Yan X Zhang
email: yanzhang@math.berkeley.edu

Four Variations on Graded Posets

Keywords: posets, combinatorics, generating functions, poset avoidance, linear algebra

We explore the enumeration of some natural classes of graded posets, including (2 + 2)-avoiding graded posets, (3 + 1)-avoiding graded posets, (2 + 2)and (3 + 1)-avoiding graded posets, and the set of all graded posets. As part of this story, we discuss a situation when we can switch between enumeration of labeled and unlabeled objects with ease, which helps us generalize a result by Postnikov and Stanley from the theory of hyperplane arrangements, answer a question posed by Stanley, and see an old result of Klarner in a new light.

) et (3+1), et l'ensemble de tous les posets gradués. En particulier, nous considérons une situation où l'énumération d'objets marqués et non marqués sont reliées de fac ¸on simple, ce qui nous permet de généraliser un résultat de Postnikov et Stanley en théorie des arrangements d'hyperplans, répondre à une question posée par Stanley, et voir sous un nouveau jour un vieux résultat de Klarner et Kreweras.

Introduction

The enumeration of posets has been a classical part of enumerative combinatorics. Graded posets form a very natural subfamily of posets to consider. (2 + 2)-avoidance and (3 + 1)-avoidance are two types of poset-avoidance frequently encountered in literature. Combining these two types of avoidance gives 4 natural families of graded posets:

• All graded posets;

• (2 + 2)and (3 + 1)-avoiding graded posets.

• (2 + 2)-avoiding graded posets;

• (3 + 1)-avoiding graded posets; This extended abstract summarizes some enumerative results on these four families. After some preliminaries in Section 2, we enumerate all graded posets in Section 3 in a way that better packages old results of Klarner [START_REF] David | The number of graded partially ordered sets[END_REF] and Kreweras [START_REF] Kreweras | Denombrement des ordres etages[END_REF]. We also develop some unifying themes for the other three families.

OGF EGF n-element sets (antichains) 1 1-x e x n-element lists (chains)

1 1-x 1 1-x
Tab. 1: The OGF and EGF for two simple classes of posets.

Then, we take a detour and explore a question of Stanley, in Section 4, obtaining the main result Theorem 4.1. Some other ramifications of these results are discussed at the end of Section 4 and Section 8, including a generalization of an observation in work of Stanley and Postnikov on hyperplane arrangements [START_REF] Postnikov | Deformations of Coxeter hyperplane arrangements[END_REF], and Yangzhou Hu's [START_REF] Hu | On the number of fixed-length semiorders[END_REF] direct application of our theorem in enumerating (2 + 2)-avoiding posets.

Finally, we go back to the families of problems listed above and discuss our results in Sections 5, 6, and 7. The last of these, Section 7, is a brief sketch of a particularly fruitful standalone work of enumerating (3 + 1)-avoiding graded posets, joint with Lewis [START_REF] Brewster | Enumeration of graded (3+ 1)-avoiding posets[END_REF]. This work was fortuitously timed shortly before the enumeration of all (3 + 1)-avoiding posets by Guay-Paquet, Morales, and Rowland [START_REF] Guay-Paquet | Structure and enumeration of (3+ 1)-free posets[END_REF]. We end with some ideas for future work in Section 8.

Preliminaries

Generating Functions

When we count combinatorial structures with a concept of "size," such as posets (where a choice of a "size function" would be the number of vertices), we frequently consider using a generating function to count the said structure for every size simultaneously. Two types of generating functions usually occur:

• the OGF (ordinary generating function), usually for unlabeled structures. If there are o n objects with size n, the OGF is

O(x) = n o n x n .
• the EGF (exponential generating function), usually for structures labeled by {1, . . . , n}. If there are e n objects with size n, the EGF is

E(x) = n e n x n n! .
OGFs and EGFs together tell the story of the symmetries of the combinatorial family and carry different information. An example of two extreme cases, with the same OGF but different EGFs, can be seen in Table 1.

Posets and Poset Avoidance

In this work, all posets we care about are finite. We represent (and refer to) all such posets by their Hasse diagrams, so we use the language of graphs. For a poset P , we use V (P ) to denote its vertices.

We say a poset P contains (avoids) another poset Q if we can (cannot) select some vertices S ⊂ V (P ) and some edges of P such that they form an isomorphic poset to Q. We are only going to be interested in the case where Q is of the form (a

1 + a 2 + • • • + a k ),

All Graded Posets

There is some ambiguity in the literature about the concept of a graded poset, so it is important that we start by being precise. In this work, we define a poset P to be weakly-graded (of rank M ) if there is a rank function r : V (P ) → Z such that if x covers y then r(x) -r(y) = 1. Furthermore, we define a poset to be strongly-graded, or simply graded, if all maximal chains have the same length. In this situation, we can consider all minimal vertices to have rank 0 and all maximal vertices to have the same rank M . See Figure 1 for a comparison. Strongly-graded posets tend to be easier to work with because they do not have problematic boundary conditions that come up with weakly-graded posets. Furthermore, the enumeration of weakly-graded posets from strongly-graded ones seems to be a routine but tedious exercise of inclusion-exclusion on the boundary conditions. Thus, we simply call strongly-graded posets graded in our work; this choice is consistent with, say, Stanley [START_REF] Richard | Enumerative combinatorics[END_REF].

Fig. 1: In our language: a non-graded poset, a weakly graded poset, and a strongly graded poset. Now, we make a simple but very important observation: the subposet induced by two adjacent ranks of a weakly-graded poset is just a bipartite graph. Thus, it makes sense to consider the EGF of the ψ w (m, n) = 2 mn counting bipartite graphs with m vertices on one side and n on the other:

Ψ w (x, y) = m,n≥0 ψ w (m, n) x m y n m!n! = m,n≥0
2 mn x m y n m!n! .

Similarly, the subposet induced by two adjacent ranks of a graded poset is just a bipartite graph with no isolated vertices. The EGF enumerating these structures is

Ψ s (x, y) = m,n≥0 ψ s (m, n) x m y n m!n! = e -x-y Ψ w (x, y)
by inclusion-exclusion. Ψ w will resurface when we talk about (3 + 1)-graded posets in Section 7. With the concession of treating the ψ s 's as "simple," we obtain the following:

Theorem 3.1 For the enumeration of all graded posets, the following hold:

• Let V = x 1 /1! x 2 /2! • • • x r /r! • • •
T be an infinite column matrix. Let D be the infinite diagonal matrix with diagonal entries matching that of V . Let A = {A ij }, where A ij = ψ s (i, j) for i, j ≥ 1. Then, the generating function for all graded posets is:

P (x) = V T (I -DAD) -1 V.
• The above calculation with infinite matrices and vectors can be made finite in the following way: take r ≥ 1 an integer and let V r be the r × 1 truncation of the first r elements of V . Similarly, let D r and A r be the r × r truncations of D and A respectively. Then the generating function for all graded posets with at most r vertices on each rank is:

P r (x) = V T r (I r -D r A r D r ) -1 V r .
A similar calculation with different V can be used to give generating functions for weakly-graded posets.

Proof idea: We can think of a graded poset with n levels as (n -1) "slices" of 2-level posets glued together at their vertices. There are ψ s (i, j) possible slices with i and j vertices at their two levels, and keeping track of how they fit together naturally gives us the result. 2

Klarner [START_REF] David | The number of graded partially ordered sets[END_REF] gave an enumeration of weakly-graded posets (which he called graded) and Kreweras [START_REF] Kreweras | Denombrement des ordres etages[END_REF] gave a similar enumeration of graded posets (which he called tiered (i) ). Both results were presented in nested sums of the form

f (n 1 , n 2 )f (n 2 , n 3 ) • • • f (n i-1 , n i )g(n i ),
where f (m, n) and g(n) are specific functions and the sum, as noted by Klarner [START_REF] David | The number of graded partially ordered sets[END_REF] himself, "extends over all compositions (n 1 , • • • , n i ) of n into an unrestricted number of positive parts." Theorem 3.1 is the most natural reformulation of these results as these sums occur naturally from matrix multiplication.

Seeds and Gardens

As an interlude, we examine four results on enumerating interval orders and semiorders, all by different authors:

• (Wine and Freund [WF57]) OGF of interval semiorders: O(x) = 1- √ 1-4x 2x . • (Stanley [Sta96]) EGF of interval semiorders: E(x) = 1- √ 4e -x -3 2(1-e -x ) .
• (Zagier [START_REF] Zagier | Vassiliev invariants and a strange identity related to the dedekind eta-function[END_REF]) EGF of interval orders:

E(x) = n n k=1 (1 -e -kx ). • (Bousquet-Melou [BMCDK10]) OGF of interval orders: O(x) = n n k=1 (1 -(1 -x) k ).
Note that for both interval orders and semiorders, we have E(x) = O(1-e -x ). Our work was motivated by Stanley's question (personal communication) of the unifying reason behind this pattern, which clearly does not hold for arbitrary combinatorial structures (recall Table 1)! First, we define some notions that capture the concept of "cloning" vertices in a poset; in fact, these notions are generalizable to other combinatorial structures, though we limit our statements to posets for clarity in this paper. Given a poset P , call two incomparable vertices x and x in V (P ) exchangeable if for all y ∈ V (P ), x < y (resp. x > y) if and only if x < y (resp. x > y). Given a poset P and a vertex x ∈ V (P ), we denote by cloning at x the process that outputs a poset P with an additional vertex x exchangeable with x. We denote by de-cloning at x the process that outputs a poset P with x removed, given that x was exchangeable with at least one other vertex x ∈ V (P ). An example of cloning can be seen in Figure 2. It is easy to see that repeated de-cloning associates each poset to exactly one seed.

Let a seed be a poset that cannot be de-cloned (i.e. it has no pair of exchangeable vertices). Let a garden be a collection of posets that result from cloning from a set S of seeds. We say it has the seed (exponential) generating function S(x) = P x |V (P )| |V (P )|! , where we sum over all seeds P . Example: the garden of isolated vertices has a single seed of size 1; this is just the top row of Figure 2.

Call a poset primitive if it has no nontrivial automorphism. Note that a primitive poset must be a seed, because otherwise exchanging two exchangeable vertices is a nontrivial automorphism, but the reverse is not true: consider the poset (2 + 2), which is the lower-left poset in Figure 2. Its automorphism group is Z 2 but no pairs of vertices are exchangeable. This property of (2 + 2) is actualy the key point of our situation:

Theorem 4.1 When all seeds in a garden are primitive (ii) , its OGF O(x), EGF E(x), and seed EGF S(x) satisfy E(x) = S(e x -1), E(x) = O(1 -e -x ), and O(x) = S(x/(1 -x)).

Whenever this situation happens, we can switch between all three generating functions at ease, as depicted below. For some more examples of this principle applied to other combinatorial structures, see Figure 6 Of particular relevance to us, however, is: Proposition 4.2 If we have a garden of posets that avoid (2+2), then all seeds in the garden are primitive.

In particular, the gardens corresponding to interval orders and interval semiorders are both (2 + 2)avoiding, which explains the numerology that we observed at the beginning of this chapter; see Figure 3. For later sections in our work, we now know that for the classes of graded posets that are (2 + 2)-avoiding, we can cheaply obtain both the EGFs and OGFs from one another, and we should not expect this nice situation to occur for the "all graded posets" and "(3 + 1)-avoiding graded posets" classes.

It turns out that this observation was not only useful for our work in this paper, but also a couple of other projects.

• Before we finished this extended abstract, Hu [START_REF] Hu | On the number of fixed-length semiorders[END_REF] borrowed our result for her work on the number of fixed-length semiorders.

• One of the main results, Theorem 7.1, in Postnikov and Stanley's work [START_REF] Postnikov | Deformations of Coxeter hyperplane arrangements[END_REF] on deformations of Coxeter hyperplane arrangements can be seen as a special case of Theorem 4.1.

• We can use Proposition 4.2 to immediately generalize our observation about interval semiorders to that of marked interval orders as found in Stanley [START_REF] Richard | Hyperplane arrangements, interval orders, and trees[END_REF]. We leave this discussion to Appendix A.

(3 + 1)-and (2 + 2)-avoiding Graded Posets

Given a sum of chains C = (x + y), we say that P grade-avoids

x[b x ] + y[b y ],
where the b i are in Z, if we cannot find an incomparable x-chain C x and y-chain C y and a fixed b ∈ Z such that for i ∈ {x, y}, the j-th element of the chain C i is in the (b i + j + b)-th rank of P . Intuitively, grade-avoidance captures avoidance of chains contiguous in the levels of P at a specified relative level.

e n : labeled interval orders o n : unlableled interval orders, ascent sequences s n : uppertriangular 0 -1 matrices with no zero rows/columns Now that we enter the territory of poset avoidance, we introduce the following lemma, which captures a very powerful idea that we use in all the remaining cases. The simple intuition underneath the technicalsounding description (and a technical proof, which we omit) is that the avoidance of a sum of two chains in weakly-graded posets is a "local condition"; in other words, if P contains a sum of two chains, we do not have to look arbitrarily far to find those chains.

s n = A138265(n) : 1, 1, 1, 2, 5, 16, 61, . . . S(x) = n k (1 -(1 + x) -k ) o n = A022493(n) : 1, 1, 2, 5, 15, 52, 217, . . . O(x) = n k (1 -(1 -x) k ) e n = A079144(n) : 1, 1, 3, 19, 207, 3451, . . . E(x) = n k (1 -e -kx )
i ∈ {x, y}, i + b i -1 ≤ S.
Our main result is: Theorem 5.2 The OGF for graded interval semiorders is

O(x) = 1 -3x + 2x 2 -x 3 (1 -x)(1 -3x + x 2 )
.

Fig. 4: A height-k poset can be decomposed into a "skeleton" plus k sets of almost-independent "slices."

e n : labeled graded interval semiorders o n : directed column-convex polyominos with area n + 2 and 2 elements in bottom row s n : variation of Fibonacci numbers

s n = A000045(n + 1)(n!) : 1, 1, 2, 6, 48, 360, . . . S(x)
= 1 + x + x 2 1-x-x 2 o n = A055588(n -1) : 1, 1, 2, 4, 9, 22, 56, 145, . . . O
(x) = 1-3x+2x 2 -x 3 (1-x)(1-3x+x 2 )
e n = ??? : 1, 3, 13, 99, 1021, 12723, . . . E(x) = • • • Fig. 5: The three generating functions that come up from graded interval semiorders.

The triplet of related generating functions can be found in Figure 5.

Proof idea: Lemma 5.1 allows us to basically only consider adjacent levels when we check for poset containment. The combination of (3 + 1) and (2 + 2) avoidance shows that there is at least one vertex on each level of the poset that is adjacent to all vertices in the levels above and below. These vertices form a "skeleton" of the poset; furthermore, each vertex not of this form belongs to a unique "slice" that spans two adjacent ranks. The problem is then basically equivalent to choosing the size of the skeleton and then picking the slices independently. See Figure 4 for intuition. 2

Our enumeration of graded interval seminorders seems to be original, although the generating function has already appeared in literature in the context of directed column-convex polyominoes [START_REF] Deutsch | A bijection between directed column-convex polyominoes and ordered trees of height at most three[END_REF]. Since we have (2 + 2)-avoidance, we also get to put the enumeration into context of Proposition 4.2. Note that looking at the seed EGF suggests some relation with Fibonacci numbers, as the generating function is somewhat similar to the classic generating function

1 1-x-x 2 .

(2 + 2)-avoiding Graded Posets

For (2 + 2)-avoiding graded posets, we have the following result; note its similarity with Theorem 3.1. Theorem 6.1 The following are true about the enumeration of graded interval orders:

• Let A = {A ij }, where A ij = ∞ l=0 x l+n l!n! m m =0 m m ψ 2+2 (n, l + m ),
where ψ 2+2 (i, j) has the exponential generating function i,j ψ 2+2

x i y j i!j! = 1 e x + e y -e x+y .

Then, the EGF for graded interval orders is:

P (x) = (e x -1)[(I -(e x -1)A) -1 ] 0,0 .
Proof idea: We again use Lemma 5.1 to reduce the set of combinatorial conditions we have to check to a finite number. We no longer have (3 + 1)-avoidance, so it is a more technical problem to count the "slices," where ψ 2+2 (i, j) plays a similar role with ψ s (i, j) in Theorem 3.1.

2

With the EGF, we may apply Theorem 4.1 to obtain the OGF. We do not include the triplet of sequences as we could not find further insight (none of the sequences are in the OEIS, for example). Maybe a keen reader could find something we did not!

(3 + 1)-avoiding Graded Posets

(3 + 1)-avoidance is ubiquitous in the study of poset-avoidance, including the Stanley-Stembridge conjecture [START_REF] Richard | On immanants of Jacobi-Trudi matrices and permutations with restricted position[END_REF], the birthday problem [START_REF] Fadnavis | A generalization of the birthday problem and the chromatic polynomial[END_REF], etc. When we started this project, the enumeration of (3 + 1)-avoiding posets had been a long-standing unsolved problem, so we thought to explore the graded version as a stepping stone. Using some intuitions developed in the earlier sections we successfully enumerated (3 + 1)-avoiding graded posets in a standalone work joint with Lewis [START_REF] Brewster | Enumeration of graded (3+ 1)-avoiding posets[END_REF], with the following main result (note the re-appearance of Ψ w ):

Theorem 7.1 ([LZ13])
The EGF for graded (3 + 1)-avoiding posets is:

F (x) = e x -1 + 2e x + (e x -2)Ψ w (x, x) 2e 2x + e x + (e 2x -2e x -1)Ψ w (x, x)
.

This story has a happy ending -soon afterwards, Guay-Paquet, Morales, and Rowland [GPMR13] obtained the complete enumeration of (3 + 1)-avoiding posets, using some techniques similar to ours (as predicted, Ψ w appears in their work also, albeit in a different form) and some more sophisticated techniques of their own.

Conclusion and Future Directions

In this work, we have basically understood the enumeration and structure of the four classes of graded posets in a fairly organized manner. However, there are many fruitful directions for future work:

• The real value of Theorem 4.1 may lie not in its modest application to generating functions, but rather in its ability to predict relationships between seemingly-unrelated objects, such as the appearance of Motzkin objects, Fibonacci numbers, and directed column-convex polyominos in Figures 6 and5.

• Theorem 4.1 can be used to juxtapose many more triplets of generating functions, including some very familiar ones. Figure 6 gives a couple of examples, and we have verified a few more that we did not include due to space. For example, compositions are just (2 + 1)-avoiding posets, which avoid (2 + 2)! We welcome the reader to explore further in this direction.

• The matrices that appear in the enumeration of all graded posets and (2 + 2)-avoiding graded posets still have potential to be simplified, maybe with something like a spectral theory of infinite matrices over the ring of formal power series.

• Lemma 5.1, while seemingly simple, is very powerful and crucial to our work. It may be extended to give a unified theory of chain-avoidance. This seems to be a useful project if posets besides (2 + 2)and (3 + 1)start to become relevant in poset-avoidance.

• It may be useful to recast the work in Section 4, which is really about cycle indices at heart, in a more algebraic language, such as species or invariant theory. Fig. 6: Top: our garden has a single seed of a single vertex. Bottom: our garden has a single seed of n distinguishable objects. Strictly speaking, we are applying Theorem 4.1 to an extended category of posets that allow distinct "colors."

• Each interval with exactly n equidistant marks: a similar situation as generalized Shi arrangements (x i -x j ∈ {-k, -k + 1, . . . , -1, 1, . . . , k})

Each such family O of marked interval orders corresponds to a region of the hyperplane arrangement A(O) of R n constructed by following: for each j, pick l 1 , l 2 , • • • , l n dependent on the interval lengths and create the hyperplane arrangement from the hyperplanes x i -x j = l 1 , . . . , l n for all i. In general, we are going to get deformations of the Braid arrangement. A slightly more general version of Theorem 4.1 fairly quickly generalizes Theorem 2.3 from [START_REF] Richard | Hyperplane arrangements, interval orders, and trees[END_REF] and the result from Postnikov and Stanley [START_REF] Postnikov | Deformations of Coxeter hyperplane arrangements[END_REF] that labeled interval semiorders correspond (with multiplicity n!) to regions of the Catalan arrangement.

Fig. 2 :

 2 Fig. 2: In each row, cloning one of the vertices repeatedly gives the posets to the right. Decloning at the cloned vertices would revert to the posets at the left.

Fig. 3 :

 3 Fig. 3: Triplets of generating functions for interval orders (top) and interval semiorders (bottom). Note the juxtoposition of not-obviously related objects.

Lemma 5. 1

 1 Let S = x+y -2. A weakly-graded poset P is (x+y)-avoiding if and only if it grade-avoids all (x[b x ] + y[b y ]), where min(b x , b y ) = 0 and for

  n = A019590 : 1, 1, 0, 0, 0, 0, 0, . . . S(x)= 1 + x o n = A000012 : 1, 1, 1, 1, 1, 1, 1, . . . O(x) = 1 1-x e n = A000012 : 1, 1, 1, 1, 1, 1, . . . E(x) = e xe n : labeled set partitions (Bell numbers) o n : sets of lists, chain gangs s n : n distinct objectss n = A000012 : 1, 1, 1, 1,1, 1, 1, . . . S(x) = e x o n = A000262 : 1, 1, 3, 13, 73, 501, . . . O(x) = e x/(1-x) e n = A000110 : 1, 1, 2, 5, 15, 52, . . . E(x) = e e x -1

Theorem A. 1

 1 Let O be any family of marked interval orders. Labeled members of O correspond to regions of A(O) and unlabeled members of O correspond (with multiplicity n!) to regions of A(O) ∪ B, where B is the Braid arrangement; their EGF E(x) and OGF O(x) satisfy E(x) = O(1 -e -x ).

  which we use to mean the union of k incomparable

	avoids (2 + 2)	yes	no	yes
	avoids (3 + 1)	yes	yes	no
		Tab. 2: Some examples of poset avoidance.
		2 + 2)-and (3 + 1)-avoiding poset is called an interval semiorder.
	Examples are given in Table 2. A standard reference on posets is Stanley [Sta11].

chains of lengths a 1 , . . . , a k ; in fact, we will basically only see (2+2) or (3+1). A (2+2)-avoiding poset is also called an interval order, and a (

(ii) A generalization, which we will not use here but is good to keep in mind, is that when all seeds in a garden have the same cycle index K, we haveE(x) = R K (x)O(1 -e -x ),where R K (x) is a rational function dependent only on the cycle index.

Acknowledgements

The author thanks Richard Stanley for motivating the enumeration of graded posets and for ever-helpful advice, and Joel Lewis, Alejandro Morales, Francois Bergeron, and Alexander Postnikov for helpful discussion. In particular, the author is grateful for Joel Lewis for the collaboration that led to the (3 + 1)avoiding graded posets work and for helping build much of the intuition in the other projects as well. The author also appreciates the high-quality reviews of the two anonymous referees. Much of this work was done at MIT, when the author was supported via an NSF Graduate Fellowship.

A An Application of Theorem 4.1 -Marked Interval Orders

We end with a generalization of a result of Stanley [START_REF] Richard | Hyperplane arrangements, interval orders, and trees[END_REF] from studying hyperplane arrangements. Consider an embedding of n intervals in R such that each interval is marked at several points. Consider two arrangements to be equivalent if for every two intervals a and b, the number of marked points of a to the left of the leftmost (not necessarily marked!) point of b are the same. Call such an equivalence class a marked interval order. This generalizes some familiar arrangements. For example:

• Each interval only marked at the right endpoint: interval orders of a particular set of lengths;

• Each interval only marked at the right endpoint, each interval having length 1: interval semiorders (x i -x j ∈ {1, -1});


Ernesto Vallejo
email: vallejo@matmor.unam.mx.

Stability of Kronecker coefficients via discrete tomography (Extended abstract)

Keywords: Kronecker coefficient, Schur function, Stability, Discrete tomography, Additivity, Transportation polytope

In this paper we give a sufficient condition for a general stability of Kronecker coefficients, which we call additive stability. Its main ingredient is the property of a matrix of being additive. This notion seems to be an important one: it appears in Discrete Tomography as a sufficient condition to uniqueness; it also appears in Manivel's study of asymptotic properties of plethysm through Borel-Weil theory. The proof sketched here combines several results of the author on integer matrices motivated by Discrete Tomography with a new idea of Stembridge, that permits to bound some sequences of Kronecker coefficients. The advantage of additivity with respect to the previous approach by Stembridge is that it is very easy to produce new examples of additive matrices and, therefore, to produce many new examples of stability of Kronecker coefficients. We also show that Murnaghan's stability property and other instances of stability discovered previously by the author are special cases of additive stability. Besides, our approach permits us to disprove a recent conjecture of Stembridge and to give a new characterization of additivity.

Résumé. Dans ce papier nous donnons une condition suffisant pour la stabilité general des coefficients de Kronecker, que nous appelons stabilité additive. Le ingrédient principal est la propriété d'une matrice d'être additif. Cette notion est apparemment d'importance: elle apparaît en Tomographie Discrète comme une condition suffisant pour unicité; elle apparaît aussi dans l'étude de Manivel de propriétés asymptotiques du plethysm par moyen de la théorie de Borel-Weil. La démonstration esquissé ici combine plusiers résultats de l'auteur sur les matrices à coefficients entières stimulés pour la Tomographie Discrète avec une nouvelle ideé de Stembridge, qui permet d'borner quelques successions des coefficients de Kronecker. L'avantage de notre méthode sur l'approche de Stembridge est qui c'est très facile de produire nouveaux exemples de matrices additives, et pourtant, nouveaux exemples d'estabilité des coefficients de Kronecker. Nous démontrons aussi que la stabilité de Murnaghan et d'autres exemples de stabilité trouvés antérieurement par l'auteur sont des cas spéciales de la stabilité additive. En plus, avec notre approche nous réfutons une conjecture de Stembridge et donnons une nouvelle caractérisation d'additivité.

Introduction

Kronecker coefficients are among the most fascinating and complex objects in algebraic combinatorics. They appear as multiplicities of various constructions involving tensor products of representations of the symmetric group S n , Schur powers of some representations of the general lineal group GL n (C) or the space of sections of a line bundle over a flag variety. They also appear in certain expansions of symmetric functions. A long sought combinatorial or geometric description of Kronecker coefficients (such as the ones existing for the Littlewood-Richardson coefficients) is still to be found.

For our purposes, given three partitions λ, µ, ν of a positive integer n, we choose to define the Kronecker coefficient g(λ, µ, ν) as the multiplicity of the complex irreducible character χ ν of S n associated to ν in the Kronecker product χ λ ⊗ χ µ of other two irreducible characters χ λ and χ µ of S n . Thus, if • , • denotes the scalar product of complex characters of S n , g(λ, µ, ν) = χ λ ⊗ χ µ , χ ν .

In the first paper on the Kronecker coefficients [START_REF] Murnaghan | The analysis of the Kronecker product of irreducible representations of the symmetric group[END_REF], published in 1938, Murnaghan stated without proof the following stability property for Kronecker coefficients (see also [START_REF] Murnaghan | On the analysis of the Kronecker product of irreducible representations of Sn[END_REF]): For each triple of partitions λ, µ, ν of the same size, the sequence of integers

{g(λ + (n), µ + (n), ν + (n))} n∈N (1)
converges. It was first proved by Littlewood [START_REF] Littlewood | Products and plethysms of characters with orthogonal, symplectic and symmetric groups[END_REF]. Since then many other proofs of this property with different flavours have appeared [START_REF] Bowman | The partition algebra and the Kronecker coefficients[END_REF][START_REF] Brion | Stable properties of plethysm: on two conjectures of Foulkes[END_REF][START_REF] Church | FI-modules: a new approach to stability for Sn-representations[END_REF][START_REF] Pak | Bounds on the Kronecker coefficients[END_REF][START_REF] Thibon | Hopf algebras of symmetric functions and tensor products of symmetric group representations[END_REF][START_REF] Vallejo | Stability of Kronecker product of irreducible characters of the symmetric group[END_REF][START_REF] Vallejo | A diagrammatic approach to Kronecker squares[END_REF]. Estimations of lower bounds L for stability can be found in [START_REF] Briand | The stability of the Kronecker product of Schur functions[END_REF][START_REF] Brion | Stable properties of plethysm: on two conjectures of Foulkes[END_REF][START_REF] Vallejo | Stability of Kronecker product of irreducible characters of the symmetric group[END_REF][START_REF] Vallejo | A diagrammatic approach to Kronecker squares[END_REF].

Recently, a generalization of Murnaghan's stability was discovered [START_REF] Vallejo | A diagrammatic approach to Kronecker squares[END_REF]Thm. 10.2]. Let d(ν) = |ν|-ν 1 denote the depth of ν. There it is shown that if i ≤ min{ (λ), (µ)}, λ i -λ i+1 ≥ d(ν) and µ i -µ i+1 ≥ d(ν) (if i = (λ), we define λ i+1 = 0), then for all n ∈ N, we have

g(λ + (n i ), µ + (n i ), ν + (ni)) = g(λ, µ, ν). (2)
The case i = 1 yields Murnaghan's stability. The stability given in equation [START_REF] Bowman | The partition algebra and the Kronecker coefficients[END_REF] was found independently in [START_REF] Pak | Bounds on the Kronecker coefficients[END_REF] (with different bounds), where it was called k-stability. Both results were inspired on a particular case of (2) (when λ = µ), that appears on the first version of [40] (i) .

The proof of Theorem 10.2 in [START_REF] Vallejo | A diagrammatic approach to Kronecker squares[END_REF] also yields a more general type of stability, which includes [START_REF] Bowman | The partition algebra and the Kronecker coefficients[END_REF]. We now describe it. Let λ, µ, ν be a triple of partitions of the same size and let d = d(ν), then for any pair of vectors (i 1 , . . . , i t ), (n 1 , . . . , n t ) ∈ N t , such that i

1 > i 2 > • • • > i t and n j ≥ d, for all j ∈ [ t ], g λ + t j=1 (n j ij ), µ + t j=1 (n j ij ), ν + t j=1 n j i j = g λ + t j=1 (d ij ), µ + t j=1 (d ij ), ν + t j=1 di j (3)
Let α be the conjugate partition of (i 1 , . . . , i t ), then

t j=1 (1 ij ) = α and t j=1 (d ij ) = dα. Thus, identity (3
) can be rewritten in the following way Another, apparently different, kind of stability was discovered sometime ago in [START_REF] Vallejo | A stability property for coefficients in Kronecker products of complex Sn characters[END_REF]Thm. 3.1]. There, it is shown that, if q, r are positive integers such that (λ) ≤ qr, (µ) ≤ q and (ν) ≤ r. Then for all n ∈ N, g(λ + (n qr ), µ + ((nr) q ), ν + ((nq) r )) = g(λ, µ, ν).

g λ + t j=1 (n j ij ), µ + t j=1 (n j ij ), ν + t j=1 n j i j = g(λ + dα, µ + dα, ν + (d|α|)). ( 4
) (i)
(

) 5
Even more recently, J. Stembridge [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF]Corollary 6.2] proved the following general stability property for Kronecker coefficients. Denote φ λ = Ind Sn S λ (1 λ ) denote the permutation character associated to λ. Let α, β, γ be partitions of the same size, such that g(α, β, γ) > 0 and

φ nα ⊗ φ nβ , χ nγ = 1, for all n ∈ N. (6)
Then, for any triple of partitions λ, µ, ν of the same size, there is a positive integer L such that, for all n ≥ L,

g(λ + nα, µ + nβ, ν + nγ) = g(λ + Lα, µ + Lβ, ν + Lγ). (7)
In this paper we will show that practically all these stability properties fit into what we call additive stability. The main contribution of this paper is the discovery of the connection between additivity of integer matrices with Stembridge's approach to stability. Additivity, in our context, is a concept coming from discrete tomography (see [START_REF] Herman | Discrete Tomography: Foundations, Algorithms, and Applications[END_REF][START_REF] Herman | Advances in Discrete Tomography and Its Applications[END_REF] for an overview of the area). In [START_REF] Fishburn | Sets uniquely determined by projections on axes II. Discrete case[END_REF][START_REF] Fishburn | Sets of uniqueness and additivity in integer lattices[END_REF] Fishburn et al. consider the problem of studying finite subsets in R n that are uniquely reconstructible from its coordinate projection counting functions. They called them sets of uniqueness, and observed that there was no loss of generality in considering them as subsets of N n . They also introduced the notion of additive sets and proved that it was sufficient for uniqueness, but not necessary. Later, the author of this paper and Torres-Cházaro considered the case n = 3, and, by viewing those sets as 3-dimensional binary matrices, gave a characterization of sets of uniqueness using the dominance order of partitions [START_REF] Torres-Cházaro | Sets of uniqueness and minimal matrices[END_REF]. Afterward, the author of this paper introduced a notion of additivity for matrices with nonnegative integer entries [START_REF] Vallejo | A characterization of additive sets[END_REF] and used it to give a characterization of additive sets in the sense of [START_REF] Fishburn | Sets uniquely determined by projections on axes II. Discrete case[END_REF]. A different but equivalent definition of additivity appears already in Manivel's study of plethysm by means of Borel-Weil theory [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF]. The first proof for our characterization of uniqueness [START_REF] Torres-Cházaro | Sets of uniqueness and minimal matrices[END_REF] used character theory of the symmetric group (this proof can be translated into a purely combinatorial one [START_REF] Brualdi | Minimal nonnegative integral matrices and uniquely determined (0, 1)-matrices[END_REF]). At that time, we thought to apply the representation theory of the symmetric group to learn about uniqueness. It turned the other way round. Direct applications of discrete tomography to Kronecker coefficients were given in [START_REF] Avella-Alaminos | Kronecker products and the RSK correspondence[END_REF][START_REF] Vallejo | Plane partitions and characters of the symmetric group[END_REF]. These are, to my best knowledge, the first applications of discrete tomography to Kronecker coefficients. Here, we apply ideas from [START_REF] Onn | Permutohedra and minimal matrices[END_REF][START_REF] Santoyo | Additivity obstructions for integral matrices and pyramids[END_REF][START_REF] Torres-Cházaro | Sets of uniqueness and minimal matrices[END_REF][START_REF] Vallejo | Reductions of additive sets, sets of uniqueness and pyramids[END_REF][START_REF] Vallejo | Plane partitions and characters of the symmetric group[END_REF][START_REF] Vallejo | A characterization of additive sets[END_REF][START_REF] Vallejo | The classification of minimal matrices of size 2 × q[END_REF][START_REF] Vallejo | Minimal matrices and discrete tomography[END_REF][START_REF] Vallejo | Uniqueness and additivity for n-dimensional binary matrices with respect to their 1-marginals[END_REF] to prove additive stability, give examples and get some related results.

In order to state our main theorem we need the following definition, which is fundamental for this paper. A matrix A = (a i,j ) of size p × q is called additive (see [START_REF] Vallejo | A characterization of additive sets[END_REF]Thm. 1]), if there exist real numbers x 1 , . . . , x p , y 1 , . . . , y q , such that

a i,j > a k,l =⇒ x i + y j > x k + y l ,
for all 1 ≤ i, k ≤ p and all 1 ≤ j, l ≤ q. Let α, β be partitions of the same size. Denote by M(α, β) the set of matrices A with nonnegative integer entries, row-sum vector α and column-sum vector β. Also denote by π(A) the sequence of entries of A arranged in weakly decreasing order. Our main result is Theorem (Additive stability) 1.1 Let α, β, γ be partitions of the same size. If there is an additive matrix A ∈ M(α, β) with π(A) = γ, then, for any triple of partitions λ, µ, ν of the same size, the sequence {g(λ + nα, µ + nβ, ν + nγ)} n∈N0 is weakly increasing and bounded from above. In particular, there is a positive integer L such that, for all n ≥ L,

g(λ + nα, µ + nβ, ν + nγ) = g(λ + Lα, µ + Lβ, ν + Lγ).
This theorem appears in [START_REF] Vallejo | Stability of Kronecker coefficients via discrete tomography[END_REF]. It is implicit in [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF] and is made explicit in [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF]. Manivel's proof uses Borel-Weil theory and is of very different nature from ours.

Triples of partitions (α, β, γ) such that the sequence {g(λ + nα, µ + nβ, ν + nγ)} n∈N0 is weakly increasing and bounded from above for all triples of partitions λ, µ, ν are called stable ( [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF]). So, the additive stability theorem can be rephrased by saying that any additive matrix A yields a stable triple, or more precisely, if A ∈ M(α, β) is an additive matrix with π(A) = γ, then (α, β, γ) is a stable triple.

The next two results are new instances of stability. They follow from the additivity of matrices B and C in Example 3.12.

Corollary 1.2 Let a, b, c be nonnegative integers. Then, the triple

(b + c + 1, 1 a ), (a + c + 1, 1 b ), (c + 1, 1 a+b ) is stable. Corollary 1.3 For any k ∈ N, the triple k+1 2 , . . . , 2 2 , k+1 2 , . . . , 2 2 , (k, (k -1) 2 , . . . , 1 k ) is stable.
Next Corollary follows from the additivity of the matrix A in Example 3.13. It also appears as Example 6.3(b) in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF].

Corollary 1.4 Let α be a partition. Then, the triple (α, α , (1 |α| )) is stable.

Stembridge's hypothesis [START_REF] Church | FI-modules: a new approach to stability for Sn-representations[END_REF] and additivity are closely related. In Section 6 we show that additivity implies condition [START_REF] Church | FI-modules: a new approach to stability for Sn-representations[END_REF], and that this condition implies the existence of certain unique additive matrix. The advantage of our hypothesis is that it is far easier to find examples of additive matrices than to find examples of partitions α, β, γ satisfying [START_REF] Church | FI-modules: a new approach to stability for Sn-representations[END_REF]. Therefore, it is much easier to find new examples of stability. Another advantage is that there is already a body of results concerning additive matrices [START_REF] Fishburn | Sets uniquely determined by projections on axes II. Discrete case[END_REF][START_REF] Fishburn | Sets of uniqueness and additivity in integer lattices[END_REF][START_REF] Onn | Permutohedra and minimal matrices[END_REF][START_REF] Santoyo | Additivity obstructions for integral matrices and pyramids[END_REF][START_REF] Vallejo | Reductions of additive sets, sets of uniqueness and pyramids[END_REF][START_REF] Vallejo | A characterization of additive sets[END_REF][START_REF] Vallejo | Minimal matrices and discrete tomography[END_REF][START_REF] Vallejo | Uniqueness and additivity for n-dimensional binary matrices with respect to their 1-marginals[END_REF].

This manuscript is an extended abstract of [START_REF] Vallejo | Stability of Kronecker coefficients via discrete tomography[END_REF]. It is organized as follows. Section 2 contains some notation. Section 3 contains a summary of all results about uniqueness and additivity required for this paper, as well as examples and one application to Kronecker coefficients needed in the proof of additive stability. In Section 4 we present a geometric characterization of additivity from [START_REF] Onn | Permutohedra and minimal matrices[END_REF]. This is fundamental in the proof of the main theorem, sketched in Section 5, and also in Section 6, where we study the relation between additivity and condition [START_REF] Church | FI-modules: a new approach to stability for Sn-representations[END_REF]. We also present a new characterization of additivity (Theorem 6.4), that might be of interest in discrete tomography. Finally, in section 7, we relate additive stability to the previous known cases of stability of Kronecker coefficients.

Notation

We assume the reader is familiar with the standard results in the representation theory of the symmetric group (see for example [START_REF] James | The representation theory of the symmetric group[END_REF][START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF][START_REF] Sagan | The Symmetric Group[END_REF][START_REF] Stanley | Enumerative Combinatorics[END_REF]). We use thoughout the following notation: N is the set of positive integers, N 0 = N ∪ {0}, and, for any n ∈ N 0 , [ n ] = {1, . . . , n}, so that [ 0 ] = ∅. If λ = (λ 1 , . . . , λ p ) is a partition (a weakly decreasing sequence of positive integers), we denote its size i∈[ p ] λ i by |λ| and its length p by (λ). If |λ| = n, we also write λ n. The notation λ = (1 a1 , 2 a2 , . . . ) means that λ has a i parts equal to i. We denote by λ the partition conjugate to λ, obtained by transposing the diagram of λ. The depth of λ is d(λ) = |λ| -λ 1 . If λ and µ are two partitions of length p and q, respectively, and p ≤ q, we denote λ p+1 = • • • = λ q = 0 and define its sum by

λ + µ = (λ 1 + µ 1 , . . . , λ q + µ q ).
Given two partitions λ, µ of n we write λ µ to indicate that λ is greater than or equal to µ in the dominance order of partitions. We write λ µ, if λ µ and λ = µ. The number of semistandard Young tableaux of shape λ and content ν is denoted by K λν .

For any partition λ n, we denote by χ λ the irreducible character of S n associated to λ, and, for any partition ν = (ν 1 , . . . , ν r ) of n, by φ ν = Ind Sn Sν (1 ν ) the permutation character associated to ν. That is, φ ν is the character induced from the trivial character of the Young subgroup S ν = S ν1 × • • • × S νr to S n .

Discrete tomography

In this section we gather several results from discrete tomography that are used to prove additive stability. These results have been obtained mainly by the author of this note and his collaborators along the years. Some of them have been recently rediscovered by Stembridge and appear in [28, §6]. Applications of discrete tomography to Kronecker coefficients have already been given before in [START_REF] Vallejo | Plane partitions and characters of the symmetric group[END_REF][START_REF] Avella-Alaminos | Kronecker products and the RSK correspondence[END_REF]. Two notions from discrete tomography are relevant for this paper: uniqueness and additivity of 3-dimensional binary matrices (see [START_REF] Fishburn | Sets of uniqueness and additivity in integer lattices[END_REF][START_REF] Vallejo | Uniqueness and additivity for n-dimensional binary matrices with respect to their 1-marginals[END_REF]). They are related to the notions of minimality, π-uniqueness [START_REF] Torres-Cházaro | Sets of uniqueness and minimal matrices[END_REF][START_REF] Vallejo | Minimal matrices and discrete tomography[END_REF] and additivity [START_REF] Vallejo | A characterization of additive sets[END_REF][START_REF] Onn | Permutohedra and minimal matrices[END_REF][START_REF] Vallejo | Minimal matrices and discrete tomography[END_REF], which are defined for matrices with nonnegative integer entries. Below we explain these concepts, the relations among them, give examples and show how they are related to Kronecker coefficients.

Integral matrices 3.1 Let A = (a i,j ) be a matrix of size p × q. We associate to A two compositions and one partition. For each i ∈ [ p ] and each j ∈ [ q ], one defines

α i = y∈[ q ] a i,y and β j = x∈[ p ] a x,j .
Then, α = (α 1 , . . . , α p ) is called the row-sum vector and β = (β 1 , . . . , β q ) the column-sum vector of A. We denote by π(A) the vector of entries of A arranged in weakly decreasing order. Then, π(A) is a partition called the π-sequence of A. We say that A is a plane partition if it has nonnegative integer entries and its rows and columns are weakly decreasing. For the applications we have in mind we assume, from now on, without loss of generality, that the row-sum and column-sum vectors are weakly decreasing. Otherwise we just permute rows and columns.

We denote by M(α, β) the set of all matrices A = (a i,j ) with nonnegative integer entries, row-sum vector α and column-sum vector β. If γ is a partition, we denote by M(α, β) γ the set of all matrices in M(α, β) with π-sequence γ.

Binary matrices 3.2 Let A = (a i,j,k ) be a matrix of size p × q × r with entries in {0, 1} (we call it a binary matrix for short). We associate to A three compositions. For each i ∈ [ p ], j ∈ [ q ] and k ∈ [ r ], one defines

α i = (y,z)∈[ q ]×[ r ] a i,y,z , β j = (x,z)∈[ p ]×[ r ] a x,j,z and γ k = (x,y)∈[ p ]×[ q ] a x,y,k .
Then, the compositions α = (α 1 , . . . , α p ), β = (β 1 , . . . , β q ), γ = (γ 1 , . . . , γ r ) are called the 1-marginals of A. See, for example [START_REF] Loera | Combinatorics and geometry of transportation polytopes: an update[END_REF], where 3-dimensional binary matrices are called three-way statistical tables.

For the applications we have in mind we assume, from now on, without loss of generality, that the 1marginals are weakly decreasing. Otherwise we just permute 2-dimensional slices.

We denote by M * (α, β, γ) the set of all 3-dimensional binary matrices with 1-marginals α, β, γ, and by m * (α, β, γ) its cardinality. A matrix X ∈ M * (α, β, γ) is called a matrix of uniqueness [START_REF] Fishburn | Sets uniquely determined by projections on axes II. Discrete case[END_REF], if m * (α, β, γ) = 1.

More on integral matrices 3.3 Let α, β be partitions of the same size and let A = (a i,j ) ∈ M(α, β). We say that A is minimal [START_REF] Torres-Cházaro | Sets of uniqueness and minimal matrices[END_REF] if there is no matrix B ∈ M(α, β) such that π(B) ≺ π(A), and we say that A is π-unique if there is no other matrix B ∈ M(α, β) such that π(B) = π(A). Suppose A has size p × q, and let r be the maximum of the entries of A. The graph of A is the 3-dimensional binary matrix

G(A) = (a i,j,k ) defined, for all (i, j, k) ∈ [ p ] × [ q ] × [ r ], by a i,j,k = 1 if 1 ≤ k ≤ a i,j ; 0 otherwise. Note that, if A ∈ M(α, β) γ , then G(A) ∈ M * (α, β, γ ). Therefore, G : M(α, β) γ -→ M * (α, β, γ )
is a well-defined injective map. If X is the image G(A) of a plane partition A, then X is called the diagram of A [START_REF] Stanley | Symmetries of plane partitions[END_REF] or pyramid [START_REF] Vallejo | Reductions of additive sets, sets of uniqueness and pyramids[END_REF].

The next theorem relates the property of uniqueness for binary matrices to properties of integral matrices. Part of it was recently rediscovered by J. Stembridge (see [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF]Thm. 6.4]).

Theorem 3.4 [31, Thm. 1] Let α, β, γ be partitions. Then m * (α, β, γ ) = 1 if and only if there is a matrix A ∈ M(α, β) γ that is minimal and π-unique. Moreover, if A ∈ M(α, β) is minimal and π-unique, then A is a plane partition.

To the best of our knowledge, the first application of uniqueness to Kronecker coefficients is Theorem 1.1 in [START_REF] Vallejo | Plane partitions and characters of the symmetric group[END_REF]. Here, we need only a particular case, which was recently rediscovered by J. Stembridge [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF]Thm. 6.4]. Theorem 3.5 [START_REF] Vallejo | Plane partitions and characters of the symmetric group[END_REF]Cor. 4.2] Let A ∈ M(α, β) γ be a matrix that is minimal and π-unique. Then, g(α, β, γ) = 1. Now we turn to the notion of additivity, which is fundamental for our main theorem. It appears already, with no name, in [START_REF] Vallejo | A characterization of additive sets[END_REF]. Definition 3.6 Let A = (a i,j ) be a matrix of size p × q. Then, A is called additive [37, § 6] if there exist real numbers x 1 , . . . , x p , y 1 , . . . , y q such that a i,j > a k,l =⇒ x i + y j > x k + y l , for all i, k ∈ [ p ] and all j, l ∈ [ q ]. Even though the definition makes sense in general, we will be concerned only with matrices having nonnegative entries. In this section we work with matrices having integer entries, in Section 4 with matrices having real entries.

In the next example we illustrate the concepts of minimal, π-unique and additive (see Example 8 in [START_REF] Vallejo | Uniqueness and additivity for n-dimensional binary matrices with respect to their 1-marginals[END_REF]).

Example 3.7 Let

A =   3 3 1 2 1 1 2 0 0   , B =   4 4 1 2 1 1 2 0 0   and C =   4 3 2 3 1 0 1 1 0   .
The three matrices A, B, C are plane partitions. The first one is minimal, but not π-unique. The matrix B is π-unique, but not minimal. Finally the matrix C is additive. To see this take (x 1 , x 2 , x 3 ) = (7, 2, 0) and (y 1 , y 2 , y 3 ) = (6, 3, 0). Now we show some relations between being additive, minimal, π-unique and a plane partition. We start with the following result. Part of it was recently rediscovered by J. Stembridge (see [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF]Prop. 6.9]).

Theorem 3.8 [35, Thm. 2] Let A ∈ M(α, β) of size 2 × q.
Then, the conditions of A being a plane partition; minimal and π-unique; and additive are equivalent. This result does not hold in general. Already, the matrix A in Example 3.7 is a plane partition that is not π-unique, and B is a plane partition that is not minimal. Obstructions for a plane partition A of size 3 × 3 to be minimal and π-unique are given in [32, §5]. These correspond to the obstructions for additivity given by the arrow diagram in [25, Fig. 3] and its transpose. It is not difficult to show that these are all obstructions to additivity for a plane partition of size 3 × 3 (see [START_REF] Santoyo | Aditividad para matrices y (0, 1)-aditividad para pirámides[END_REF]Ch. 2]). Compare with Proposition 6.11 in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF]. We have however one implication. Theorem 3.9 [35, Thm. 6.1] Let A be an additive matrix, then A is minimal and π-unique.

The converse is not true (see [START_REF] Vallejo | Minimal matrices and discrete tomography[END_REF]Ex. 7.4] and [25, §3]).

Example 3.10 The matrix

A =       5 5 5 4 4 5 5 5 3 3 3 3 1 1 0 2 1 1 1 0 2 1 0 0 0      
is minimal and π-unique but not additive.

Next result follows from Theorems 3.4, 3.5 and 3.9. Part of it also appears in [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF].

Theorem 3.11 Let A ∈ M(α, β) γ be additive. Then A is a plane partition and

g(α, β, γ) = 1.
Examples 3.12 The following are additive matrices

A =    r • • • r . . . . . . . . . r • • • r    , B =      c + 1 1 • • • 1 1 0 • • • 0 . . . . . . . . . . . . 1 0 • • • 0      , C =        k k -1 • • • 2 1 k -1 k -2 • • • 1 0 . . . . . . . . . . . . . . . 2 1 • • • 0 0 1 0 • • • 0 0        .
The graph of A is a 3-dimensional box of size p × q × r; the graph of B is called tripod, it has size (a + 1) × (b + 1) × (c + 1) and 1-marginals

(b + c + 1, 1 a ), (a + c + 1, 1 b ), and (a + b + 1, 1 c ). The graph of C is a pyramid of size k × k × k.
Example 3.13 Let α be a partition. Let A be the only binary matrix in M(α, α ). Then A is additive. For example, if α = (4, 2, 1), then

A =   1 1 1 1 1 1 0 0 1 0 0 0   .
To see that A is additive take x i = α i and y j = (α ) j .

Symmetries of plane partitions 3.14

There is a group T of order 12 of operations on plane partitions. These operations are very natural if we look at the graph or diagram of a plane partition. The group is generated by a rotation of order 3, a transposition of order 2 and a complementation of order 2, see [26, § 2]. It is not difficult to prove that these operations preserve additivity. Therefore, given any additive matrix, we can generate up to 12 different additive matrices.

Definition 3. [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF] We say that (α, β, γ) is an additive triple if there is an additive matrix A ∈ M(α, β) γ .

Using rotations and transpositions on G(A) we obtain that (γ , α, β ), (β, γ , α ), (β, α, γ), (α, γ , β ) and (γ , β, α ) are also additive triples. Theorem 1.1 can be reformulated by saying that any additive triple is stable. The converse is not true since the triple ((2, 2), (2, 2), (2, 2)) is stable [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF], but not additive. Remark 3. [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF] The additive triples (α, β, γ) and (γ , α, β ) yield different stability properties. The identity g(λ, µ, ν) = g(λ, µ , ν ) and other symmetries of Kronecker coefficients are not enough to prove that they yield the same stability property, because, in general (µ + β) = µ + β . Obstructions to additivity 3.18 There are also ways of proving that a matrix is not additive. In [START_REF] Santoyo | Additivity obstructions for integral matrices and pyramids[END_REF]Thm. 3.8] we showed that certain arrow diagrams are obstructions to additivity. While the case of two rows is fairly simple (Theorem 3.8), the general case is much more complex. We showed in [25, § 5] that there are infinitely many essentially different obstructions needed for deciding additivity of plane partitions with three rows.

Geometry of additive matrices

In this section we explain a geometric characterization of additivity from [START_REF] Onn | Permutohedra and minimal matrices[END_REF]. It will be central in our proof of additive stability. We start by extending some notions defined for objects with integer entries to objects with real entries. For a vector a = (a 1 , . . . , a m ) ∈ R m , we denote by π(a) = (a [ 1 ] , . . . , a [ m ] ) the vector formed by the entries of a arranged in weakly decreasing order. We say that a is majorized by b = (b 1 , . . . , b m ) (see [START_REF] Hardy | Inequalities[END_REF][START_REF] Marshall | Inequalities: Theory of majorization and its applications[END_REF]), and denote it by a b, if

m i=1 a i = m i=1 b i , and k 1=1 a [ i ] ≤ k i=1 b [ i ] , for all k ∈ [ m ].
If a b and π(a) = π(b), then we write a ≺ b.

Let α, β be two partitions of the same size. Denote by T(α, β) the set of all matrices with nonnegative real entries, row-sum vector α and column-sum vector β. T(α, β) is called a transportation polytope. We say that a matrix A ∈ T(α, β) is real-minimal [START_REF] Onn | Permutohedra and minimal matrices[END_REF], if there is no other matrix B ∈ T(α, β) such that π(B) ≺ π(A). Let a ∈ R m and ρ ∈ S m . Denote by a ρ the vector (a ρ(1) , . . . , a ρ(m) ). The permutohedron determined by a is the convex hull of the set of all vectors obtained by permuting the entries of a:

P(a) = conv{a ρ | ρ ∈ S m }.
In order to state our next results, we denote, by M p,q the set of all matrices with real entries of size p × q and define a linear isomorphism Φ : M p,q -→ R pq , for each A = (a i,j ), by Φ(A) = (a 11 , a 12 , . . . , a 1q , a 21 , a 22 , . . . , a 2q , . . . , a p1 , a p2 , . . . , a pq ).

We have the following characterization of real-minimality.

P(Φ(A)) ∩ Φ(T(α, β)) = {Φ(A)}.
The following example disproves Conjecture 6.7 in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF].

Relations between additive stability and other stability known results

In this section we explore the relation of previous stability known results to additive stability. Let us consider the additive triples ((|α|), α, α) and (α, α , (1 |α| )), see Example 3.17. Practically, all known stability properties follow from Theorem 1.1 applied to some instance of one of these triples, together with some symmetry of Kronecker coefficients. In other words, every known stability property is related, up to symmetry of its graph (Paragraph 3.14), to some particular case of the additive matrix from Example 3.17, or equivalently, to some particular case of the additive matrix from Example 3.13. The power of additive stability is that we can construct very easily many other examples of additive matrices, thus producing new instances of stability. However, the method of proof of this theorem does not produce an explicit bound L for stability.

Murnaghan's stability (1) follows from α = (1). The stability from equation ( 4), which includes (2) as a particular case, follows, in case n 1 = • • • = n t , from Theorem 1.1 applied to the additive triple ((|α|), α, α) together with the symmetry g(ζ, η, θ) = g(η, θ, ζ). The diagrammatic method from [START_REF] Vallejo | A diagrammatic approach to Kronecker squares[END_REF] provides better results in the case of the triple ((|α|), α, α), because it gives very precise bounds for stability and also because it permits to consider independent parameters n 1 , . . . , n t in equation ( 4).

Finally, the stability property from equation ( 5) follows from Theorem 1.1 applied to the additive triple (α, α , (1 |α| )) in the particular case in which α = (q r ) is a rectangular partition, together with a symmetry of Kronecker coefficients. Formula (5) gives, under some conditions, a very good bound for stability, which is useful in some applications (see [39, §5]).

Example 3 . 17

 317 Let β = (β 1 , . . . , β b ) n. Then, B = β 1 • • • β b is an additive matrix and ((n),β, β) is an additive triple. Hence, also (β, β , (1 n )) is an additive triple. This is Example 3.13.

Theorem 4 . 1

 41 [START_REF] Onn | Permutohedra and minimal matrices[END_REF] Thm. 6.2] Let A ∈ T(α, β). Then A is additive if and only if A is real-minimal.

Proposition 4 . 2

 42 [START_REF] Onn | Permutohedra and minimal matrices[END_REF] Cor. 5.2] Let A ∈ T(α, β). Then A is real-minimal if and only ifP(Φ(A)) ∩ Φ(T(α, β)) = {Φ(A)}.Next result is fundamental in our proof of additive stability. It is a consequence of Theorem 4.1 and Proposition 4.2. Theorem 4.3 Let A ∈ T(α, β). Then A is additive if and only if

Acknowledgements

I would like to thank to Richard Brualdi, Jesús De Loera, Laurent Manivel, Shmuel Onn, Igor Pak, Greta Panova and Adolfo Torres-Cházaro for fruitful conversations.

The matrix X comes from the obstruction to additivity given in [START_REF] Santoyo | Additivity obstructions for integral matrices and pyramids[END_REF]Fig. 4]. Let α = [START_REF] Sagan | The Symmetric Group[END_REF][START_REF] Onn | Permutohedra and minimal matrices[END_REF][START_REF] Fishburn | Sets uniquely determined by projections on axes II. Discrete case[END_REF][START_REF] Brualdi | Minimal nonnegative integral matrices and uniquely determined (0, 1)-matrices[END_REF][START_REF] Briand | The stability of the Kronecker product of Schur functions[END_REF], β = [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF][START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF][START_REF] Herman | Advances in Discrete Tomography and Its Applications[END_REF][START_REF] Fishburn | Sets of uniqueness and additivity in integer lattices[END_REF][START_REF] Loera | Combinatorics and geometry of transportation polytopes: an update[END_REF]. Since the row-sum and column-sum vectors of X are zero, A and A - 1 2 X are elements of T(α, β). We know, by Example 3.10, that A is minimal and π-unique. One easily checks that π(A -1 2 X) ≺ π(A). Therefore A is not real minimal. Hence, P(Φ(A)) ∩ Φ(T(α, β)) has only one integer point, but it is not 0-dimensional.

Sketch of the proof of additive stability

For each convex polytope P let us denote by #P the number of integer points in P .

Next result follows from Theorem 4.3 and a close analysis of the convergence of the sequence of polytopes P(

Theorem 5.1 Let A ∈ M(α, β) γ be additive. Then, for any triple of partitions λ, µ, ν of the same size, the sequence of integers

is weakly increasing and bounded from above.

The proof of additive stability combines Theorem 5.1 and Proposition 5.2 in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF] and a well-known formula for the Kronecker product of two permutation characters.

Relation between Stembridge's hypothesis and additivity

In this Section we show the relation between Stembridge's hypothesis (6) and additivity. Theorem 6.1 Let A ∈ M(α, β) γ be an additive matrix. Then, φ nα ⊗ φ nβ , χ nγ = 1, for all n ∈ N.

If φ nα ⊗ φ nβ , χ nγ = 1, for all n ∈ N, we cannot assure the existence of an additive matrix in M(α, β) γ (see Example 6.3). But we can prove. Theorem 6.2 Let α, β, γ be partitions of the same size. If φ nα ⊗ φ nβ , χ nγ = 1, for all n ∈ N. Then, there is a unique matrix A ∈ P(γ) ∩ M(α, β). Besides A is additive and K γ,π(A) = 1.

Compare with Theorem 6.1 in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF]. Example 6.3 Let α = (7, 1), β = (5, 3) and γ = (4, 4). We will show that φ nα ⊗ φ nβ , χ nγ = 1, for all n ∈ N, and that there is no additive matrix in M(α, β) γ . First, note that M(α, β) γ = ∅. Hence, the second assertion holds. For each t ∈ N, let X t = 4n+t 3n-t n-t t

. Then, M(nα, nβ) = {X t | 0 ≤ t ≤ n}. The only t for which, π(X t ) nγ, is t = 0. The first assertion follows from K nγ,n(4,3,1) = 1.

We also obtain a new characterization of additivity. Theorem 6.4 Let A be a plane partition. Then A is additive if and only if nA is minimal and π-unique, for all n ∈ N. Example 6.5 Let A and X be defined as in Example 4.4. We now that A is minimal and π-unique, but not additive (Example 3.10). Besides 2A is not minimal because π(2A -X) ≺ π(2A).


Héctor Blandin
email: hectorblandin@gmail.com.

Generalized Polarization Modules (extended abstract)

Keywords: Algebraic Combinatorics, symmetric functions, diagonally symmetric polynomials, representation theory, polarization operators

This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest Sn-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in sets of n variables xij with 1 ≤ i ≤ et 1 ≤ j ≤ n. Given a Sn-stable family of homogeneous polynomials in the variables xij the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains F is the polarization module generated by the family F . These polarization modules are all representation of the direct product Sn × GL (C). In order to study the decomposition into irreducible submodules, we compute the graded Frobenius characteristic of these modules. For several cases of Sn-stable families of homogeneous polynomials in n variables, for every n ≥ 1, we show general formulas for this graded characteristic in a global manner, independent of the value of .

Résumé. Ce travail s'inscrit dans la lignée de recherche des travaux de M. Haiman sur le théorème de l'opérateur (exconjecture de l'opérateur). Ce théorème affirme que le plus petit Sn-module clos par dérivation partielle et clos par l'action des opérateurs de polarisation qui contient le déterminant de Vandermonde est l'espace des polynômes harmoniques diagonaux. On commence par généraliser le contexte du théorème de l'opérateur au contexte de polynômes à ensembles de n variables xij avec 1 ≤ i ≤ et 1 ≤ j ≤ n. Étant donnée une famille Sn-stable F des polynômes homogènes en les variables xij, le plus petit espace vectoriel MF clos par dérivation partielle et clos par l'action des opérateurs de polarisation contenant F est le module de polarisation engendré par la famille F . Les modules MF sont tous des représentations du produit direct Sn×GL (C). Dans le but d'étudier la décomposition en sous-modules irréductibles on calcule la caractéristique de Frobenius graduée de ces modules. Pour plusieurs cas de familles homogènes Sn-stables constituées des polynômes homogènes à n variables, pour tout n ≥ 1, on démontre des formules générales pour cette caractéristique graduée de fac ¸on globale, indépendante de la valeur de .

Introduction

This work is inspired by the Operator Theorem (ex-operator conjecture) of M. Haiman (see [START_REF] Haiman | Vanishing theorems and character formulas for the hilbert scheme of points of the plane[END_REF]). This theorem states that the smallest subspace of C[x 1 , . . . , x n ] closed under taking partial derivatives ∂ ∂xi , closed under the action of generalized polarization operators, E p = n j=1 y j ∂ p ∂x p j , that contains the Vandermonde determinant ∆ n (x) := 1≤i<j≤n (x i -x j ), coincides with the space D n of diagonal harmonics polynomials of S n . The space D n consist of all polynomials f in the variables x 1 , . . . , x n , y 1 , . . . , y n killed by the power sum differential operators

n j=1 ∂ h ∂x h j ∂ k ∂y k j with 1 ≤ h + k ≤ n.
In others words this space is generated by the Vandermonde determinant as a module over the algebra of operators C ∂ ∂x1 , . . . , ∂ ∂xn , E 1 , . . . , E n-1 . In this work we generalize this construction by considering polynomials in sets of n variables, that is, polynomials in the variables x 11 , . . . , x 1n , . . . , x 1 , . . . , x n . We are interested in the decomposition into irreducible submodules of these spaces.

We start by generalizing the context of the Operator Theorem to the context of polynomials in the matrix variables X = (x ij ), with 1 ≤ i ≤ and 1 ≤ j ≤ n. The diagonal action of S n on this polynomials in sets of n variables is defined by permuting the columns of X, that is, for any permutation σ ∈ s n we replace the variable x ij by x iσ(j) . We say that a family F of homogeneous polynomials (in X = (x ij )) is S n -stable if F is closed under the diagonal action of S n . Given any such a family F , we define the polarization module generated by the family F as the smallest vector space closed under taking partial derivatives ∂ ∂xij , closed under the action of generalized polarization operators

E (p) i,k = n j=1 x ij ∂ p ∂x p kj
, that contains F . The diagonal action of S n makes M F an S n -module. The closure by the action of polarization operators E (p) i,k is equivalent to the closure by the action x ij -→ k=1 m ik x kj where M = (m ij ) ∈ GL (C) (see [START_REF] Kfrat | Classical Invariant Theory A primer[END_REF][START_REF] Procesi | Lie Groups An Approach through Invariants and Representations / Claudio Procesi[END_REF]). Then, with this action M F is also a polynomial representation of GL (C). The two actions of S n and GL (C) commutes and this implies that M F is a representation of the direct product S n × GL (C). In particular, when the family F is the orbit of a single homogeneous polynomial f , that is, F = {σ • f | σ ∈ S n } we denote the polarization module generated by F simply as M f . Also, we call M f the polarization module generated by f . Particular cases of this construction correspond to certain important spaces in Combinatorics (see [START_REF] Bergeron | Multivariate diagonal coinvariant spaces for complex reflection groups[END_REF]) and Algebraic Geometry (see [START_REF] Geramita | Hilbert functions of S n -stable artinian gorenstein ideals[END_REF]). For instance, when = 1 (1 set of variables x = x 1 , . . . , x n ) and f = ∆ n (x) we get the space H n of harmonics polynomials of the group S n , that is, the space of polynomials zeros of the power sum differential operators [START_REF] Bergeron | Algebraic combinatorics and coinvariants spaces[END_REF]). It's remarkable that the dimension of the space H n is n! (see [START_REF] Humphreys | Reflections groups and Coxeter groups[END_REF]). As we already said, the case = 2 and f = ∆ n (x) gives us the space D n of diagonal harmonics polynomials. It's also a remarkable fact that dim(D n ) = (n + 1) n-1 (the number of parking functions) (see [START_REF] Haiman | Vanishing theorems and character formulas for the hilbert scheme of points of the plane[END_REF]). The space of trivariate diagonal harmonics D The goal of this paper is to study the decomposition into irreducible submodules under the action of S n × GL (C) of polarization modules M F . To do this we compute explicitly the graded Frobenius characteristic of M F in the form

n i=1 ∂ ∂x k i with k such that 1 ≤ k ≤ n, (see
M F (q, w) = λ n |µ|≤d b λ,µ s µ (q)s λ (w) (1)
where b λ,µ ∈ N, q = q 1 , q 2 , . . . , q , w = w 1 , w 2 , . . ., and d is the maximal degree of polynomials in F .

Here the Schur functions s µ (q) encodes irreducible for GL (C) while s λ (w) encodes irreducible for S n .

The coefficients b λ,µ are the multiplicities of irreducible submodules under the action of S n × GL (C).

A theorem of F. Bergeron shows that this coefficients are independent of , in others words, the value of only affects the formula (1) giving the numbers of q variables appearing in s µ (q) (see [START_REF] Bergeron | Multivariate diagonal coinvariant spaces for complex reflection groups[END_REF]). Furthermore, he shows in (2) that µ has at most n parts. This leads us to obtain a general formula for (1) that holds for any ≥ 1 if we can compute it for every ≤ n.

In this paper we completely describe the decomposition into irreducible submodules of the polarization modules generated by each of the polynomials p d 1 , p d , and e d for any d ≥ 1. We construct an explicit linear basis of each module and then we compute the graded Frobenius characteristic of M p d 1 , M p d and M e d . We propose a conjectural formula for the graded Frobenius series of the polarization module generated by the monomial symmetric function m (2,1 d-2 ) , in any degree d. For instance, we also conjecture that, for d ≥ 5 we must have

M e d-1,1 ∼ = M m (2,1 d-2 ) as S n × GL (C)-modules.
We believe that these are modules are the elementary building blocks for a general classification of polarization modules generated by a given homogeneous symmetric polynomial in any number of variables n. Experimental evidence has shown that the last assertion is true up to degree 5 and n ≤ 6. In particular, we completely determine the classification of polarization modules generated by a single homogeneous symmetric polynomial when the degree is 2 or 3. For the case of degree 4 and 5 we have a conjectural classification that seems to be complete. This framework lead us to think that the Hilbert series of the polarization modules generated by a single homogeneous symmetric polynomial, in any degree, is always h-positive (see Conjecture 5.8). Obviously, we have M f ∼ = M k•f for every scalar k. In order to classify up to isomorphism polarization modules generated by a given homogeneous symmetric polynomial of any degree d, we identify any non zero homogeneous symmetric polynomial f of degree d, written in the monomial basis as f = λ d c λ m λ , with a point in the real projective space RP p(d)-1 , where p(d) is the number of integer partitions of d. The homogeneous coordinates of the corresponding point are ordered according to the following order on integer partitions of d: (d), (d -1, 1), (d -2, 2), (d -2, 1, 1),. . ., (1, 1, . . . , 1). In degree 2, we show that there are two types of polarization modules M p 2 1 and M p2 up to isomorphism. More precisely, if

[a : b] ∈ RP 1 and f = a • m 2 + b • m 11 then M f ∼ = M p 2 1 when [a : b] = [1 : 2], while [a : b] = [1 : 2] implies M f ∼ = M p2 .
Notice that the last statement is independent on the number of variables n. The situation when the degree is 3 is more complicated, in this case, we will need to introduce the notion of n-exception to completely classify these polarization modules.

Let n ≥ 3, a point [a : b : c] ∈ RP 2 is a n-exception if and only if [a : b : c] = [1 : 3 : 6] and 6a(2b+(n-2)c) = 4(n-1)b 2 . When n = 2, [a : b : c] is a 2-exception if
= a • m 3 + b • m 21 + c • m 111 . If [a : b : c] = [1 : 3 : 6] then M f ∼ = M p 3 1 ; if [a : b : c] is a n-exception then M f ∼ = M p3 ; otherwise, M f ∼ = M h3 .
These results are valid for any (the number of sets of n variables). Also, we will see that n-exceptions appear in any degree d ≥ 3. Characterize n-exceptions for degrees higher than 3 is a problem for the future.

For particulars S n -stable families of homogeneous polynomials we completely describe the graded Frobenius characteristic of the associated polarization module. In particular, we have open problems about the constraints for the multiplicities of irreducible submodules of polarization modules generated by any S n -stable family consisting of homogeneous polynomials of degree at most 2. Similarly, for any S n -stable families of homogeneous polynomials of degree at most 3. More precisely, we compute the graded Frobenius characteristic of polarization modules generated by the family T d of all monomials of degree d, when d = 2, 3 in any number of variables n. If F is any set of polynomials of degree d at most, then M F ⊆ M T d . This implies that the multiplicities of irreducible S n × GL (C)-modules of M F are least or equal to the corresponding multiplicities in M T d .

Preliminaries

Let X := (x ij ) i,j be a × n matrix of commuting and independent variables x ij For any i, we call the i th -row of X, denoted by x i := (x i1 , . . . , x in ), the i th set of variables. For any j, X j denotes the j th column of X. The same convention is adopted for any × n non negative integer matrix of exponents A. Then the monomials are defined as follows:

X A := x a11 11 • • • x aij ij • • • x a n n . (2)
these monomials form a linear basis of the C-vector space R ( )

n = C[X]
of polynomials in sets of n variables. The (vector) degree deg X A lies in N and is given by deg

X A := n j=1 a 1j , . . . , n j=1 a j . For each d ∈ N , we denote by R ( ) n,d the span of degree d monomials in R ( ) n . Then R ( ) n is a N -graded, that is, R ( ) n = d∈N R ( ) n,d . We will consider homogeneous subspace V of R ( ) n , the degree d homo- geneous component of V is denoted by V d . Recall that, V d := V ∩ R ( )
n,d and the Hilbert series of V is V(q) := d∈N dim(V d )q d , where q := (q 1 , . . . , q ) and q

d := q d1 1 • • • q d .
On V we consider two linear group actions:

1. The (left) diagonal action of S n , σ • X A := x a11 1σ(1) • • • x aij iσ(j) • • • x a n σ(n) , ∀σ ∈ S n . 2. The (right) action of GL (C), X A •M := (M X) A , ∀M ∈ GL (C), that is, x ij -→ k=1 m ik x kj , for every matrix M = (m ij ) ∈ GL (C).
These two group actions on V commute and then we can consider V as a representation of the direct product S n × GL (C), with the (left) action, (σ, M ) • X A := σ • (M -1 X) A . Then we have a direct sum decomposition of the form (see [START_REF] Sagan | The Symmetric Group Representations, Combinatorial Algorithms, and Symmetric Functions[END_REF][START_REF] Fulton | Representation Theory A First Course[END_REF][START_REF] Procesi | Lie Groups An Approach through Invariants and Representations / Claudio Procesi[END_REF])

V = λ n µ b λ,µ W µ ⊗ U λ , (3)
where b λ,µ ∈ N, the U λ are irreducible S n -modules and the W µ are irreducible polynomial representations of GL (C). The graded Frobenius characteristic of V is defined as follows

V(q, w) := d∈N 1 n! σ∈Sn χ V d (σ) p λ(σ) (w) q d , (4)
where χ V d is the S n -character of V d and p λ(σ) (w) := p 1 (w) c1(σ) • • • p n (w) cn(σ) . It's well known (see [START_REF] Bergeron | Multivariate diagonal coinvariant spaces for complex reflection groups[END_REF]) that the graded Frobenius characteristic of V has the following form (not depending on ):

V(q, w) = λ n µ b λ,µ s µ (q)s λ (w), (5)
where b λ,µ are the multiplicities in (3). (µ) ≤ n (see [START_REF] Bergeron | Multivariate diagonal coinvariant spaces for complex reflection groups[END_REF] for more details). The Schur functions s µ (q) encode the irreducible polynomial representations of GL (C) in V of type µ, and the Schur functions S λ (w) encodes the irreducible S n -modules in V of type λ. Recall that the Hilbert Series of V is obtained by replacing s λ (w) by f λ (the Hook length formula) for every λ in formula 5.

Definitions and discussions

We denote the partial derivative operator on R i,k depend on the choice of (i, k, p). For p = 1 we simply write E i,k := E

i,k (see, [START_REF] Weyl | The Classical Groups: Their Invariants and Representations[END_REF][START_REF] Hunziker | Classical invariant theory for finite reflection groups[END_REF], and ( 13) for more details).

Generalized Polarization Modules

We say that a subspace V of R ( )

n is closed under derivatives if for every g ∈ V we have ∂ ij (g) ∈ V, for all (i, j) such that 1 ≤ i ≤ and 1 ≤ j ≤ n. We say that V is closed under polarization if E (p) i,k (g) ∈ V
for all g ∈ V and all suitable triple (i, k, p). Since any intersection of closed under derivatives subspaces is closed under derivatives we define the derivative closure D(V) of V as the smallest subspace of R ( ) n closed under derivatives that contains V. Similarly, we define the polarization closure E(V) of V. It's not hard to show the following result (proofs will be avaible in a complete version of this paper, see (4)): Lemma 3.1 Let V be a homogeneous subspace of R ( )

n then E(D(V)) = D(E(V)). We set P(V) := E(D(V)). A subset F of R ( )
n is called a homogeneous stable family if the following conditions holds:

1. F consist only of homogeneous polynomials, 2. F is stable (or S n -stable w.r.t. the diagonal action of S n ), that is, for any permutation σ ∈ S n we have σ • g ∈ F , for all g ∈ F .

Definition 3.1 For a given homogeneous stable family F , we set M F to be the smallest R-vector space closed under derivatives and closed under polarization containing the family F . We call the vector space M F the polarization module generated by the family F .

We can describe the polarization module generated by F as M F := P R • F , where R • F denotes the real vector space spanned by F .

M f := P R • {σ • f | σ ∈ S n } .

Properties of Polarization Modules

The closure under polarization operators property implies that any polarization modules is a polynomial representation of GL (C) with the (right) action f (X) • M := f (M X). In fact

Lemma 3.2 (see C. Procesi, (13)) A subspace V of R ( )
n is closed under the action of polarization operators E i,k (when p = 1) if and only if V is a GL (C)-module with the action

X A • M = (M X) A .
For any g ∈ R ( )

n we have the identities: σ • E (p) i,k (g) = E (p) i,k (σ • g) and σ • ∂ ij (g) = ∂ i,σ(j) (σ • g).
These identities implies the following assertions for any family F Lemma 3.3 M F is a representation of S n with the diagonal action of S n .

Recall that the two actions of S n and GL (C) on R ( ) n commutes and so, we can assert that Lemma 3.4 M F is a S n × GL (C)-module with the action (σ, M ) • f (X) := f (M -1 X σ ), where X σ := (x iσ(j) ) i,j .

Frobenius characteristics of some polarization modules

In the following lines we use the notation of Macdolnald's book [START_REF] Macdonald | Symmetric functions and Hall polynomials Second edition[END_REF]. The proofs of the following results will be available in a complete version of this paper (see ( 4)).

M e d 1 (q, w) =   d j=0 s j (q)   s n (w) =   d j=0 h j (q)   h n (w). (6)
M p d (q, w) =   m j=0 s j (q)   s n (w) +   m-1 j=1 s j (q)   s n-1,1 (w) (7) = 1 + h d (q) h n (w) +   d-1 j=1 h j (q)   h n-1,1 (w). ( 8
)
M e d (q, w) = d/2 i=0   d-i j=i s j (q)   s n-i,i (w) (9) = d 2 i=0 h n-i,i (w) h i (q) + d i= d 2 +1 h n-d+i,d-i (w) h i (q). ( 10
)
Theorem 4.2 The graded Frobenius characteristic of the families A and B are given by

M A (q, w) =   d j=0 s j (q)   s n (w) +   d j=1 s j (q)   s n-1,1 (w). M B (q, w) =   d-1 j=0 s j (q)   s n (w) +   d j=1 s j (q)   s n-1,1 (w).
Theorem 4.3 Let f (x 1 ) be a symmetric polynomial of degree 2 in n ≥ 2 variables x 1 = x 11 , x 12 , . . . , x 1n . Suppose that f (x 1 ) is given in the monomial basis as follows:

f (x 1 ) = a • m 2 (x 1 ) + b • m 1,1 (x 1 ), (11)
then the Frobenius characteristic of the space M f is given by one of the following two cases:

M f (q, w) =    1 + h 1 (q) + h 2 (q) • h n (w) if a : b = 1 : 2 , 1 + s 1 (q) + s 2 (q) • s n (w) + s 1 (q) • s n-1,1 (w) = 1 + h 2 (q) • h n (w) + h 1 (q) • h n-1,1 (w)
otherwise.

(

) 12
Corollary 4.1 If f is a homogeneous symmetric polynomial of degree 2, then the associated S n -module M f is isomorphic as an S n × GL (K)-module to one of the two modules M p 2 1 , M p2 .

Exceptions

Recall that, we identify

f = a • m 3 + b • m 21 + c • m 111 , (a,b,c in R) with it's homogeneous coordinates [f ] := [a : b : c] ∈ RP 2 . For instance p 3 1 is the point [1 : 3 : 6], p 21 is [1 : 1 : 0] and h 3 is the point [1 : 1 : 1]. Definition 4.1 We say that an homogeneous symmetric polynomial f in R[x 11 , . . . , x 1n ] is a n-exception if dim R{∂ 11 (f ), . . . , ∂ 1n (f ), E (2) 1,1 (f )} = n.
In others words, f is a n-exception if the dimension of the real linear span of its first order partial derivatives ∂ 11 (f ), . . . , ∂ 1n (f ) and the polynomial E given by one of the following three cases:

If [f ] = [1 : 3 : 6] then dim R{∂ 11 (f ), . . . , ∂ 1n (f ), E (2) 1,1 (f )} ≥ n. Theorem 4.5 Suppose that n ≥ 3 and f (x 1 ) = a • m 3 (x 1 ) + b • m 21 (x 1 ) + c • m 111 (x 1 ), (a,b,c dans R). Then f is a n-exception if and only if [a : b : c] = [1 : 3 : 6] and 6a(2b + (n -2)c) = 4(n -1)b 2 . When n = 2, [a : b : c] is a 2-exception if and only if b = 0 or b = 3a.
               1 + h 1 (q) + h 2 (q) + h 3 (q) • h n (w) if a : b : c = 1 : 3 : 6 , 1 + h 3 (q) • h n (w) + h 1 (q) + h 2 (q) h n-1,1 (w) = (1 + s 1 (q) + s 2 (q) + s 3 (q)) • s n (w) + (s 1 (q) + s 2 (q)) • s n-1,1 (w) if a : b : c is an n-exception, 1 + h 2 (q) + h 3 (q) • h n (w) + h 1 (q) + h 2 (q) • h n-1,1 (w) = (1 + s 1 (q) + 2s 2 (q) + s 3 (q)) • s n (w) + (s 1 (q) + s 2 (q)) • s n-1,1 (w) otherwise. (13)
Corollary 4.2 Le f be a homogeneous symmetric polynomial of degree 3. There are three types of polarization modules M p 3 1 , M p3 or M h3 .

Open problems

In the following lines we present several open problems for the future Problem 5.1 For any degree d ≥ 6, f = p 2 p d-2

1 is a (d + 1)-exception (this is already settled for d = 2, 3, 4, 5).

In the following, we will write s j := s j (q) for short.

M m (2,1 d-2 ) (q, w) =   1 + s 1 + 2 • d-1 j=2 s j + s d   s n (w) + d 2 -1 i=1   s i + 2 • d-i-1 j=i+1 s j + s d-i   s n-i,i (w) +   d-d 2 j= d 2 s j   s n-d 2 , d 2 (w).
Also, if d ≥ 5, we believe that the following isomorphism as a representation of S n × GL (C) holds

M m2,1,1,1,•••1 ∼ = M e d-1,1 .
Consider the following S n -stable families of homogeneous polynomials in the variables x 11 , . . . , x 1n

A := x d 1j 1 ≤ j ≤ n , B := x d 1,i -x d 1,j 1 ≤ i < j ≤ n and C := a∈A x 1,a A ⊆ [n], |A| = d . Problem 5.3
The graded Frobenius characteristic of the family C is

M C (q, w) =   d j=0 s j (q)   s n (w) + n/2 i=1   min(d,n-i) j=i s j (q)   s n-i,i (w).
In particular, the Hilbert series is

M C (q) = d k=0 n k s k (q).
Problem 5.4 Let T 2 be the family of all monomials of total degree 2 in the variables x 11 , . . . , x 1n . Then the graded Frobenius characteristic of its associated polarization module is

M T2 (q, w, n) =          (1 + s 1 + s 2 )s 1 (w) if n = 1, (1 + s 1 + 2 s 2 )s 2 (w) + (s 1 + s 2 )s 1,1 (w) if n = 2, (1 + s 1 + 2 s 2 )s 3 (w) + (s 1 + 2 s 2 )s 2,1 (w) if n = 3, (1 + s 1 + 2 s 2 ) s n (w) + (s 1 + 2 s 2 ) s n-1,1 (w) + s 2 s n-2,2 (w) ∀ n ≥ 4.
When the total degree 3, we have the following conjecture verified up to n = 6 and = 3: Problem 5.5 Le T 3 be the family of all monomials of total degree 3 in the variables x 11 , . . . , x 1n . Then the graded Frobenius characteristic of the associated polarization module is given by table 1 Tab. 1: Frobenius series for the family of monomials of degree 3

(1 + s 1 + s 2 + s 3 )s 1 (w) n = 1 (1 + s 1 + 2 s 2 + 2 s 3 )s 2 (w) + (s 1 + s 2 + s 1,1 + 2 s 3 )s 1,1 (w) n = 2 (1 + s 1 + 2 s 2 + 3 s 3 )s 3 (w) + (s 1 + 2 s 2 + s 1,1 + 3 s 3 )s 2,1 (w) +(s 1,1 + s 3 )s 1,1,1 (w) n = 3 (1 + s 1 + 2 s 2 + 3 s 3 ) s 4 (w) + (s 1 + 2 s 2 + s 1,1 + 4s 3 ) s 3,1 (w) +(s 2 + s 3 )s 2,2 (w) + (s 1,1 + s 3 )s 2,1,1 (w) n = 4 (1 + s 1 + 2 s 2 + 3 s 3 ) s 5 (w) + (s 1 + 2 s 2 + s 1,1 + 4s 3 ) s 4,1 (w) +(s 2 + 2 s 3 )s 3,2 (w) + (s 1,1 + s 3 )s 3,1,1 (w) n = 5 (1 + s 1 + 2 s 2 + 3 s 3 ) s n (w) + (s 1 + 2 s 2 + s 1,1 + 4s 3 ) s n-1,1 (w) +(s 2 + 2 s 3 )s n-2,2 (w) + (s 1,1 + s 3 )s n-2,1,1 (w) + s 3 s n-3,3 (w) ∀ n ≥ 6.
Corollary 4.1, Corollary 4.2 and independent verifications with Maple and Sage, lead us to establish the following questions about the Frobenius characteristic of M f when f is homogeneous of degree 4 or 5:

Problem 5.6 The classification given by tables 2 and 3 is complete (up to n-exceptions), that is, if f is any homogeneous diagonally symmetric polynomial of degree 4 (respectively, degree 5) then the Frobenius characteristic of the module M f is one of the formulas in the table 2 (respectively, table3).

Looking at tables 2 and 3 we are lead to think that Problem 5.7 If f is an homogeneous symmetric polynomial of degree d. Then, there exist a monomorphism ϕ : M f -→ M hm .

A description of Hilbert series of polarization modules and further research directions are available in (4). The next conjecture is already settled for degree 2 and 3, in higher degree, we believe that Problem 5.8 Let f be any homogeneous symmetric polynomial. The Hilbert series of the module M f is h-positive, that is, there are a µ ∈ N such that M f (q) = µ a µ h µ (q). where the sum runs over the set of partitions µ of integers less or equal to deg(f ).

Problem 5.9 Let f be a homogeneous symmetric polynomial of degree

d in R[x 11 , . . . , x 1n ]. If f is not a scalar multiple of p d 1 then dim R{∂ 11 (f ), . . . , ∂ 1n (f ), E (2)
1,1 (f )} ≥ n.

6 Tables for the graded Frobenius characteristic

( 3 )

 3 n consist of polynomials zeros of the polarized power sums operatorsn j=1 ∂ a ∂xj a ∂ b ∂yj b ∂ c ∂zj c (see (3)), where 1 ≤ a + b + c ≤ n. M. Haiman conjectured that this space coincides with the polarization module generated by the Vandermonde determinant when = 3 (see[START_REF] Haiman | Conjectures on the quotient ring by diagonal invariants[END_REF]) this is still an open problem. Also, M. Haiman conjectured that dim(D(3) n ) = 2 n (n + 1) n-2 . F.Bergeron extended this conjecture to the space H (r) n of higher diagonal harmonics polynomials which has dimension dim(H (r) n ) = (r + 1) n (rn + 1) n-2 ((3)). Also, F. Bergeron extend Haiman's conjecture concerning to the identification of the space D ( ) n of multivariate diagonal harmonics polynomials with polarization module generated by the Vandermonde determinant for > 3.

  and only if b = 0 or b = 3a. There are three types of polarization modules generated by a single polynomial of the form f

x

  ij ∂ p kj where 1 ≤ i, k ≤ and p ≥ 1. Clearly the operators E (p)

Remark 3 . 1

 31 When the family F consist of only one homogenous polynomial f ∈ R ( ) n , we denote by M f the polarization module generated by the family {σ • f | σ ∈ S n }. In symbols,

Theorem 4 . 1

 41 Let d be a positive integer. The following formulas holds for any ≥ 1

  (f ) has dimension n. For example, the points [1 : 0 : 0] ( f = p 3 ) and [0 : 0 : 1] ( f = e 3 ) are n-exceptions for every n ≥ 2. For instance, [3 : 3 : -2] is a 3-exception, [9 : 21 : 28] is a 4-exception, [2 : 3 : 2] is a 5-exception, [4 : -3 : 4] is a 5-exception. Another example is [1, 1, 0] which is a 4-exception, because E (2) 1,1 p 21 = 4 j=1 ∂ 1j p 21 . For instance, for every n ≥ 3, the point [1 : 1 : 1] (f = h 3 ) is not an n-exception, the point [0 : 1 : 0] (f = m 21 ) is not an n-exception. En degree 4, the point [5 : 14 : 21 : 28 : 35] is a 11-exception. Theorem 4.4 Let f be a homogeneous symmetric polynomial of degree 3 in R[x 11 , . . . , x 1n ].

Theorem 4 . 6

 46 Let f be a homogeneous symmetric polynomial of degree 3 in n ≥ 2 variables. Suppose that f ∈ R[x 11 , . . . , x 1n ] and [f ] = [a : b : c], then the Frobenius characteristic of the S n -module M f is

Problem 5 . 2

 52 Let d ≥ 3 be an integer. Suppose that m 211•••1 is the monomial symmetric function indexed by the partition of d µ = (2, 1 d-2 ). If n ≥ d we have

  Tab. 2: Frobenius characteristic for degree 4(1 + s 1 + s 2 + s 3 + s 4 )s n (w) p 4 1 (1 + s 1 + s 2 + s 3 + s 4 )s n (w) + (s 1 + s 2 + s 3 )s n-1,1 (w) p 4 (1 + s 1 + s 2 + s 3 + s 4 )s n (w) + (s 1 + s 2 + s 3 )s n-1,1 (w) + s 2 s n-2,2(w) e 4(1 + s 1 + 2s 2 + 2s 3 + s 4 )s n (w) + (s 1 + 2s 2 + s 3 )s n-1,1 (w) e 31 (1 + s 1 + 2s 2 + 2s 3 + s 4 )s n (w) + (s 1 + 2s 2 + s 3 )s n-1,1 (w) + s 2 s n-2,2 (w) + 2s 2 + 2s 3 + s 21 + s 4 )s n (w) + (s 1 + s 2+ s 11 + s 3 )s n-1,1 (w) + 2s 2 + 2s 3 + s 21 + s 4 )s n (w) + (s 1 + 2s 2 + s 11 + s 3 )s n-1,1 (w) + 2s 2 + s 3 + s 21 + s 4 )s n (w) + (s 1 + 2s 2 + s 11 + s 3 )s n-1,1 (w) + s 2 s n-2,2 (w) m 22 (1 + s 1 + 2s 2 + 2s 3 + s 21 + s 4 )s n (w) + (s 1 + 2s 2 + s 11 + s 3 )s n-1,1 (w) + s 2 s n-2,2(w) Frobenius characteristic for degree 5(1 + s 1 + s 2 + s 3 + s 4 + s 5 )sn(w) p51(1 + s 1 + s 2 + s 3 + s 4 + s 5 )sn(w) + (s 1 + s 2 + s 3 + s 4 )s n-1,1(w) p 5 (1 + s 1 + s 2 + s 3 + s 4 + s 5 )sn(w) + (s 1 + s 2 + s 3 + s 4 )s n-1,1 (w) + (s 2 + s 3 )s n-2,2 (w) e 5 (1 + s 1 + 2s 2 + 2s 3 + 2s 4 + s 5 )sn(w) + (s 1 + 2s 2 + 2s 3 + s 4 )s n-1,1 (w) + (s 2 + s 3 )s n-2,2 (w) + 2s 2 + 2s 3 + s 21 + 2s 4 + s 31 + s 5 )sn(w) + (s 1 + 2s 2 + s 11 + 2s 3 + s 21 + s 4 )s n-1,1 (w) +(s 2 + s 3 )s n-2,2 (w) s 221 (1 + s 1 + 2s 2 + 2s 3 + s 21 + 2s 4 + s 31 + s 5 )sn(w) + (s 1 + 2s 2 + s 11 + 2s 3 + s 21 + s 4 )s n-1,1 (w) m 41 p 41 (1 + s 1 + 2s 2 + 3s 3 + s 21 + 2s 4 + s 31 + s 5 )sn(w) + (s 1 + 2s 2 + s 11 + 3s 3 + s 21 + s 4 )s n-1,1 (w) +(s 2 + s 3 )s n-2,2 (w) + 2s 2 + 2s 3 + s 21 + 2s 4 + s 31 + s 5 )sn(w) + (s 1 + 2s 2 + s 11 + 3s 3 + s 21 + s 4 )s n-1,1 (w) +(s 2 + s 3 )s n-2,2 (w) + 2s 2 + 2s 3 + s 21 + 2s 4 + s 31 + s 5 )sn(w) + (s 1 + s 2 + s 11 + s 3 + s 21 + s 4 )s n-1,1 (w) + 2s 2 + 3s 3 + s 21 + 2s 4 + s 31 + s 5 )sn(w) + (s 1 + 2s 2 + s 11 + 2s 3 + s 21 + s 4 )s n-1,1 (w)

	Tab. 3: m 2111
	s 2111
	e 41
	(1 + s 1 h 5
	h 41
	h 32
	h 221
	p 221
	s 41
	s 32
	s 311 s 211 e 221 h 22 m 311
	m 211 (1 + s 1 p 32 e 32 (1 + s 1 p 211 m 32 e 211 m 221 h 211 (1 + s 1 h 31 (1 + s 1 p 2111 h 2111 m 31 e 2111 p 31 (1 + s 1 e 311 h 311
	(1 + s 1 s 4 p 311
	s 31
	s 22
	e 22
	p 22

Acknowledgements

Thanks to Franc ¸ois Bergeron and Franco Saliola for their advice during my Ph.D. studies. Thanks to Franco Saliola and Eduardo Blazek for helpful discussions and many independent computations Sage. Thanks to Adolfo Rodríguez for his help with Maple. Thanks to Yannic Vargas, Alejandro Morales, Marco Pérez and Luis Serrano for many valuable suggestions to reach the final version of this paper. To my fiancee Oana-Andreea Kosztan for love and support with L A T E X during my Ph.D. Thesis. I want to thank Francois Bergeron for showing me this beautiful subject.