different from zero only if the diagram L consists of n distinct cells in the positive quadrant. In this case ∆ L is bihomogeneous of degree |p| = p 1 + • • • + p n in X and of degree |q| = q 1 + • • • + q n in Y . To insure that this definition associates a unique determinant to L we require that the list of lattice cells is given with respect to the lexicographic order with priority to the second entry that is to say:

(p 1 , q 1 ) < (p 2 , q 2 ) ⇐⇒ q 1 < q 2 or [q 1 = q 2 and p 1 < p 2 ].

(1.2) Definition 1.3 For a polynomial P (Z) ∈ Q[Z], the vector space spanned by all the partial derivatives of P of all orders is denoted L ∂ [P ], i.e.:

L ∂ [P ] = Q[∂Z], (1.3)
where for a polynomial Q in Q[Z], Q(∂) = Q(∂Z) denotes the differential operator obtained by substituting x i and y i respectively by ∂x i and ∂y i in the expression of Q. Next we define

M L = L ∂ [∆ L (X, Y )] (1.4)
the vector space associated to the lattice diagram L.

A permutation σ ∈ S n acts diagonally on a polynomial P (X, Y ) ∈ Q[X n , Y n ] as follows: σP (X, Y ) = P (x σ1 , x σ2 , . . . , x σn , y σ1 , y σ2 , . . . , y σn ). Under this action, ∆ L (X, Y ) is clearly an alternant. It follows that for any lattice diagram L with n cells, the vector space M L is an S n -module. Since ∆ L (X, Y ) is bihomogeneous, this module affords a natural bigrading.

The most general problem considered in [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF] and [START_REF] Bergeron | Multiple Left Regular Representations Associated with Alternants of the Symmetric Group[END_REF] concerns the space M L . The main question is to decide whether this space is S n -isomorphic to a sum of left regular representations or not. In [START_REF] Bergeron | Multiple Left Regular Representations Associated with Alternants of the Symmetric Group[END_REF], the case where all the lattice cells of L lies on a single axis is solved. In the particular case where L corresponds to a partition µ the question leads to the "n! conjecture" which asserts that the space M µ is a single copy of the left regular representation. Many efforts to prove this conjecture were only sufficient to obtain it in some special cases (see [START_REF] Allen | The decomposition of a bigraded left regular representation of the diagonal action of S n[END_REF], [START_REF]Monomial bases related to the n! conjecture[END_REF], [START_REF] Garsia | Some natural bigraded S n -modules and q, t-Kostka coefficients[END_REF], [START_REF] Garsia | Orbit Harmonics and Graded Representations[END_REF] for example).

The next class of lattice diagrams that is of interest is obtained by removing a single cell from a partition diagram. Its interest comes in part from the fact that it gives a possible recursive approach for the n! conjecture, with the statement of a conjectural "four term recurrence." If µ is a partition of n + 1, we denote by µ/ij the lattice diagram obtained by removing the cell (i, j) from the Ferrers diagram of µ. We refer to the cell (i, j) as the hole of µ/ij. It is conjectured in [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF] that the number of copies of the left regular representations in M µ/ij is equal to the cardinality (which we denote by s µ (i, j) or by s if there is no ambiguity) of the (i, j)-shadow, where the shadow of a cell (i, j), as shown on the figure below is:

S µ ((i, j)) = {(i ′ , j ′ ) ∈ µ : i ′ ≥ i, j ′ ≥ j}. i,j
A study of the subspace M µ/ij (X) of M µ/ij consisting of elements of 0 Ydegree can be found in [START_REF] Aval | Lattice diagram polynomials in one set of variables[END_REF], in which the corresponding "four term recursion" is proven by using the construction of explicit bases.

The aim of this article is to propose a generalization for the n! conjecture. The space that we consider is defined as follows. Let µ be a partition of n + k. This partition is fixed and does not appear in the following notations. Definition 1.4 Let M k i,j denote the following sum of vector spaces

M k i,j = M k i,j (X, Y ) = (a1,b1),...,(a k ,b k ) M µ/{(a1,b1),...,(a k ,b k )} , (1.5)
where the sum is over all the k-tuples of cells in the shadow of (i, j).

We first observe that because of the "shift" operators (see [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF], Proposition I.3 or Section 2 in this paper) we have M µ/ij = M 1 i,j (equation 2.20). Hence this space M k i,j is a possible generalization of M µ/ij if we want to make k holes in the Ferrers diagram. The object of this paper is to show the interest of the space M k i,j and to give support to the Conjecture 3.8 that dim M k i,j = s k n!. The organization of the article is the following. In the second section we introduce some "shift" operators which are useful to move the holes and the cells in the diagrams. The third section is devoted to the proof of an upper bound ( s k n!) for the dimension of M k i,j , that is conjectured to be optimal. In the fourth section we study M k i,j (X), the subspace of M k i,j (X, Y ) consisting of elements of 0 Y -degree, for which we obtain explicit bases.

The "shift" operators

In this paragraph, we want to describe the action of some special symmetric differential operators on the determinants ∆ L . We recall the following definitions as stated in [START_REF] Macdonald | Symmetric Functions and Hall Polynomials, Second Edition[END_REF]: Definition 2.1 For each integer r ≥ 1, the r-th power sum P r (X) (we do not use the classical notation p r to avoid a possible confusion with the biexponents) is defined by P r (X) = x r i .

(2.1)

For each integer r ≥ 0, the r-th elementary symmetric function e r (X) is the sum of all products of r distinct variables x i , so that e 0 = 1 and for r ≥ 1:

e r (X) = i1<•••<ir x i1 • • • x ir . (2.2)
For each integer r ≥ 0, the r-th complete symmetric function h r (X) is the sum of all monomials of total degree r in the variables x i , so that:

h r (X) = i1≤•••≤ir x i1 • • • x ir . (2.3)
For the sake of simplicity, we only state the following propositions for X-shifts.

Of course similar results also hold for Y -shifts. The only difference concerns the signs. The choice of the lexicographic order 1.2 is made to simplify the results and the proofs for X-shifts.

Proposition 2.2 Let L be a lattice diagram. Then for any integer k ≥ 1 we have

P k (∂X)∆ L (X, Y ) = n i=1 ±ǫ(L, P k (i; L))∆ P k (i;L) (X, Y ), (2.4)
where P k (i; L) is the diagram obtained by replacing the i-th biexponent (p i , q i ) by (p i -k, q i ) and the coefficient ǫ(L, P k (i; L)) is a positive integer. The sign in 2.4 is the sign of the permutation that reorders the obtained biexponents with respect to the lexicographic order 1.2.

Proof. This is a particular case of Proposition I.1 in [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF], but we shall give here a simple proof because some ingredients will be useful later.

If the diagram L consists of the cells L = {(p 1 , q 1 ), . . . , (p n , q n )}, we can develop the determinant ∆ L with respect to the j-th column and write:

∆ L (X, Y ) = n i=1 x pi j y qi j .A i,j (2.5)
where A i,j denotes the cofactor (i, j). Let us remark that this cofactor is a polynomial where the variable x j does not appear. Thus when we derive 2.5, we obtain:

∂x k j ∆ L (X, Y ) = n i=1 c k i x pi-k j y qi j .A i,j (2.6)
where

c k i = p i (p i -1) • • • (p i -k + 1
). Next we sum 2.6 over j to get:

P k (∂X)∆ L (X, Y ) = n i=1 c k i n j=1 x pi-k j y qi j .A i,j . (2.7)
Thus we obtain 2.4 by recognizing in 2.7 the development (up to sign) of ∆ P k (i;L) . As a biproduct we observe that ǫ(L, P k (i;

L)) = c k i = p i (p i -1) • • • (p i - k + 1
) and that this coefficient does not depend on the operator P k .

Remark 2.3 The diagram P k (i; L) is the diagram obtained by pushing down the i-th cell of L: its biexponent (p i , q i ) is replaced by (p i -k, q i ) which corresponds to k steps down. The other biexponents are unchanged. This duality between the substractions on the set of biexponents and the movements of cells in the diagram will be extensively employed throughout this article, explicitly or implicitly.

Observe also that since ∆ L ′ = 0 only if L ′ consists of n distinct cells in the positive quadrant, we can forget all the terms in the sum 2.4 but those relative to such diagrams. Proposition 2.4 Let L be a lattice diagram. Then for any integer k ≥ 1 we have

e k (∂X)∆ L (X, Y ) = 1≤i1<i2<•••<i k ≤n ǫ(L, e k (i 1 , . . . , i k ; L))∆ e k (i1,...,i k ;L) (X, Y ) (2.8)
where e k (i 1 , . . . , i k ; L) is the lattice diagram obtained by replacing the biexponents (p i1 , q i1 ), . . . , (p i k , q i k ) by (p i1 -1, q i1 ), . . . , (p i k -1, q i k ) and where the coefficient ǫ(L, e k (i 1 , . . . , i k ; L)) is a positive integer.

Proof. The proof is almost the same as for the previous proposition. We write

e k (X) = 1≤j1<•••<j k ≤n x j1 . . . x j k .
(2.9)

We develop the determinantal form of ∆ L with respect to the columns j 1 , . . . , j k to obtain the following expression where ∆ i1,...,i k L denotes the lattice diagram determinant relative to the biexponents i 1 , . . . , i k of L and A i1,...,i k ;j1,...,j k the cofactor:

∆ L = 1≤i1<•••<i k ≤n ∆ i1,...,i k L (x j1 , . . . , x j k )A i1,...,i k ;j1,...,j k .
(2.10)

Next we derive 2.10 to obtain

∂(x j1 . . . x j k )∆ L = 1≤i1<•••<i k ≤n c i1,...,i k ;j1,...,j k ∆ i1,...,i k e k (i1,...,i k ;L) (x j1 , . . . , x j k ) ×A i1,...,i k ;j1,...,j k , (2.11)
where c i1,...,i k ;j1,...,j k is a positive integer. We see that c i1,...,i k ;j1,...,j k is equal to p i1 • • • p i k and thus does not depend on j 1 , . . . , j k . Therefore we can omit the subscript j 1 , . . . , j k . Thus we get

e k (∂X)∆ L = (1≤i1<•••<i k ≤n) (1≤j1<•••<j k ≤n) c i1,...,i k ∆ i1,...,i k e k (i1,...,i k ;L) (x j1 , . . . , x j k ) ×A i1,...,i k ;j1,...,j k . (2.12)
By recognizing in 2.12 the development of ∆ e k (i1,...,i k ;L) , we finally obtain the expected formula. The sign in front of the coefficient ǫ(e k ; i 1 , . . . , i k ; L) should be the sign of the permutation that reorders the obtained biexponents in increasing lexicographic order. In fact the choice of the lexicographic order 1.2 is such that this permutation is always the identity: each cell stays in its original column and no one of them "jumps" over another one, so that the order is unchanged.

: if L = {(p 1 , q 1 ), . . . , (p n , q n )} and L ′ = {(p ′ 1 , q ′ 1 ), . . . , (p ′ n , q ′ n )},
ǫ is given by the following formula:

ǫ(L, L ′ ) = n i=1 p i !q i ! n i=1 p ′ i !q ′ i ! . (2.13)
This coefficient is a positive integer that appears (up to sign) as the coefficient of ∆ L ′ in the expression of P (∂X)∆ L , where P is a power sum or an elementary symmetric function ; we shall see in the next proposition that it is also the case for homogeneous symmetric functions. Another important remark is that we have to be careful when we apply products of differential operators. Indeed in this case multiplicities may appear in the formulas. Let P (∂) and Q(∂) be two differential operators such that formulas like 2.4 or 2.8 hold for P (∂) and Q(∂) with ǫ given by 2.13. We first observe that ǫ is multiplicative, i.e.

ǫ(L, L ′ ) = ǫ(L, L ′′ )ǫ(L ′′ , L ′ ), (2.14)
for L, L ′′ and L ′ three diagrams. Thus the coefficient of ∆ L ′ in P (∂)Q(∂)∆ L is a multiple (up to sign for power sums) of ǫ(L, L ′ ). This multiplicity corresponds to the number of choices in the order of the different shifts, that is to say the number of diagrams L ′′ such that L ′′ appears in Q(∂)∆ L and L ′ appears in P (∂)∆ L ′′ . This multiplicity is denoted by c P,Q (L ′ L ′ ) Let us take an example: if we apply e 1 (∂X)e 1 (∂X) to the determinant of the diagram L = {(1, 0), (1, 1)}, we obtain a single diagram L ′ = {(0, 0), (0, 1)}, with ǫ(L, L ′ ) = 1, but

e 1 (∂X)e 1 (∂X)∆ L = 2∆ L ′ .
(2.15)

The multiplicity 2 correponds to the fact that we can either first move down the cell (1, 0) and next the cell (1, 1) or do it in the reverse order. All these observations are crucial to well understand the proof of the following proposition. Now, to state the next proposition, we need to introduce some notation. For a lattice diagram L, we denote by L its complement in the positive quadrant (it is an infinite subset). Again we order L = {(p 1 , q 1 ), (p 2 , q 2 ), . . . } using the lexicographic order 1.2. Proposition 2.6 Let L be a lattice diagram. Then for any integer k ≥ 1 we have

h k (∂X)∆ L (X, Y ) = 1≤i1<i2<•••<i k ǫ(L, h k (i 1 , . . . , i k ; L))∆ h k (i1,...,i k ;L) (X, Y ) (2.

16)

where h k (i 1 , . . . , i k ; L) is the lattice diagram with the following complement diagram. Replace the biexponents (p i1 , q i1 ), . . . , (p i k , q i k ) of the complement L with (p i1 + 1, q i1 ), . . . , (p i k + 1, q i k ) and keep the other unchanged. The coefficient ǫ(L, h k (i 1 , . . . , i k ; L)) is a positive integer, given by formula 2.13.

Proof. We shall prove this proposition by induction on k. If k = 1, then h 1 = e 1 and the result is true since moving down a cell is equivalent to moving up a hole. Assume the result is true up to k -1. Then we use the fact that

h k = e 1 h k-1 -e 2 h k-2 + • • • + (-1) k e k-1 h 1 + (-1) k+1 e k .
Each term e l h k-l for 1 ≤ l ≤ k gives a linear combination of ∆ L ′ , whose coefficients are multiple of ǫ(L, L ′ ) according to Remark 2.5. The problem is to compute the alternating sum of all these coefficients to get the result of h k (∂X)∆ L .

Let L ′ be one of the diagrams created by the terms e l h k-l . The coefficient of ∆ L ′ in e l (∂X)h k-l (∂X) is equal to c e l ,h k-l (L, L ′ )ǫ(L, L ′ ). In this proof let us denote c e l ,h k-l (L, L ′ ) simply by c l (L, L ′ ). The question is to compute 1≤l≤k (-1) l+1 c l (L, L ′ ).

(2.17)

Let k ′ ≤ k be the number of distinct holes moving between L and L ′ and d ≤ k ′ the number of those which have a moving hole below them. Each of these d holes has to move with h k-l (∂X) because the hole below it is able to move up with e l (∂X) only if it has a cell above itself. The choice therefore comes from the k ′ -d other holes which can either move up with h k-l (∂X) or not: we choose k -l -d among them to move with h k-l (∂X). Thus we get

c l (L, L ′ ) = k ′ -d k -l -d = k ′ -d l -(k -k ′ ) . (2.18)
And the sum in 2.17 becomes

k l=1 (-1) l+1 k ′ -d l -(k -k ′ ) = 1 if k ′ = k, 0 if k ′ < k. (2.19)
Thus we get the desired formula 2.16.

Remark 2.7 One efficient application of the previous proposition is to give a necessary condition that tests if a partial symmetric operator belongs to the vanishing ideal of a lattice diagram determinant (see [START_REF] Aval | Vanishing ideals of lattice diagram determinants[END_REF]). An example of the strength of this principle is to give immediate proofs of Propositions 1-2-3-4 of [START_REF]Monomial bases related to the n! conjecture[END_REF] (these propositions provide a Groebner basis of the vanishing ideal of ∆ µ when µ is a hook). The previous proofs in [START_REF]Monomial bases related to the n! conjecture[END_REF] were recursive and intricate but the results now become simple applications of Proposition 2.6.

Remark 2.8

The shift operators are also useful to reduce the sum 1.5 defining M k i,j . In the special case of one hole, it is now easy to see that

M 1 i,j = M µ/i,j . (2.20)
Indeed we have that for any integer k and l e k (∂X)e l (∂Y )∆ µ/i,j = c.∆ µ/i+k,j+l ,

with c an integer different from zero. This implies M 1 i,j ⊆ M µ/i,j , and the reverse inclusion is obvious.

In the particular case of two holes, let k and l be positive integers and let us use the following notations: for two cells h 1 and h 2 , ǫ i h1,h2 = ǫ µ/{(i, j), (i + 1, j)}, µ/{h 1 , h 2 } and ǫ j h1,h2 = ǫ µ/{(i, j), (i, j + 1)}, µ/{h 1 , h 2 } . If we are careful of the different signs by applying Propositions 2.2 and 2.4, then we get the following identities P l (∂Y )e k-1 (∂X)∆ µ/{(i,j),(i+1,j)} = P l (∂Y ) ǫ i (i,j),(i+k,j) ∆ µ/{(i,j),(i+k,j)} = (-1) b+h ǫ i (i+k,j),(i,j+l) ∆ µ/{(i+k,j),(i,j+l)} +(-1) b+v+1 ǫ i (i,j),(i+k,j+l) ∆ µ/{(i,j),(i+k,j+l)} (2.22) and = (-1) b+h+1 (-1) h ǫ j (i+k,j),(i,j+l) ∆ µ/{(i+k,j),(i,j+l)} +(-1) v ǫ j (i,j),(i+k,j+l) ∆ µ/{(i,j),(i+k,j+l)} ,

P k (∂X)e l-1 (∂Y )∆ µ/{(i,j),(i,j+1)} =P k (∂X) (-1) b+h+1 ǫ j (i,
where h, v and b are respectively the numbers of cells with horizontal, vertical and both horizontal and vertical stripes in the figure above (we have to compute the sign of the permutation which reorders the cells in the lexicographic order 1.2). By observing that the product of the signs of the four coefficients in 2.22 and 2.23 is (-1) 2(2b+2h+v+1)+1 = (-1) we have that exactly three coefficients in 2.22 and 2.23 are of the same sign, whence ∆ µ/{(i,j),(i+k,j+l)} and ∆ µ/{(i+k,j),(i,j+l)} are in M µ/{(i,j),(i,j+1)} + M µ/{(i,j),(i+1,j)} .

Next, by Proposition 2.6 we can move simultaneously the two holes. This implies that for any couple of holes (h 1 , h 2 ) in the shadow of (i, j) then ∆ µ/{h1,h2} ∈ M µ/{(i,j),(i,j+1)} + M µ/{(i,j),(i+1,j)} thus M 2 i,j = M µ/{(i,j),(i,j+1)} + M µ/{(i,j),(i+1,j)} .

(2.24)

The question of whether the obvious generalization of the previous result 2.24 is true when k ≥ 3 appears naturally. Is it sufficient to take only the diagrams such that the holes form a partition of origin (i, j)? The answer is negative. For example it is easy to check (by computer) that when µ = (3, 2), ∆ µ/{(0,0),(1,0),(0,2)} ∈ M µ/{(0,0),(1,0),(0,1)} + M µ/{(0,0),(0,1),(0,2)} .

(2.25)

3 The upper bound

Definition 3.1 Let M be a vector subspace of Q[Z]
where Z is a subalphabet of (X n+k , Y n+k ). We define its vanishing ideal as the following ideal:

I M = {P ∈ Q[Z] : ∀Q ∈ M, P (∂)Q = 0}. ( 3

.1)

If P (∂)Q = 0, we shall say that P "kills" Q.

If M = L ∂ [P ] then we denote its vanishing ideal simply by I P . In the case of M k i,j we denote I M k i,j by I k i,j .

We recall the following important result ([8], Proposition 1.1):

Proposition 3.2 For M a subspace of Q[Z],
we have

M = I ⊥ M = {P ∈ Q[X n , Y n ] : ∀Q ∈ I M , P, Q = 0}, (3.2)
where the scalar product is defined by P, Q = L 0 (P (∂)Q) and where L 0 is the linear form that associates to a polynomial its term of degree 0.

About ideals

We want here to prove the following Proposition 3.3

I k i,j = (a1,b1),...,(a k ,b k ) I ∂x a 1 n+1 ∂y b 1 n+1 •••∂x a k n+k ∂y b k n+k ∆µ ∩ Q[X n , Y n ] def = I, (3.3)
where the intersection is over the k-tuples of different cells in the shadow of (i, j) that we assume to be ordered in lexicographic order.

Proof. Let (a 1 , b 1 ), . . . (a k , b k ) be k cells in S µ ((i, j)), the shadow of (i, j) in µ.

By expanding ∆ µ with respect to the last k columns, we obtain:

∆ µ (X n+k , Y n+k ) = (a ′ 1 ,b ′ 1 ),...,(a ′ k ,b ′ k ) ±∆ {(a ′ 1 ,b ′ 1 ),...,(a ′ k ,b ′ k )} ( Xn , Ȳn ) ×∆ µ/{(a ′ 1 ,b ′ 1 ),...,(a ′ k ,b ′ k )} (X n , Y n ), (3.4)
where Xn = {x n+1 , . . . , x n+k } and Ȳn = {y n+1 , . . . , y n+k }. Thus we get:

∂(x a1 n+1 y b1 n+1 • • • x a k n+k y b k n+k )∆ µ (X n+k , Y n+k ) = c∆ µ/{(a1,b1),...,(a k ,b k )} (X n , Y n ) + C (3.5)
where c is a rational constant (different from 0) and C a linear combination with coefficients in

Q[x n+1 , y n+1 , . . . , x n+k , y n+k ] of polynomials ∆ µ/{(a ′ 1 ,b ′ 1 ),...,(a ′ k ,b ′ k )} (X n , Y n ), with: ∀ 1 ≤ l ≤ k, (a ′ l , b ′ l ) ∈ S µ ((i, j)). (3.6) Indeed ∆ {(a ′ 1 ,b ′ 1 ),...,(a ′ k ,b ′ k )} ( Xn , Ȳn ) is not killed by ∂(x a1 n+1 y b1 n+1 • • • x a k n+k y b k n+k )
only if there exists at least a permutation σ ∈ S k , the symmetric group on k elements, such that

(a ′ σ(l) , b ′ σ(l) ) ∈ S µ ((a l , b l )), ∀ 1 ≤ l ≤ k. (3.7)
This follows easily from the definition of the ∆ {(a ′

1 ,b ′ 1 ),(a ′ 2 ,b ′ 2 ),...,(a ′ k ,b ′ k )} as a de- terminant: ∆ {(a ′ 1 ,b ′ 1 ),(a ′ 2 ,b ′ 2 ),...,(a ′ k ,b ′ k )} = σ∈S k sgn(σ) x a ′ σ(1) n+1 y b ′ σ(1) n+1 x a ′ σ(2) n+2 y b ′ σ(2) n+2 • • • x a ′ σ(k) n+k y b ′ σ(k)
n+k .

Taking the partial derivative ∂(

x a1 n+1 y b1 n+1 • • • x a k n+k y b k n+k ), we get 3.7. For all 1 ≤ l ≤ k, we have S µ (a l , b l ) ⊆ S µ (i, j). Consequently (a ′ σ(l) , b ′ σ(l) ) ∈ S µ (i, j), ∀ 1 ≤ l ≤ k.
Because σ is a permutation, 3.6 is now obvious.

To illustrate the equation 3.5, we give the following example:

µ = (3, 2), n = 3, k = 2, (a 1 , b 1 ) = (0, 0), (a 2 , b 2 ) = (1, 0), then ∂(x 0 4 y 0 4 x 1 5 y 0 5 )∆ µ (X 5 , Y 5 ) = ∆ µ/{(0,0)(1,0)} (X 3 , Y 3 ) + y 5 ∆ µ/{(0,0)(1,1)} (X 3 , Y 3 ) -y 4 ∆ µ/{(1,0)(0,1)} (X 3 , Y 3 ) + (-x 4 y 5 + x 4 y 4 )∆ µ/{(1,0)(1,1)} (X 3 , Y 3 ) -y 2 4 ∆ µ/{(1,0)(0,2)} (X 3 , Y 3 ) + y 4 y 5 ∆ µ/{(0,1)(1,1)} (X 3 , Y 3 ) -y 2 4 y 5 ∆ µ/{(1,1)(0,2)} (X 3 , Y 3 ).
Hence we get what we want because:

• I ⊆ I k i,j : let P be a polynomial in I. Since P kills the left-hand side of 3.5, it kills the constant term in Q[ Xn , Ȳn ] of the left-hand side which is

∆ µ/{(a1,b1),...,(a k ,b k )} (X n , Y n ). Thus P is in I k i,j .
• I k i,j ⊆ I: let P be a polynomial in I k i,j . By 3.6, P kills all the terms of the right-hand side of 3.5; thus it kills the left-hand side. This implies P ∈ I.

Sets of points and vanishing ideals

The reasoning is inspired from [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF], Theorem 4.2.

Let µ be a partition of n + k, l = µ 1 its length and h its height (the number of its positive parts). We consider two sets α = (α 0 , . . . , α h-1 ) and β = (β 0 , . . . , β l-1 ) of distinct rational numbers. To any injective tableau T of shape µ with entries {1, . . . , n + k}, we associate a point (a(T ), b(T )) in Q 2(n+k) by the following process:

∀ 1 ≤ t ≤ n + k, a t (T ) = α rt(T ) and b t (T ) = β ct(T ) (3.8)
where r t (T ) (resp. c t (T )) is the number of the row (resp. column) of T where the entry t lies in T . We think useful to recall here that the convention introduced in the Definition 1.1 is that the first row and column are indexed by 0. We define ρ as the orbit of (a, b) when T varies over the (n + k)! injective tableaux of shape µ. Let us observe that, since the α t 's and β t 's are distinct, two different tableaux give two different points, i.e. T → (a(T ), b(T )) is an injective map. We introduce J ρ the ideal of polynomials that are zero over all the orbit. We recall that the operator gr is the operator that associates to a polynomial its term of maximum degree and that the gr of an ideal is the ideal generated by the gr of its elements. Next we define I = grJ ρ and H = I ⊥ . We will use the following important result (cf. [START_REF] Garsia | Some natural bigraded S n -modules and q, t-Kostka coefficients[END_REF], Theorem 1.1):

Proposition 3.4 For any choice of α t 's and β t 's, if I is the graded ideal associated to the vanishing ideal of ρ then we have the inclusion

I ⊆ I ∆µ . (3.9)
We now look at another set, this time in Q 2n . We consider the set of tableaux T of shape µ with n entries and k white cells such that the k white cells are in the shadow of (i, j). Let us denote this set of tableaux by T k i,j . By the same process as described in 3.8, we define a set ρ k in Q 2n . We recall here that we denote by s µ (i, j) or simply by s the cardinality of the shadow of the cell (i, j) in µ. Since the cardinality of T k i,j is s k n! and the process 3.8 is still injective, the set ρ k has s k n! points. We introduce J ρ k the ideal of polynomials that are zero all over ρ k , and

I k = grJ ρ k and H k = (I k ) ⊥ .
The first information is given by the following equation:

dim H k = s k n!. (3.10)
This comes from the fact that dim

H k = dim Q[X, Y ]/J ρ k = #ρ k = s k n!.
We shall not develop this point, extensively treated in [START_REF] Garsia | Orbit Harmonics and Graded Representations[END_REF].

We want to prove that M k i,j ⊆ H k and by Proposition 3.2, it is equivalent to prove that I k ⊆ I.

Inclusion

We want here to obtain the next proposition: Proposition 3.5 We have the inclusion:

I k ⊆ I. (3.11)
Proof. Let P be a polynomial in J ρ k . Let us consider

Q(X n+k , Y n+k ) = P (X n , Y n ) × i-1 i ′ =0 (x n+1 -α i ′ ) • • • i-1 i ′ =0 (x n+k -α i ′ ) × j-1 j ′ =0 (y n+1 -α j ′ ) • • • j-1 j ′ =0 (y n+k -α j ′ ). (3.12)
We want to check that this polynomial is an element of J ρ . We take an element (a, b) = (a(T ), b(T )) of ρ. If its projection on Q 2n (by keeping the first n entries of a and b) is in ρ k then Q(a, b) = 0 because of P . If not, the tableau T must have at least one entry between n + 1 and n + k in the complement of the shadow of (i, j), i.e. in the first i rows or the first j columns and we have still

Q(α, β) = 0. Thus Q ∈ J ρ , hence gr(Q) ∈ I ∆µ .
Next by looking at the term of maximal degree we get: gr(P ) ∈ I ∂x i+1 n+1 ∂y j+1 n+1 •••∂x i+1 n+k ∂y j+1 n+k ∆µ . For any set of k cells {(a 1 , b 1 ), . . . , (a k , b k )} in the shadow of (i, j), we observe that ∀r, 1 ≤ r ≤ k, a r ≥ i and b r ≥ j. Hence gr(P ) is in I, which was to be proved.

Conclusion

The main result is now a consequence of all what precedes: Theorem 3.6 If µ is a partition of n + k and s the cardinality of the shadow of the cell (i, j), then we have: Numerical examples and the fact that the construction described in the previous subsection affords the "good" upper bound in the case of one set of variables (see the next section) support the following conjecture, which was first stated by F. Bergeron. The goal of this section is to obtain an explicit basis for M k i,j (X), the subspace of M k i,j (X, Y ) of elements of 0 Y -degree.

dim M k i,j ≤ s k n!. ( 3

Construction

We first recall results about M µ (X) the subspace of M µ of elements of 0 Ydegree (which was denoted by M 0 µ in [START_REF]Monomial bases related to the n! conjecture[END_REF] and [START_REF] Aval | Lattice diagram polynomials in one set of variables[END_REF]). When µ is a partition of n, we have

dim M µ (X) = n!/µ!, (4.1)
where

µ! = µ 1 ! • • • µ k !.
This space has been studied in [START_REF]Monomial bases related to the n! conjecture[END_REF], [START_REF] Garsia | Some natural bigraded S n -modules and q, t-Kostka coefficients[END_REF], [START_REF] Bergeron | On certain spaces of harmonic polynomials[END_REF]. Let M(µ) be a set of monomials whose cardinality is n!/µ!, such that the set B µ = {M (∂)∆ µ : M ∈ M(µ)} is a basis for the space M µ (X). By the work in [START_REF]Monomial bases related to the n! conjecture[END_REF] we know such a set exists. Now let µ be a partition of n + k. Next we choose in the Ferrers diagram µ, k cells which are simultaneously in the shadow of (i, j) and such that any circled cell has either a cell outside the partition on its right or a circled cell (see the figure below). A circled cell satisfying this condition is said to be "Right".

We denote by F k µ the set of the obtained objects, which we call Right diagrams (associated to µ).

We are now going to associate to each Right diagram two objects: a partition and a diagram with (at most) k holes in the shadow of (i, j). The Figure 1 illustrates this construction. In this figure, the chosen cells in the Right diagram are cells with a circle, in the cell (i, j) appears a + sign and the holes are as usual cells with crosses (×). In this example n = 142 and k = 10.

To a Right diagram F in F k µ we first associate µ F the partition of n obtained by pushing up the circled cells and by removing the corresponding cells (see Figure 1).

We also define a diagram µ k F with k holes by proceeding as follows. We look at the columns where a circled cell appears. In our example we have 8 such columns. For a column j ′ ≥ j where a circled cell appears, we denote by h(j ′ ) the number of places where we could have put a circled cell (of course a Right one) below the lowest circled cell of this column. In our example, we have: h(3) = 1, h(5) = 0, h(6) = 0, h(7) = 1, . . . , h(13) = 0. Next for any column j ′ with a circled cell, we do the following. We denote the positions of the circled cells in this column by (c(j ′ ), j ′ ), (c(j ′ ) + a 1 , j ′ ), . . . , (c(j ′ ) + a d , j ′ ), with (c(j ′ ), j ′ ) the position of the lowest one, 0 < a 1 < • • • < a d , and d + 1 the number of circled cells in the column j ′ . We then place holes in cells (i + h(j ′ ), j ′ ), (i + h(j ′ ) + a 1 , j ′ ), . . . , (i + h(j ′ ) + a d , j ′ ). Doing this for all columns gives the diagram µ k F . This construction is illustrated in Figure 1. The crucial idea is to apply the monomials associated to µ F to the determinant associated to µ k F and we are now able to state the main result of this section. is a basis for M k i,j (X).

The object of the end of the article is to prove this theorem. We will obtain an upper bound for the dimension of M k i,j (X), next verify that the cardinality of B k i,j (X) is equal to this upper bound, and prove that the family B k i,j (X) is linearly independent. of this column (we look only at the cells above the i-th row) and l + h is the height of the "next" column (i.e. the first on the left). We want to prove that if we put a circled cell either in this column or in the next one, its depth will be greater or equal to the preceeding ones, and that its position is unambiguous if its depth is given.

h l m c
The depth of the lowest circled cell is p = l -c -m. The highest depth that could be obtained in this column is l -c if m = 0 and l -c -1 if m > 0. In the next column the lowest depth is (it corresponds to put a circle at the top of the column): l + h -1 -c -h + 1 = l -c. Thus there is no ambiguity for the position of the next circle if its depth is given, which proves the lemma.

Remark 2 . 5 A

 25 useful observation is the fact that in the Propositions 2.2 and 2.4, the coefficient ǫ(L, L ′ ) only depends on the original diagram L and on the final diagram L ′ , but not on the differential operator. Let us clearly define this coefficient

.13) Remark 3 . 7

 37 If we recall the proof of Theorem 1.1 of[START_REF] Garsia | Some natural bigraded S n -modules and q, t-Kostka coefficients[END_REF], we observe that the previous reasoning implies the following fact. If equality holds in Theorem 3.6, then M k i,j decomposes as s k times the left regular representation.

Conjecture 3 . 8 Remark 3 . 9

 3839 With the notations of the previous theorem: When k = 1, this conjecture reduces to Conjecture I.2 of[START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF] and when s = k or k = 0 to the n! conjecture.4 Case of one set of variablesDefinition 4.1 Let M = M (X, Y ) be a subspace of Q[X, Y ]. Then we denote by M (X) the subspace of M consisting of elements of 0 Y -degree. We also denote the vanishing ideal of M (X) by I M (X).

Theorem 4 . 2

 42 With the previous notationsB k i,j (X) = {M (∂)∆ µ k F : M ∈ M(µ F ), F ∈ F k µ } (4.2)

Figure 1 :

 1 Figure 1: A Right diagram F and its associated partition µ F and diagram with k holes µ k F .

2 -d 0 1 . . . h 1 (∂) d 0 k -d 0 k- 1 to

 1101 Now let us complete the proof of Proposition 4.7. If we have a non-trivial dependence relation between the elements of B k i,j (X), we consider the greatest k-tuple of depths with respect to the lexicographic order which appears in this relation: (d 0 1 , d 0 2 , . . . , d 0 k ). This k-tuple is relative to a Right diagram F 0 . We then apply the differential operator h k (∂) d 0 1 .h k-1 (∂) d 0 the dependence relation. It kills all the terms but those which come from the single Right diagram F 0 . These terms give polynomials which are in B = {M (∂).∆ µ F 0 : M ∈ M(µ F 0 )}. They are independent since B is a basis of M µ F 0 (X).The proof of Proposition 4.7 and as a consequence of Theorem 4.2 are now complete.

Acknowledgement. The author would like to thank François Bergeron for indicating him this problem and for numerous valuable suggestions and also the referees for their efforts to improve the quality of this article.

Upper bound

Definition 4.3 We denote by T k i,j the set of injective, row-increasing tableaux of shape µ with n entries {1, . . . , n} and k white cells (without any entry) such that the k white cells are in the shadow of (i, j) and not on the left side of an entry 1, . . . , n.

We can also see T k i,j as the set of injective, row-increasing tableaux with entries 1, . . . , n of shapes all the Right diagrams F of F k µ . The following lemma will be useful in the proof of the next proposition.

and M (X) its subspace of elements of 0 Y -degree. We suppose that M is stable under derivation. Then we have the following relation between vanishing ideals:

The reverse inclusion is obtained as follows. If P is an element of I(X) and Q a polynomial in M (X, Y ), we look at the monomials of Q in Y with coefficients in Q[X]. These coefficients are elements of M (X) because M is supposed to be stable under derivation. Thus these coefficients are killed by P and so is Q itself.

The next proposition gives the analogue upper bound to Theorem 3.6 in the case of one set of variables.

Proposition 4.5 The dimension of M k i,j (X) satisfies the following inequality:

Proof. From the Proposition 3.3 and the Lemma 4.4 applied to M k i,j (X, Y ), which is of course stable under derivation, we deduce that

We consider the projection of the set ρ k on Q n , i.e. we associate to each injective tableau T of shape µ with n entries {1, . . . , n} and k holes in the shadow of (i, j) the point a |n (T ) following the process defined in 3.8. Let ρ k 0 denote this set of points and J 0 ρ k its vanishing ideal. From the definition of a |n (T ) it is clear that two tableaux give the same point if and only if they have the same entries on each line. It is therefore equivalent to associate a point a |n (T ) to each tableau T in T k i,j . In this case the correspondance is one-to-one and the number of points in ρ k 0 is precisely #T k i,j , which is also the dimension of gr(J 0 ρ k ) ⊥ (this is the analogue of 3.10).

It remains to prove the following inclusion to justify Proposition 4.5:

Let P be a polynomial in J 0

P is also in the vanishing ideal of ρ k , thus gr(P ) ∈ I k i,j and next gr(P

X). Hence we have gr(J 0 ρ k ) ⊆ I k i,j (X) and the equation 4.4 is now a consequence of Proposition 3.2.

Cardinality

We claim that: Proposition 4. [START_REF] Bergeron | Multiple Left Regular Representations Associated with Alternants of the Symmetric Group[END_REF] We have the following equality

Proof. Let h be the height of the partition µ. For a fixed Right diagram F in F k µ , the number of associated elements in

where the r t 's are the lengths of the rows of µ F because of 4.1 and 4.2. By Definition 4.3 the number of elements in

where the s t 's are the lengths of the rows of F .

It is therefore sufficient to observe that we do not change the cardinality by pushing up the circled cells. We look at the example of the lines 9, 10 and 11 of the previous example.

We observe that the lengths of the lines before the transformation are 5, 7, 6 and after the transformation 7, 6, 5. Thus the set of the lengths is unchanged. It is easy to see that it is always the case: the operation that pushes the holes up only permutes the lengths of the rows.

Independence

We want here to conclude the proof of the Theorem 4.2 by proving the independence of the set B k i,j (X).

Proposition 4.7 The set of polynomials B k i,j (X) defined in Theorem 4.2 is linearly independent. Thus in particular equality holds in Proposition 4.5.

Proof. Assume that we have a non-trivial dependence relation.

We define the depth of a hole to be the number of cells (different from holes) that are above this hole. We look at the k-tuples of the depths of the k holes of

The crux of the proof is the following result: Proof. We want to prove that the depth of the holes increases from the right to the left and from top to bottom, and that two different Right diagrams F and F ′ of F k µ give two different k-tuples of depths. We look at the circled cells with respect to this order. We refer to the next figure and look at the columns from the right to the left. In this figure, c denotes the number of circled cells in the column that we consider, m the number of positions below the lowest circled cell where we could put a circle (these cells appear with a square), l the height


Jean-Christophe Aval
email: aval@math.u-bordeaux.fr

On certain spaces of lattice diagram polynomials

. We recall that MD denotes the space of all partial derivatives of ∆D. In this paper, we want to study the space M k i,j (X, Y ) which is the sum of MD spaces where the lattice diagrams D are obtained by removing k cells from a given partition, these cells being in the "shadow" of a given cell (i, j) of the Ferrers diagram. We obtain an upper bound for the dimension of the resulting space M k i,j (X, Y ), that we conjecture to be optimal. These upper bounds allow us to construct explicit bases for the subspace M k i,j (X) consisting of elements of 0 Y -degree.

1 Introduction Definition 1.1 A lattice diagram is a finite subset of N × N. For µ 1 ≥ µ 2 ≥ • • • ≥ µ k > 0, we say that µ = (µ 1 , µ 2 , . . . , µ k ) is a partition of n if n = µ 1 + • • • + µ k . We associate to a partition µ its Ferrers diagram {(i, j) : 0 ≤ i ≤ k -1, 0 ≤ j ≤ µ i+1 } and we use the symbol µ for both the partition and its Ferrers diagram.

Most definitions and conventions we use are similar to [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF]. For example, given the partition (4, 2, 1), its partition diagram is 2,0 1,0 1,1 0,0 0,1 0,2 0,3 . It consists of the lattice cells {(0, 0), (0, 1), (0, 2), (0, 3), [START_REF] Allen | The decomposition of a bigraded left regular representation of the diagonal action of S n[END_REF]0), [START_REF] Allen | The decomposition of a bigraded left regular representation of the diagonal action of S n[END_REF][START_REF] Allen | The decomposition of a bigraded left regular representation of the diagonal action of S n[END_REF], (2, 0)}. Definition 1.2 Given a lattice diagram D = {(p 1 , q 1 ), (p 2 , q 2 ), . . . , (p n , q n )} we define the lattice determinant

∆ D (X; Y ) = det x pj i y qj i n i,j=1 ,
where X = X n = {x 1 , x 2 , . . . , x n } and Y = Y n = {y 1 , y 2 , . . . , y n }.

The polynomial ∆ D (X; Y ) is bihomogeneous of degree |p| = p 1 + • • • + p n in X and of degree |q| = q 1 + • • • + q n in Y . To insure that this definition associates a unique polynomial to D we require that the list of lattice cells be given with respect to the following "pseudo-lexicographic" order: {(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (0, 2), (0, 3)}.

For a polynomial P (X; Y ), the vector space spanned by all the partial derivatives of P of all orders is denoted L ∂ [P ]. A permutation σ ∈ S n acts diagonally on a polynomial P (X; Y ) as follows: σP (X; Y ) = P (x σ1 , x σ2 , . . . , x σn ; y σ1 , y σ2 , . . . , y σn ). Under this action, ∆ D (X; Y ) is clearly an alternant. It follows that for any lattice diagram D with n cells, the vector space

M D = L ∂ [∆ D (X; Y )] is an S n -module. Since ∆ D (X; Y ) is bihomogeneous, this module affords a natural bigrading.
The most general problem exposed in [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF] and [START_REF] Bergeron | Multiple Left Regular Representations Associated with Alternants of the Symmetric Group[END_REF] concerns the space M D . The main question is to decide whether this space is S n -isomorphic to a sum of left regular representations or not. In the particular case where D corresponds to a partition µ the question leads to the "n! conjecture" which asserts that the space M µ is a single copy of the left regular representation. Many efforts to prove this conjecture were only sufficient to obtain it in some special cases (see [START_REF] Allen | The decomposition of a bigraded left regular representation of the diagonal action of S n[END_REF], [START_REF]Monomial bases related to the n! conjecture[END_REF], [START_REF] Garsia | Some natural bigraded S n -modules and q, t-Kostka coefficients[END_REF], [START_REF] Garsia | Orbit Harmonics and Graded Representations[END_REF] for example). In [START_REF] Bergeron | Multiple Left Regular Representations Associated with Alternants of the Symmetric Group[END_REF], the case where all the lattice cells of D lies on a single axis is solved.

The next class of lattice diagrams that is of interest is obtained by removing a single cell from a partition diagram. Its interest comes in part from the fact that it gives a possible recursive approach for the n! conjecture. If µ is a partition of n + 1, we denote by µ/ij the lattice diagram obtained by removing the cell (i, j) from the Ferrers diagram of µ. We refer to the cell (i, j) as the hole of µ/ij. It is conjectured in [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF] that the number of copies of the left regular representations in M µ/ij is equal to the cardinality (which we denote by s) of the (i, j)-shadow, that is the cardinality of

{(i ′ , j ′ ) ∈ µ : i ′ ≥ i, j ′ ≥ j}.
A study of the subspace M µ/ij (X) of M µ/ij consisting of elements of 0 Ydegree can be found in [START_REF] Aval | On certain spaces of quasi-harmonic polynomials[END_REF], in which the corresponding "four term recursion" is proved by using the construction of explicit bases.

Here we study the following problem. Let µ be a partition of n + k. This partition is fixed and does not appear in the following notations. Definition 1.3 Let M k i,j denote the following sum of vector spaces

M k i,j (X, Y ) = (a1,b1),...,(a k ,b k ) M µ/{(a1,b1),...,(a k ,b k )} ,
where the sum is over all the k-tuples of cells in the shadow of (i, j).

Because of the "shift" operators (see [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF], Proposition I.3 or section 2 in this paper) we have M µ/ij = M 1 i,j . Hence this space M k i,j is a possible generalization of M µ/ij if we want to make k holes in the Ferrers diagram. That is precisely this space that we want to study. In fact we obtain an upper bound for the dimension of this object, that we conjecture to be optimal.

In the second section we introduce some "shift" operators which are useful to move the holes and the cells in the diagrams. The third section is devoted to the proof of an upper bound for the dimension of M k i,j that is conjectured to be optimal. In the fourth section we study M k i,j (X), the subspace of M k i,j (X, Y ) consisting of elements of 0 Y -degree, for which we obtain explicit bases.

The "shift" operators

For the sake of simplicity, we only settle the following propositions for X-shifts. Of course similar results also hold for Y -shifts. Proposition 2.1 Let L be a lattice diagram. Then for any integer k ≥ 1 we have

p k (∂X)∆ L (X, Y ) = n i=1 ǫ(p k (i; L))∆ p k (i,L) (X, Y )
where p k (i, L) is obtained by replacing the i-th biexponent p i , q i by p i -k, q i and the coefficient ǫ(p k (i; L)) is different from zero only if the resulting diagram consists of n distinct cells in the positive quadrant. Its sign is the sign of the permutation that reorders the obtained biexponents in increasing pseudolexicographic order.

Proof 1 This is a particular case of Proposition I.1 in [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF].

Remark 1. The diagram p k (i, L) is the diagram obtained by pushing down the i-th cell of L: its biexponent (p i , q i ) is replaced by (p i -k, q i ) which corresponds to k steps down. The other biexponents are unchanged. This duality between the substractions on the set of biexponents and movements of cells of the diagram will be extensively employed all along this article, explicitly or implicitly.

Proposition 2.2 Let L be a lattice diagram. Then for any integer k ≥ 1 we have

e k (∂X)∆ L (X, Y ) = 1≤i1<i2<•••<i k ≤n ǫ(e k (i 1 , . . . , i k ; L))∆ e k (i1,...,i k ;L) (X, Y )
where e k (i 1 , . . . , i k ; L) is obtained by replacing the biexponents (p i1 , q i1 ), . . . , (p i k , q i k ) by (p i1 -1, q i1 ), . . . , (p i k -1, q i k ) and where the coefficient ǫ(e k (i 1 , . . . , i k ; L)) is a nonnegative integer which is different from zero only when the resulting diagram consists of n distinct cells in the positive quadrant.

Proof 2

The proof is almost the same as for the previous proposition. We write

e k (X) = 1≤j1<•••<j k ≤n x j1 . . . x j k .
We develop the determinantal form of ∆ L with respect to the columns j 1 , . . . , j k to obtain the following expression where ∆ i1,...,i k L denotes the lattice diagram polynomial relative to the biexponents i 1 , . . . , i k of L and A i1,...,i k ;j1,...,j k the cofactor:

∆ L (X) = 1≤i1<•••<i k ≤n ∆ i1,...,i k L (x j1 , . . . , x j k )A i1,...,i k ;j1,...,j k .
Next we derive to obtain

∂(x j1 . . . x j k )∆ L (X) = 1≤i1<•••<i k ≤n (c i1,...,i k ;j1,...,j k ∆ i1,...,i k e k (i1,...,i k ;L) (x j1 , . . . , x j k ) × A i1,...,i k ;j1,...,j k ),
where c i1,...,i k ;j1,...,j k is a nonnegative integer. In fact c i1,...,i k ;j1,...,j k appears to be independent of j 1 , . . . , j k ; therefore we can omit the subscript j 1 , . . . , j k . Thus we get

e k (∂X)∆ L (X) = 1≤i1<•••<i k <n 1<j1≤•••<j k ≤n (c i1,...,i k ∆ i1,...,i k e k (i1,...,i k ;L) (x j1 , . . . , x j k ) × A i1,...,i k ;j1,...,j k ).
By recognizing the developpement of ∆ e k (i1,...,i k ;L) , we finally obtain the expected formula. The sign of the coefficient ǫ(e k (i 1 , . . . , i k ; L)) is the sign of the permutation that reorders the obtained biexponents in increasing pseudo-lexicographic order; in fact with the choice of the order this permutation is always the identity.

Let D be a lattice diagram. Let us denote by D ′ its complementary in the positive quadrant (it is an infinite subset). We take the convention that for any diagram D, ∆ D ′ = ∆ D (the prime name D ′ will denote the complementary of a lattice diagram D).

Proposition 2.3 With the previous notations, for any lattice diagram L and for any integer k ≥ 1, we have

h k (∂X)∆ L (X, Y ) = i1<i2<•••<i k ǫ(h k (i 1 , . . . , i k , L ′ ))∆ h k (i1,...,i k ,L ′ ) (X, Y )
where h k (i 1 , . . . , i k , L ′ ) is obtained by replacing the biexponents (p i1 , q i1 ), . . . , (p i k , q i k ) of L ′ by (p i1 + 1, q i1 ), . . . , (p i k + 1, q i k ) and the coefficient ǫ(h k (i 1 , . . . , i k , L ′ )) is a nonnegative integer which is different from zero only when the resulting L ′ consists of distinct cells in the positive quadrant.

Proof 3 We shall prove this proposition by induction on k. If k = 1, then h 1 = e 1 and the result is true since moving down a cell is equivalent to moving up a hole. Assume the result is true up to k -1. Then we use the fact that 1) k+1 e k . Indeed h k-1 pushes up (k -1) different holes. And e 1 moves down one cell, that is pushes up a hole. But if this hole has already been pushed up by h k-1 , we can view this situation as k -2 holes which have made one "step" and one which has made two steps. But this situation is killed by the term -h k-2 e 2 . Thus by looking successively at the terms in the previous formula for h k , we get what we want.

h k = h k-1 e 1 -h k-2 e 2 + • • • + (-1) k h 1 e k-1 + (-
Remark 2. One efficient application of Proposition 2.3 is to give immediate proofs of Propositions 1-2-3-4 of [START_REF]Monomial bases related to the n! conjecture[END_REF] (these propositions provide a Groebner basis of the annulator ideal of ∆µ when µ is a hook. The previous proofs were recursive and intricate but the results now become simple applications of Proposition 2.3. Remark 3. In the particular case of two holes, we obtain

e k-1 (X)p l (Y )(∂)∆ µ/{(i,j),(i+1,j)} = c 1 ∆ µ/{(i,j),(i+k,j+l)} + c 2 ∆ µ/{(i+l,j),(i,j+k)} , e l-1 (Y )p k (X)(∂)∆ µ/{(i,j),(i,j+1)} = c 3 ∆ µ/{(i,j),(i+k,j+l)} + c 4 ∆ µ/{(i+l,j),(i,j+k)}
where c 1 , c 2 , c 3 , c 4 are rational constants such that c 1 and c 2 are of the same sign and c 3 and c 4 are of opposite sign.

By Proposition 2.3 we can move simultaneously the two holes. This implies that for any couple of holes (h 1 , h 2 ) in the shadow of (i, j) then ∆ µ/{h1,h2} ∈ M µ/{(i,j),(i,j+1)} + M µ/{(i,j),(i+1,j)} thus M 2 i,j = M µ/{(i,j),(i,j+1)} + M µ/{(i,j),(i+1,j)} .

The question of the generalization of the previous result when k ≥ 3 appears spontaneously. Is it sufficient to take only the diagrams such that the holes form a partition of origin (i, j) ? The answer is negative. For example it is easy to check that when µ = (3, 2) ∆ µ/{(0,0),(1,0),(0,2)} ∈ M µ/{(0,0),(1,0),(0,1)} + M µ/{(0,0),(0,1),(0,2)} .

The upper bound

We define the annulator ideal of a vector subspace M of Q[X n , Y n ] as the following ideal:

I M = {P ∈ Q[X n , Y n ] : ∀Q ∈ M, P (∂)Q = 0}. If M = L ∂ [P ]
then we denote its annulator ideal simply by I P . In the case of M k i,j we denote I M k i,j by I k i,j . We recall the following important result ([7], Proposition 1.1): M = I ⊥ M , where the scalar product is defined by (P, Q) = L 0 (P (∂)Q) and where L 0 is the linear form that associates to a polynomial its term of degree 0.

About ideals

We want here to prove the following Proposition 3.1

I k i,j = (a1,b1),...,(a k ,b k ) I ∂x a 1 n+1 ∂y b 1 n+1 •••∂x a k n+k ∂y b k n+k ∆µ ∩ Q[X n , Y n ] def = I,
where the sum is over the k-tuples of different cells in the shadow of (i, j) that we assume to be ordered in lexicographic order.

Proof 4 By expanding ∆ µ with respect to the last k columns, we obtain:

∆ µ (X n+k , Y n+k ) = (a1,b1),...,(a k ,b k ) ± ∆ {(a1,b1),...,(a k ,b k )} (x n+1 , . . . , x n+k , y n+1 , . . . , y n+k ) × ∆ µ/{(a1,b1),...,(a k ,b k )} (X n , Y n ).
Thus for example:

∂(x a1 n+1 y b1 n+1 • • • x a k n+k y b k n+k )∆ µ (X n+k , Y n+k ) = c∆ µ/{(a1,b1),...,(a k ,b k )} (X n , Y n ) + C
where c is a rational constant (different from 0) and C a linear combination with coefficients in

Q[x n+1 , y n+1 , . . . , x n+k , y n+k ] of ∆ µ/{(a ′ 1 ,b ′ 1 ),...,(a ′ k ,b ′ k )} (X n , Y n ) with (a ′ j , b ′ j )
in the shadow of (a j , b j ). Hence we get what we want because:

• I ⊂ I k i,j
follows from what precedes by looking at the constant term in Q[x n+1 , y n+1 , . . . , x n+k , y n+k ];

• I k i,j ⊂ I follows directly from what precedes.

Orbits

The reasonning is inspired from [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF], Theorem 4.2. We consider two families α = (α 1 , . . . , α h ) and β = (β 1 , . . . , β l ). To any injective tableau T of shape µ with entries 1, . . . , n + k, we associate a point (a(T ), b(T )) in C 2n by the classical process, ie:

a i (T ) = α ri(T ) , b i (T ) = β ci(T )
where r i (T ) (resp. c i (T )) is the number of the row (resp. column) of T where the entry i lies in T . We define ρ as the orbit of (a, b) when T varies over the (n + k)! injective tableaux of shape µ. We introduce J ρ the ideal of polynomials that are zero all over the orbit, and I =grJ ρ and H = I ⊥ (we recall that gr is an operator that associate to a polynomial its term of maximum degree). It is now a classical result (cf. [START_REF] Garsia | Some natural bigraded S n -modules and q, t-Kostka coefficients[END_REF], Theorem 1.1) that I ⊂ I ∆µ .

We now look at another orbit in C 2n . We consider the set of tableaux T of shape µ with n entries and k white cells such that the k white cells are in the shadow of (i, j) (we denote this set of tableaux by T k i,j ). By the same process as described above, we define an orbit ρ k in C 2n . Since the cardinality of T k i,j is s k n! the orbit ρ k has this cardinality. We introduce J ρ k the ideal of polynomials that are zero all over the orbit, and I k =grJ ρ k and H k = (I k ) ⊥ . We have of course dim H k = s k n!. We want to prove that M k i,j ⊂ H k and by [START_REF] Garsia | Some natural bigraded S n -modules and q, t-Kostka coefficients[END_REF], Proposition 1.1, it is equivalent to prove that I k ⊂ I.

Inclusion

We want here to obtain the next proposition: Proposition 3.2 We have the inclusion:

I k ⊂ I. Proof 5 Let P be a polynomial in J ρ k . Let us consider Q(X n+k , Y n+k ) = P (X n , Y n ) i i ′ =1 (x n+1 -α i ′ ) • • • i i ′ =1 (x n+k -α i ′ ) j j ′ =1 (y n+1 -α j ′ ) • • • j j ′ =1 (y n+k -α j ′ ).
We want to check that this polynomial is an element of J ρ . We take an element (α, β) of ρ. If its projection on C 2n is in ρ k then Q(α, β) = 0 because of P . If not it must have at least one entry between n + 1 and n + k in the first i rows or the first j columns and we have still Q(α, β) = 0. Thus gr(P ) ∈ I ∂x i n+1 ∂y j n+1 •••∂x i n+k ∂y j n+k ∆µ . For any set of k cells {(a 1 , b 1 ), . . . , (a k , b k )} in the shadow of (i, j), we observe that ∀h, 1 ≤ h ≤ k, a h ≥ i and b h ≥ j. Hence gr(P ) is in I, which was to be proved.

Conclusion

The main result is now a consequence of all what precedes: Theorem 3.1 If µ is a partition of n + k and s the cardinal of the shadow of the cell (i, j), then we have: Numerical examples and the fact that the construction described in the previous subsection affords the "good" upper bound in the case of one set of variables (see the next section) support the following conjecture.

dim M k i,j ≤ s k n!.

Case of one set of variables

The goal of this section is to obtain an explicit basis for M k i,j (X), ie the subspace of M k i,j (X, Y ) consisting of elements of 0 Y -degree.

Construction

We recall that in [START_REF]Monomial bases related to the n! conjecture[END_REF] is constructed a basis for M 0 µ made of monomial derivatives M S (∂)∆ µ of ∆ µ , where the objects S varies over a set S(µ) which depends on µ. The corresponding construction can be found in [START_REF]Monomial bases related to the n! conjecture[END_REF]. The cardinality of S(µ)

is equal to n!/µ! where µ! = µ 1 !µ 2 ! • • • µ k !.
This cardinality is the number of injective, row-increasing tableaux of shape µ.

Then we choose in the Ferrers diagram µ k cells which are simultaneously in the shadow of (i, j) and on the right edges of µ. We denote by F k µ the set of these objects. In the next figure, the chosen cells are cells with a circle and in the cell (i, j) appears a + sign. In this example n = 142 and k = 10.

Then to an object F in F k µ we associate µ F the partition of n obtained by pushing up the circled cells and by removing the corresponding cells. We also define a partition µ k F with k holes as follows. We look at the circled cells from the right to the left and from bottom to top. For a circled cell in column j ′ ≥ j such that there are l possible places where this circle could have been placed in the same column and under this one, we put a hole in the cell (i + l, j ′ ). The following figure illustrates this construction for the previous F (the holes are as usual cells with crosses: ×).

B k i,j (X) = {M S (∂)∆ µ k F ; S ∈ S(µ F ), F ∈ F k µ }
is a basis for M k i,j (X).

Upper bound

Definition 4.1 We denote by T k i,j the set of injective, row-increasing tableaux of shape µ with entries 1, . . . , n (whence each of these tableaux has k white cells) such that the k white cells are in the shadow of (i, j). dim M k i,j (X) ≤ #T k i,j .

Proof 6 Let I k i,j (X) denote the annulator ideal of the space M k i,j (X). It is classical that by looking at the monomials in Y n on Q[X n ], we obtain I k i,j (X) =

I k i,j ∩ Q[X n ].
From the Proposition 3.1, we deduce that

I k i,j (X) = I k i,j ∩ Q[X n ] = I ∩ Q[X n ].
We look at the projection of the orbit ρ k on C n ; it is equivalent to associate a point to each tableau in T k i,j . Let J 0 ρ k denote the annulator ideal of this orbit. The question is to prove the following inclusion: gr(J 0 ρ k ) ⊂ I k i,j (X). The exactly same reasonning as in the case of two alphabets can be applied successfully.

Cardinality

We claim that: Proposition 4.2 We have the following equality #B k i,j (X) = #T k i,j .

Proof 7 It is sufficient to observe that we do not change the cardinality by pushing up the circled cells. We look at the example of the lines 9, 10 and 11 of the previous example.

We observe that the lengths of the lines before the transformation are 5, 7, 6 and after the transformation 7, 6, 5. Thus the set of the lengths is unchanged. It is easy to see that it is always the case: the operation that pushes the holes up only permutes the lengths of the rows.

Remark 4 .

 4 Recall the proof of Theorem 1.1 of[START_REF] Garsia | Some natural bigraded S n -modules and q, t-Kostka coefficients[END_REF]: the previous reasonning implies that if equality holds in Theorem 3.1, then M k i,j decomposes as s k times the left regular representation.

Conjecture 1 .Remark 5 .

 15 (F.Bergeron) With the notations of the previous theorem: When k = 1, this conjecture reduces to Conjecture I.2 of[START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF] and when s = k or k = 0 to the n! conjecture.

Theorem 4 . 1

 41 With the previous notations

Proposition 4 . 1

 41 We have the following upper bound for the dimension of M k i,j (X):

Acknowledgement. The author would like to thank François Bergeron for indicating him this problem and for numerous valuable suggestions.

Independence

We want here to conclude by proving that Proposition 4. [START_REF] Aval | On certain spaces of quasi-harmonic polynomials[END_REF] The family defined in the paragraph Construction is linearly independent. Thus in particular equality holds in Proposition 4.1.

Proof 8 Assume that we have a non-trivial dependence relation.

We define the depth of a hole to be the number of cells (different from holes) that are above this hole. We look at the k-tuples of the depths of the k holes of the µ k F :

The crux of the proof is the following result:

Proof 9 We want to prove that the depth of the holes increases from the right to the left and from top to bottom, and that two different µ k F give two different k-tuples of depths. We look at the circled cells with respect to this order. We look at the next figure where c denotes the number of circled cells in the column we consider, m the number of positions under the lowest circled cell where we could put a circle (these cells appear with a square), l the height of this column (we look only at the cells above the i-th row) and l + h is the height of the "next" column (ie the first on the left). The depth of the lowest circled cell is p = l -c -m. The highest depth that could be obtained in this column is l -c if m = 0 and l -c -1 if m > 0. And in the next column the lowest depth is (it corresponds to putting a circle at the top of the column): l + h -1 -c -h + 1 = l -c. These observations are sufficient to conclude the proof of this lemma.

Then if we have a non-trivial dependence relation, we consider the greatest ktuple of depths with respect to the lexicographic order that appear in this relation (relative to an object F 0 ):

It kills all the terms but those which come from the single object F 0 . These terms give in fact terms in B = {M S (∂).∆ µ F 0 ; S ∈ S(µ F 0 )}, which are independent since B is a basis of M 0 µ F 0 .


J.-C Aval

QUASI-INVARIANT AND SUPER-COINVARIANT POLYNOMIALS FOR THE GENERALIZED SYMMETRIC GROUP

 in the case of S n . The polynomials invariant under this action are called quasi-invariant, and we define super-coinvariant polynomials as polynomials orthogonal, with respect to a given scalar product, to the quasi-invariant polynomials with no constant term. Our main result is the description of a Gröbner basis for the ideal generated by quasi-invariant polynomials, from which we dedece that the dimension of the space of super-coinvariant polynomials is equal to m n C n where C n is the n-th Catalan number.

Résumé. Le but de ce travail est d'étendre l'étude des polynômes super-coinvariants (définis dans [2]), au cas du groupe symétrique généralisé G n,m , défini comme le produit en couronne C m ≀ S n du groupe symétrique par le groupe cyclique. Nous définissons ici une action quasi-symétrisante de G n,m sur Q[x 1 , . . . , x n ], analogue à celle définie dans [12] dans le cas de S n . Les polynômes invariants sous cette action sont dits quasi-invariants, et les polynômes super-coinvariants sont les polynômes orthogonaux aux polynômes quasi-invariants sans terme constant (pour un certain produit scalaire). Notre résultat principal est l'obtention d'une base de Gröbner pour l'idéal engendré par les polynômes quasi-invariants. Nous en déduisons alors que la dimension de l'espace des polynômes super-coinvariants est m n C n où C n est le n-ième nombre de Catalan.

Introduction

Let X denote the alphabet in n variables (x 1 , . . . , x n ) and C[X] denote the space of polynomials with complex coefficients in the alphabet X. Let G n,m = C m ≀ S n denote the wreath product of the symmetric group S n by the cyclic group C m . This group is sometimes known as the generalized symmetric group (cf. [START_REF] Osima | On the representations of the generalized symmetric group[END_REF]). It may be seen as the group of n × n matrices in which each row and each column has exactly one non-zero entry (pseudo-permutation matrices), and such that the non-zero entries are m-th roots of unity. The order of G n,m is m n n!. When m = 1, G n,m reduces to the symmetric group S n , and when m = 2, G n,m is the hyperoctahedral group B n , i.e. the group of signed permutations, which is the Weyl group of type B (see [START_REF] Lusztig | Irreducible representationsof finite reflections groups[END_REF] for example for further details). The group G n,m acts classically on C[X] by the rule (1.1) ∀g ∈ G n,m , ∀P ∈ C[X], g.P (X) = P (X. t g),

where g is the transpose of the matrix g and X is considered as a row vector. Let

Inv n,m = {P ∈ C[X] / ∀g ∈ G n,m , g.P = P }
denote the set of G n,m -invariant polynomials. Let us denote by Inv + n,m the set of such polynomials with no constant term. We consider the following scalar product on C[X]:

(1.2) P, Q = P (∂X)Q(X) | X=0
where ∂X stands for (∂x 1 , . . . , ∂x n ) and X = 0 stands for

x 1 = • • • = x n = 0. The space of G n,m -coinvariant polynomials is then defined by Cov n,m = {P ∈ C[X] / ∀Q ∈ Inv n,m , Q(∂X)P = 0} = Inv + n,m ⊥ ≃ C[X]/ Inv + n,m
where S denotes the ideal generated by a subset S of C[X].

A classical result of Chevalley [START_REF] Chevalley | Invariants of finite groups generated by reflections[END_REF] states the following equality:

(1.3) dim Cov n,m = |G n,m | = m n n!
which reduces when m = 1 to the theorem of Artin [1] that the dimension of the harmonic space H n = Cov n,1 (cf. [START_REF] Garsia | Orbit Harmonics and Graded Representations[END_REF]) is n!.

Our aim is to give an analogous result in the case of quasi-symmetrizing action. The ring Qsym of quasi-symmetric functions was introduced by Gessel [START_REF] Gessel | Multipartite P -partitions and products of skew Schur functions[END_REF] as a source of generating functions for P -partitions [START_REF] Stanley | Enumerative Combinatorics[END_REF] and appears in more and more combinatorial contexts [START_REF] Bergeron | Pieri Operations on Posets[END_REF][START_REF] Stanley | Enumerative Combinatorics[END_REF][START_REF] Stanley | [END_REF]. Malvenuto and Reutenauer [START_REF] Malvenuto | Duality between quasi-symmetric functions and the Solomon descent algebra[END_REF] proved a graded Hopf duality between QSym and the Solomon descent algebras and Gelfand et. al. [START_REF] Gelfand | Noncommutative symmetric functions[END_REF] defined the graded Hopf algebra NC of non-commutative symmetric functions and identified it with the Solomon descent algebra.

In [START_REF] Aval | Catalan Paths and Quasi-Symmetric Functions[END_REF][START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF], Aval et. al. investigated the space SH n of super-coinvariant polynomials for the symmetric group, defined as the orthogonal (with respect to (1.2)) of the ideal generated by quasi-symmetric polynomials with no constant term, and proved that its dimension as a vector space equals the n-th Catalan number:

(1.4) dim SH n = C n = 1 n + 1 2n n .
Our main result is a generalization of the previous equation in the case of supercoinvariant polynomials for the group G n,m .

In Section 2, we define and study a "quasi-symmetrizing" action of G n,m on C[X]. We also introduce invariant polynomials under this action, which are called quasiinvariant, and polynomials orthogonal to quasi-invariant polynomials, which are called super-coinvariant. The Section 3 is devoted to the proof of our main result (Theorem 2.4), which gives the dimension of the space SCov n,m of super-coinvariant polynomials for G n,m : we construct an explicit basis for SCov n,m from which we deduce its Hilbert series.

A quasi-symmetrizing action of G n,m

We use vector notation for monomials. More precisely, for ν = (ν 1 , . . . , ν n ) ∈ N n , we denote X ν the monomial (2.1)

x ν 1 1 x ν 2 2 • • • x νn n . For a polynomial P ∈ Q[X],
we further denote [X ν ] P (X) as the coefficient of the monomial X ν in P (X).

Our first task is to define a quasi-symmetrizing action of the group G n,m on C[X], which reduces to the quasi-symmetrizing action of Hivert (cf. [START_REF] Hivert | Hecke algebras, difference operators, and quasi-symmetric functions[END_REF]) in the case n = 1. This is done as follows. Let A ⊂ X be a subalphabet of X with l variables and K = (k 1 , . . . , k l ) be a vector of positive (> 0) integers. If B is a vector whose entries are distinct variables x i multiplied by roots of unity, the vector (B) < is obtained by ordering the elements in B with respect to the variable order. Now the quasisymmetrizing action of g ∈ G n,m is given by

(2.2) g • A K = w(g) c(K) (A. t |g|) < K
where w(g) is the weight of g, i.e. the product of its non-zero entries, |g| is the matrix obtained by taking the modules of the entries of g, and the oefficient c(K) is defined as follows:

c(K) = 0 if ∀i, k i ≡ 0 [m] 1 if not.
Example 2.1. If m = 3 and n = 3, and we denote by j the complex number j = e 2iπ 3 , then for example

  0 0 j 1 0 0 0 j 0   • (x 2 1 x 2 ) = (j 2 ) 1     0 0 1 1 0 0 0 1 0   . (x 1 , x 2 )   (2,1) < = j 2 (x 3 , x 1 ) < (2,1) = j 2 (x 1 , x 3 ) (2,1) = j 2 x 2 1 x 3 .
It is clear that this defines an action of the generalized symmetric group G n,m on C[X], which reduces to Hivert's quasi-symmetrizing action (cf. [START_REF] Hivert | Hecke algebras, difference operators, and quasi-symmetric functions[END_REF], Proposition 3.4) in the case m = 1.

Let us now study its invariant and coinvariant polynomials. We need to recall some definitions.

A composition α = (α 1 , α 2 , . . . , α k ) of a positive integer d is an ordered list of positive integers (> 0) whose sum is d. For a vector ν ∈ N n , let c(ν) represent the composition obtained by erasing zeros (if any) in ν. A polynomial P ∈ Q[X] is said to be quasi-symmetric if and only if, for any ν and µ in N n , we have

[X ν ]P (X) = [X µ ]P (X)
whenever c(ν) = c(µ). The space of quasi-symmetric polynomials in n variables is denoted by Qsym n .

The polynomials invariant under the action (2.2) of G n,m are said to be quasiinvariant and the space of quasi-invariant polynomials is denoted by QInv n,m , i.e.

P ∈ QInv n,m ⇔ ∀g ∈ G n,m , g • P = P.
Let us recall (cf. [START_REF] Hivert | Hecke algebras, difference operators, and quasi-symmetric functions[END_REF], Proposition 3.15) that QInv n,1 = QSym n . The following proposition gives a characterization of QInv n,m . Proposition 2.2. One has • P ) = P imply that every exponents appearing in P are multiples of m. Thus there exists a polynomial Q ∈ C[X] such that P (X) = Q(X m ). To conclude, we note that S n ⊂ G n,m implies that P is quasi-symmetric, whence Q is also quasi-symmetric. The reverse implication is obvious.

P ∈ QInv n,m ⇔ ∃Q ∈ QSym n / P (X) = Q(X m ) where Q(X m ) = Q(x m 1 , . . . , x m n ).
Let us now define super-coinvariant polynomials:

SCov n,m = {P ∈ C[X] / ∀Q ∈ QInv n,m , Q(∂X)P = 0} = QInv + n,m ⊥ ≃ C[X]/ QInv + n,m
with the scalar product defined in (1.2). This is the natural analogous to Cov n in the case of quasi-symmetrizing actions and SCov n,m reduces to the space of superharmonic polynomials SH n (cf. [START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF]) when m = 1.

Remark 2.3. It is clear that any polynomial invariant under (2.2) is also invariant under (1.1), i.e. Inv n,m ⊂ QInv n,m . By taking the orthogonal, this implies that SCov n,m ⊂ Cov n,m . These observations somewhat justify the terminology.

Our main result is the following theorem which is a generalization of equality (1.4).

Theorem 2.4. The dimension of the space Scov n,m is given by

(2.3) dim SCov n,m = m n C n = m n 1 n + 1 2n n .
Remark 2.5. In the case of the hyperoctahedral group B n = G n,2 , C.-O. Chow [START_REF] Chow | Noncommutative Symmetric Functions of Type B[END_REF] defined a class BQSym(x 0 , X) of quasi-symmetric functions of type B in the alphabet (x 0 , X). His approach is quite different from ours. In particular, one has the equality:

BQSY m(x 0 , X) = QSym(X) + QSym(x 0 , X).
In the study of the coinvariant polynomials, it is not difficult to prove that the quotient C[x 0 , X]/ BQSym + is isomorphic to the quotient C[X]/ QSym + studied in [START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF]. To see this, we observe that if G is the Gröbner basis of QSym + constructed in [START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF] (see also the next section), then the set {x 0 , G} is a Gröbner basis (any syzygy is reducible thanks to Buchberger's first criterion, cf. [START_REF] Cox | Ideals, Varieties, and Algorithms[END_REF]).

The next section is devoted to give a proof of Theorem 2.4 by constructing an explicit basis for the quotient C[X]/ QInv + n,m .

Proof of the main theorem

Our task is here to construct an explicit monomial basis for the quotient space C[X]/ QInv + n,m . Let us first recall (cf. [START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF]) the following bijection which associates to any vector ν ∈ N n a path π(ν) in the N × N plane with steps going north or east as follows. If ν = (ν 1 , . . . , ν n ), the path π(ν) is

(0, 0) → (ν 1 , 0) → (ν 1 , 1) → (ν 1 + ν 2 , 1) → (ν 1 + ν 2 , 2) → • • • → (ν 1 + • • • + ν n , n -1) → (ν 1 + • • • + ν n , n).
For example the path associated to ν = (2, 1, 0, 3, 0, 1) is

π(ν) =
We distinguish two kinds of paths, thus two kinds of vectors, with respect to their "behavior" regarding the diagonal y = x. If the path remains above the diagonal, we call it a Dyck path, and say that the corresponding vector is Dyck. If not, we say that the path (or equivalently the associated vector) is transdiagonal. For example η = (0, 0, 1, 2, 0, 1) is Dyck and ε = (0, 3, 1, 1, 0, 2) is transdiagonal.

ε η
We then have the following result which generalizes Theorem 4.1 of [START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF] and which clearly implies the Theorem 2.4.

Theorem 3.1. The set of monomials

B n,m = {(X n ) m η+α / π(η) is a Dyck path, 0 ≤ α i < m} is a basis for the quotient C[X n ]/ QInv + n,m .
To prove this result, the goal is here to construct a Gröbner basis for the ideal J n,m = QInv + n,m . We shall use results of [START_REF] Aval | Catalan Paths and Quasi-Symmetric Functions[END_REF][START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF]. Recall that the lexicographic order on monomials is

(3.1) X ν > lex X µ iff ν > lex µ,
if and only if the first non-zero part of the vector ν -µ is positive.

For any subset S of Q[X] and for any positive integer m, let us introduce S m = {P (X m ) , P ∈ S}. If we denote by G(I) the unique reduced monic Gröbner basis (cf. [START_REF] Cox | Ideals, Varieties, and Algorithms[END_REF]) of an ideal I, then the simple but crucial fact in our context is the following. Proof. This is a direct consequence of Buchberger's criterion. Indeed, if for every pair g, g ′ in G( S ), the syzygy S(g, g ′ ) reduces to zero, then the syzygy S(g(X m ), g ′ (X m )) also reduces to zero in G( S m ) by exactly the same computation.

Let us recall that in [START_REF] Aval | Catalan Paths and Quasi-Symmetric Functions[END_REF] is constructed a family G of polynomials G ε indexed by transdiagonal vectors ε. This family is constructed by using recursive relations of the fundamental quasi-symmetric functions and one of its property (cf. [START_REF] Aval | Catalan Paths and Quasi-Symmetric Functions[END_REF]) says that the leading monomial of G ε is: LM(G ε ) = X ε . Since G is a Gröbner basis of J n,1 , the following result is a consequence of Propositions 2.2 and 3.2.

Proposition 3.3. The set G m is a Gröbner basis of the ideal J n,m .

To conclude the proof of Theorem 3.1, it is sufficient to observe that the monomials not divisible by a leading monomial of an element of G m , i.e. by a X mε for ε transdiagonal, are precisely the monomials appearing in the set B n,m .

As a corollary of Theorem 3.1, one gets an explicit formula for the Hilbert series of SCov n,m . For k ∈ N, let SCov Let us recall that in [START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF] is given an explicit formula for F n,1 :

(3.5)

F n,1 (t) = F n (t) = n-1 k=0 n -k n + k n + k k t k
using the number of Dyck paths with a given number of factors (cf. [START_REF] Kreweras | Sur les éventails de segments[END_REF]). The Theorem 3.1 then implies the

  Proof. Let P be an element of QInv n,m . Let us denote by ζ the m-th root of unity ζ = e 2iπ m and by g j the element of G n,m whose matrix is the ζ in place j. Then we observe that the identities ∀j = 1, . . . , n, 1 m (P + g j • P + g 2 j • P + • • • + g m-1 j

Proposition 3 . 2 .

 32 With the previous notations,(3.2) G( S m ) = G( S ) m .

  SCov (k) n,m = SCov n,m ∩ Q (k) [X]where Q(k) [X] is the vector space of homogeneous polynomials of degree k together with zero.Let us denote by F n,m (t) the Hilbert series of SCov n,m , i.e.(3.4)F n,m (t) = k≥0 dim SCov (k) n,m t k .

Corollary 3 . 4 .

 34 With the notations of (3.5), the Hilbert series of SCov n,m is given by

F

  n,m (t) = 1 -t m 1 -t F n (t m )from which one deduces the close formulan F n,m (t) x n = (1-t) -(1-t)(1 -t -4t m x(1 -t m )) -2x(1-t m ) (1 -t)(2t m -1) -x(1 -t m ) .

Research financed by EC's IHRP Programme, within the Research Training Network "Algebraic Combinatorics in Europe," grant HPRN-CT-2001-00272.

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