Cédric Lecouvey

Cristian Lenart

C Édric

ATOMIC DECOMPOSITION OF CHARACTERS AND CRYSTALS

come

Introduction

The starting point of this paper is a result of Lascoux on the (type A) Kostka-Foulkes polynomials K λ,µ (t), which are well-known t-analogues of the Kostka numbers K λ,µ , i.e., the number of semistandard Young tableaux of shape λ and content µ. Lascoux [START_REF] Lascoux | Cyclic permutations on words, tableaux and harmonic polynomials[END_REF] stated the decomposition of the Kostka-Foulkes polynomials into so-called atomic polynomials. Some arguments of the proof in [START_REF] Lascoux | Cyclic permutations on words, tableaux and harmonic polynomials[END_REF] remained elusive, and it was not until the work of Shimozono [START_REF] Shimozono | Multi-atoms and monotonicity of generalized Kostka polynomials[END_REF] that the type A atomic decomposition was completely accepted, this time in larger generality (for the so-called generalized Kostka-Foulkes polynomials). However, the latter proof involves several intricate combinatorial arguments and related concepts, such as plactic monoid, cyclage, and catabolism. The main goal of this paper is to provide a simpler, more conceptual approach, which has the additional advantage of extending beyond type A.

Lusztig defined a remarkable t-analogue K λ,µ (t) of the multiplicity of a weight µ in the irreducible representation with highest weight λ of a seimisimple Lie algebra [START_REF] Lusztig | Singularities, character formulas, and a q-analog of weight multiplicities. Analyse et topologie sur les espaces singuliers (II-III)[END_REF]. For dominant weights µ, these polynomials generalize the type A ones mentioned above, and are therefore also called Kostka-Foulkes polynomials. They have remarkable properties:

• they are essentially affine Kazhdan-Lusztig polynomials [START_REF] Kato | Spherical functions and a q-analogue of Kostants weight multiplicity formula[END_REF];

• they record the Brylinski filtration of weight spaces [START_REF] Brylinski | Limits of weight spaces, Lusztig's q-analogs and fiberings of adjoint orbits[END_REF];

• they are the coefficients in the expansion of an irreducible character in terms of Hall-Littllewood polynomials; • they are related to the energy function coming from solvable lattice models [START_REF] Lecouvey | Lusztig's q-analogue of weight multiplicity and one-dimensional sums for affine root systems[END_REF][START_REF] Nakayashiki | Kostka-Foulkes polynomials and energy function in sovable lattice models[END_REF].

In classical Lie types, when the rank increases, these polynomials exhibit a stabilization property, so they have stable versions [START_REF] Lecouvey | Stabilization of the Brylinsky filtration and limit of Lusztig q-analogues[END_REF]. We refer to [START_REF] Nelsen | Kostka-Foulkes polynomials and Macdonald spherical functions[END_REF]35] for more information on Kostka-Foulkes polynomials.

There are two combinatorial formulas for type A Kostka-Foulkes polynomials: one due to Lascoux-Schützenberger, in terms of the charge statistic on semistandard Young tableaux [START_REF] Lascoux | Sur une conjecture de H. O. Foulkes[END_REF], and one in terms of the corresponding Kashiwara crystal graphs [START_REF] Hong | Introduction to quantum groups and crystal bases[END_REF][START_REF] Kashiwara | Crystal bases of modified quantized enveloping algebra[END_REF][START_REF] Kashiwara | On crystal bases[END_REF], due to Lascoux-Leclerc-Thibon [START_REF] Lascoux | Crystal graphs and q-analogue of weight multiplicities for the root system An[END_REF]. Similar partial combinatorial descriptions in types B-D, in terms of the corresponding 2010 Mathematics Subject Classification. 05E10, 17B10.

Kashiwara-Nakashima tableaux [START_REF] Hong | Introduction to quantum groups and crystal bases[END_REF], and a conjectured charge statistic in type C were constructed by the first author in [START_REF] Lecouvey | Kostka-Foulkes polynomials, cyclage graphs and charge statistic for the root system Cn[END_REF][START_REF] Lecouvey | Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems Bn, Cn, and Dn[END_REF]. In our previous paper [START_REF] Lecouvey | Combinatorics of generalized exponents[END_REF], we prove the first general formula beyond type A, namely a formula for K λ,0 (t) of type C. This extends the Lascoux-Leclerc-Thibon formula by using a simpler approach, and can be expressed in terms of King tableaux.

In Section 2.2, following Lascoux [START_REF] Lascoux | Cyclic permutations on words, tableaux and harmonic polynomials[END_REF], we formulate the t-atomic decomposition property in arbitrary Lie type as a nonnegative expansion for both a Kostka-Foulkes polynomial, and a t-analogue χ + λ (t) of the dominant part of an irreducible character (defined in terms of Kostka-Foulkes polynomials). Here a character expansion is in terms of so-called layer sum polynomials, which record all weights of some irreducible representation with multiplicity 1. The t-atomic decomposition property is a strengthening of the monotonicity of Kostka-Foulkes polynomials, which holds in the full generality of affine Kazhdan-Lusztig polynomials [1, Corollary 3.7].

For t = 1, the atomic decomposition was also considered from a purely algebraic point of view in [START_REF] Dhillon | The Weyl-Kac weight formula[END_REF][START_REF] Rasmussen | Layer structure of irreducible Lie algebra modules[END_REF][START_REF] Schützer | A new character formula for Lie algebras and Lie groups[END_REF]. Some partial results were given, for instance for the inverse expansion (of layer sum polynomials in terms of characters). However, the atomic decomposition itself is less understood, even for t = 1; for instance, it does not always exist, in the sense that the mentioned character expansion sometimes fails to be nonnegative, contrary to what was claimed in Theorem 2.2 of [START_REF] Schützer | A new character formula for Lie algebras and Lie groups[END_REF] (see Example 2.4). Nevertheless, these failures seem to be mild.

As opposed to the algebraic approach mentioned above, in Section 2.3 we define a stronger t-atomic decomposition property, at the combinatorial level of the highest weight crystal B(λ). This property involves a partition of the dominant part B(λ) + of B(λ), and a statistic c( • ) on B(λ) + . The combinatorial t-atomic decomposition leads to combinatorial formulas for both the Kostka-Foulkes polynomials and the atomic polynomials (into which the former decompose).

In Section 3, we consider the case when λ goes to infinity, in types A n-1 , B n , C n , D n , and G 2 . We introduce the notion of a t-atomic decomposition of the crystal B(∞), and discuss how it can be realized.

Next we introduce the two main ingredients for constructing a t-atomic decomposition of a finite highest weight crystal: the partial order on dominant weights, and a modified crystal graph structure. The natural poset structure on dominant weights is discussed in Section 4, by recalling some important properties which hold in arbitrary Lie type [START_REF] Stembridge | The partial order of dominant weights[END_REF]. We need extra information about this poset, namely the structure of its small intervals, which was only known in type A [START_REF] Brylawski | The lattice of integer partitions[END_REF]. The second ingredient, namely a modified crystal graph structure on B(λ) + , is discussed in Section 5.1; this structure can also be viewed as a poset, with covers corresponding to the modified crystal edges. The associated modified crystal operators are obtained by conjugating the ordinary Kashiwara operator f1 under the crystal action of the Weyl group. There are two main differences between the original and the modified crystal operators: (1) the latter are indexed by arbitrary positive roots;

(2) the original B(λ) is a connected graph/poset, whereas the modified crystal graph on B(λ) + is disconnected in general. An interesting question is whether the modified crystal operators for λ become those for B(∞) in Section 3 when λ goes to infinity.

In Section 6, we start by showing that type A crystals admit a t-atomic decomposition, thus realizing combinatorially the classical result, while also providing a relatively simple and conceptual proof of it. Here the desired partition of the modified crystal poset B(λ) + is given by its components, which are shown to admit unique minimal and maximal vertices. The relevant statistic c( • ) is the Lascoux-Schützenberger charge [START_REF] Lascoux | Sur une conjecture de H. O. Foulkes[END_REF]. Similarly, given a fixed dominant weight λ of type C or D, and assuming a large enough rank (depending on λ), we show that the corresponding modified crystal graph B(λ) + gives an atomic decomposition (i.e., in the case t = 1).

In Section 7 we conjecture that the latter decomposition is, in fact, a t-atomic decomposition, for appropriate choices of the statistic c( • ), which are related to the results in [START_REF] Lecouvey | Kostka-Foulkes polynomials, cyclage graphs and charge statistic for the root system Cn[END_REF][START_REF] Lecouvey | Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems Bn, Cn, and Dn[END_REF] and [START_REF] Lecouvey | Combinatorics of generalized exponents[END_REF]. We also explain how to obtain a t-atomic decomposition of the crystal B( α) + of the adjoint representation of g in any type. This is useful because it clarifies the type and rank restrictions we considered in Section 6: they are needed to ensure that the covering relations in the poset B(λ) + make only appear roots in the orbit of the simple root α 1 . Under this assumption, we are indeed able to establish fine commutation relations satisfied by the modified crystal operators, which are required to derive the atomic decomposition of crystals (notably, the existence of unique maximal and minimal vertices). For the simply laced types D (with no large rank assumption) and E, the atomic decomposition of the characters can only hold with some restrictions on the dominant weights considered, but we expect that it can be derived similarly from the same modified crystal operators and a detailed analysis of the corresponding dominant weight poset. In the non-simply laced case of general rank, the situation becomes more complicated, but it is certainly possible to again derive relevant atomic decompositions of crystals by using modified crystal operators; this time, they would be defined using the Weyl group conjugation of two ordinary crystal operators, associated to one simple root of each length. We will address the mentioned problems in future work.

A geometric interpretation of Lascoux's atomic decomposition of the type A Kostka-Foulkes polynomial was given in [START_REF] Brockman | Nilpotent orbit varieties and the atomic decomposition of the q-Kostka polynomials[END_REF] in terms of nilpotent orbit varieties. In Section 8, we provide a different, type-independent interpretation of an atomic decomposition of χ + λ , in terms of the geometric Satake correspondence.

Acknowledgments:

The second author gratefully acknowledges the partial support from the NSF grant DMS-1362627. He thanks Institut des Hautes Études Scientifiques (IH ÉS) for its hospitality during July-August 2018, when this work was completed. Both authors are grateful to Arthur Lubovsky and Adam Schultze for the computer tests (based on the Sage [START_REF]The Sage-Combinat community. Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics[END_REF] system) related to this work; they also received support from the NSF grant mentioned above.

2. The atomic decomposition: background, definitions, and basic facts 2.1. Characters and t-deformations. Let g be a simple Lie algebra over C of rank r with triangular decomposition

g = α∈R + g α ⊕ h ⊕ α∈R + g -α ,
so that h is the Cartan subalgebra of g and R + its set of positive roots. The root system R = R + (-R + ) of g is realized in a real Euclidean space E of dimension n with inner product • , • . For any α ∈ R, we write α ∨ = 2α α,α for its coroot. Let S ⊂ R + be the subset of simple roots and Q + the Z + -cone generated by S. The set P of integral weights for g consists of elements β satisfying β, α ∨ ∈ Z for any α ∈ R. We write P + = {β ∈ P | β, α ∨ ≥ 0 for any α ∈ S} for the cone of dominant weights of g, and denote by ω 1 , . . . , ω r its fundamental weights. Let W be the Weyl group of g generated by the reflections s α with α ∈ S, and write ( • ) for the corresponding length function. The dominance order ≤ on P + is defined by α < β if and only if β -α decomposes as a sum of positive roots (or equivalently, simple roots) with nonnegative integer coefficients. Many interesting properties of this order were studied in [START_REF] Stembridge | The partial order of dominant weights[END_REF], and some of them will be used in this paper; for instance, each component of this poset is a lattice.

Let χ g

λ be the Weyl character associated to the finite-dimensional irreducible representation V (λ) of g with highest weight λ, namely

χ g λ = γ∈P K g λ,γ e γ ,
where K g λ,γ is the dimension of the weight space γ in V (λ). For simplicity, we remove the superscript g when the context makes it clear. By the Weyl character formula we have [START_REF] Braden | From moment graphs to intersection cohomology[END_REF] χ λ = w∈W (-1) (w) e w(λ+ρ)-ρ α∈R

+ (1 -e -α )
.

This formula expresses the weight multiplicities K λ,γ as follows:

K λ,γ = w∈W (-1) (w) P(w(λ + ρ) -(γ + ρ)) ,
where ρ is the half sum of positive roots, and P ( • ) is the Kostant partition function, defined by

α∈R + 1 1 -e α = β∈Q + P(β) e β .
When γ = µ is dominant, the multiplicity K λ,µ has an interesting t-analogue K λ,µ (t), also known as a Kostka-Foulkes polynomial. This was introduced by Lusztig [START_REF] Lusztig | Singularities, character formulas, and a q-analog of weight multiplicities. Analyse et topologie sur les espaces singuliers (II-III)[END_REF], who defined

(2) K λ,µ (t) := w∈W (-1) (w) P t (w(λ + ρ) -(µ + ρ)) ;
here the t-analogue of the Kostant partition function P t ( • ) is given by

α∈R + 1 1 -te α = β∈Q + P t (β) e β .
We have K λ,µ (1) = K λ,µ . Moreover, K λ,µ (t) is essentially an affine Kazhdan-Lusztig polynomial, which implies that it has nonnegative integer coefficients. More precisely, we have

(3) K λ,µ (t) = t λ-µ,ρ ∨ P wµ,w λ (t -1 ) ,
where w λ denotes the longest element of W t λ W , and t λ is the translation by λ in the extended affine Weyl group [START_REF] Kato | Spherical functions and a q-analogue of Kostants weight multiplicity formula[END_REF] (see also [35, Section 4]); note that λ -µ, ρ ∨ is the number of simple roots in the decomposition of λ -µ, counted with multiplicity. Based on (3), we let (4) K λ,µ (t) := t λ-µ,ρ ∨ K λ,µ (t -1 ) , so K λ,µ (t) = P wµ,w λ (t) .

To each irreducible representation V (λ) is associated an abstract Kashiwara crystal B(λ) (see [START_REF] Hong | Introduction to quantum groups and crystal bases[END_REF][START_REF] Kashiwara | Crystal bases of modified quantized enveloping algebra[END_REF][START_REF] Kashiwara | On crystal bases[END_REF] for background on crystals), and we have [START_REF] Brylinski | Limits of weight spaces, Lusztig's q-analogs and fiberings of adjoint orbits[END_REF]

χ λ = b∈B(λ) e wt(b) ,
where wt(b) is the weight of the vertex b ∈ B(λ). The crystal B(∞) is defined as the direct limit of the crystal B(λ) when λ goes to infinity in the interior of the Weyl chamber. It corresponds to the crystal of the positive part of the quantum group U q (g) associated to g. One can then prove that

(6) char B(∞) = b∈B(∞) e wt(b) = α∈R + 1 1 -e -α .

2.2.

The definition of the atomic decomposition. Let us denote by P (λ) the set of weights of V (λ), i.e., the set of γ such that K λ,γ > 0. Also set P + (λ) = P (λ) ∩ P + . Recall we have in fact P + (λ) = {µ ∈ P + | µ ≤ λ}. Since we have K λ,γ = K λ,w(γ) for any w ∈ W , the character χ λ is completely determined by its dominant part

χ + λ := µ∈P + (λ) K λ,µ e µ .
We also define the t-analogue of χ + λ by [START_REF] Dhillon | The Weyl-Kac weight formula[END_REF] χ + λ (t) :=

µ∈P + (λ) K λ,µ (t) e µ .
For any dominant weight µ, define the layer sum polynomials by Observe that we have

w µ = ν∈P + (µ) m ν , where m ν = γ∈W ν e γ .
Similarly, define

(9) w + µ (t) := ν∈P + (µ) t µ-ν,ρ ∨ e ν = ν≤µ t µ-ν,ρ ∨ e ν .
There exists a Weyl-type formula for the polynomials w µ , which follows from Brion's formula [START_REF] Brion | Points entiers dans les polyèdres convexes[END_REF] counting the cardinality of the intersection between a convex polygon and a lattice, see [START_REF] Postnikov | Permutohedra, associahedra, and beyond[END_REF]Theorem 4.3]; this formula was rederived in [START_REF] Schützer | A new character formula for Lie algebras and Lie groups[END_REF] from the axioms of root systems. It is stated as follows:

(10)

w µ = w∈W e w(µ)
α∈S (1 -e -w(α) )

; here, for any simple root α and any Weyl group element w, we set

1 1 -e -w(α) = +∞ k=0 e kw(α) if w(α) ∈ R + , and 1 1 -e -w(α) = - e w(α) 1 -e w(α) = - +∞ k=0 e (k+1)w(α) if w(α) ∈ -R + .
Observe w + µ (t) coincides with the dominant part in the expansion e µ α∈S (1 -te -α ) .

Definition 2.1. The character χ λ admits an atomic decomposition if there exist nonnegative integer coefficients A λ,µ such that

(11) χ λ = µ∈P + (λ) A λ,µ w µ , or equivalently χ + λ = µ∈P + (λ) A λ,µ w + µ .
Similarly, we say that χ + λ (t) admits a t-atomic decomposition if there exist polynomials A λ,µ (t) ∈ Z ≥0 [t], called atomic polynomials, such that

(12) χ + λ (t) = µ∈P + (λ) A λ,µ (t) w + µ (t) .
We will list several properties equivalent to [START_REF] Kashiwara | On crystal bases[END_REF]. To this end, by analogy with (7) and using (4), we define

χ + λ (t) := µ∈P + (λ) K λ,µ (t) e µ .
Proposition 2.2. The following are equivalent.

(1) χ + λ (t) admits a t-atomic decomposition in terms of A λ,µ (t), cf. (12). (2) There exist polynomials

A λ,µ (t) ∈ Z ≥0 [t] such that K λ,ν (t) = ν≤µ≤λ t µ-ν,ρ ∨ A λ,µ (t) . (3) There exist polynomials A λ,µ (t) ∈ Z ≥0 [t] such that χ + λ (t) = µ∈P + (λ) A λ,µ (t) w + µ ,
where we recall that w + µ := w + µ (1). (4) There exist polynomials

A λ,µ (t) ∈ Z ≥0 [t] such that K λ,ν (t) = ν≤µ≤λ A λ,µ (t) .
In addition, the polynomials A λ,µ (t) in (1)-( 2) and A λ,µ (t) in (3)-(4) satisfy

A λ,µ (t) = t λ-µ,ρ ∨ A λ,µ (t -1 ) .
Proof. The t-atomic decomposition (12) can be written

χ + λ (t) = µ∈P + (λ) A λ,µ (t)   ν≤µ t µ-ν,ρ ∨ e ν   = ν≤λ   ν≤µ≤λ t µ-ν,ρ ∨ A λ,µ (t)   e ν .
By comparing with [START_REF] Dhillon | The Weyl-Kac weight formula[END_REF], we can see that (1) and (2) are equivalent. A completely similar reasoning proves the equivalence of ( 3) and ( 4).

The equivalence of ( 2) and ( 4), as well as the last statement in the proposition, can be seen in the following way:

K λ,ν (t) = t λ-ν,ρ ∨ K λ,ν (t -1 ) = t λ-ν,ρ ∨ ν≤µ≤λ t ν-µ,ρ ∨ A λ,µ (t -1 ) = ν≤µ≤λ t λ-µ,ρ ∨ A λ,µ (t -1 ) = ν≤µ≤λ A λ,µ (t) .
Remarks 2.3. (1) Lascoux [START_REF] Lascoux | Cyclic permutations on words, tableaux and harmonic polynomials[END_REF] made a statement very closely related to the one in Proposition 2.2 (4), for type A. The slight difference consists in the definition of K λ,µ (t), for given partitions λ, µ, namely K λ,µ (t) := t n(µ) K λ,µ (t -1 ), where n(µ) := i (i -1)µ i .

(2) The t-atomic decomposition, as stated in Proposition 2.2 (4), implies the monotonicity of the Kostka-Foulkes polynomials, which holds in the full generality of Kazhdan-Lusztig polynomials for finite and affine Weyl groups [START_REF] Braden | From moment graphs to intersection cohomology[END_REF]Corollary 3.7]. Indeed, the latter says that, for x ≤ y ≤ z in such a Weyl group, the difference of Kazhdan-Lusztig polynomials P x,z (t

) -P y,z (t) is in Z ≥0 [t].
For z = w λ , y = w µ , and x = w ν , with ν ≤ µ ≤ λ, this follows based on (4) and the fact that the atomic polynomials in the decomposition of K λ,µ (t) are among those in the decomposition of K λ,ν (t).

(3) We shall see that the t-atomic decomposition is always true in type A, as mentioned in Section 1. On another hand, the expansions in Definition 2.1 always exist, and we have A λ,λ = A λ,λ (1) = 1. However, even the atomic decomposition (i.e., the positivity in [START_REF] Kashiwara | Crystal bases of modified quantized enveloping algebra[END_REF]) might fail beyond type A, unlike it was claimed in [31, However, for the same λ we obtain a positive expansion in type D 5 .

Set t i := e -α i ∈]0, 1[, for i = 1, . . . , r, and consider a sequence (λ (k) ) k≥0 of dominant weights such that lim k→+∞ λ (k) , α i = +∞, for any i = 1, . . . , r. We shall then write λ → +∞ for short. Proposition 2.5. Under the previous assumption we have

lim λ→+∞ e -λ χ λ = α∈R + 1 1 -e -α
and lim

λ→+∞ e -λ w λ = α∈S 1 1 -e -α .
Proof. We have by the Weyl character formula

e -λ (k) χ λ (k) = w∈W (-1) (w) e w(λ (k) +ρ)-ρ-λ (k) α∈R + (1 -e -α )
.

Let t = max(t 1 , . . . , t r ) ∈]0, 1[. Since w(λ (k) + ρ) -ρ -λ (k) ∈ -Q + , we can set w(λ (k) + ρ) -ρ -λ (k) = - r i=1 a i (λ (k) ) α i ,
where a i (λ (k) ) ∈ Z ≥0 for any i = 1, . . . , r. The hypothesis lim k→+∞ λ (k) , α i = +∞ for any i = 1, . . . , r implies that lim k→+∞ r i=1 a i (λ (k) ) = +∞ for any w ∈ W such that w = 1. So we get for any such w

lim k→+∞ e w(λ (k) +ρ)-ρ-λ (k) ≤ lim k→+∞ t r i=1 a i (λ (k) ) = 0 .
Since for w = 1 we have (-1) (w) e w(λ (k) +ρ)-ρ-λ (k) = 1, we get our first limit.

From [START_REF] Hong | Young tableaux and crystal B(∞) for finite simple Lie algebras[END_REF], we can write

e -λ (k) w λ (k) = w∈W e w(λ (k) )-λ (k)
α∈S (1 -e -w(λ) )

and by using similar arguments, only w = 1 contributes when we consider the limit.

Now set

α∈R + \S 1 1 -te -α = β∈Q + M t (β) e -β .
In particular, M 1 (β) is the number of decompositions of β as a sum of nonsimple positive roots. We get ( 13)

α∈R + 1 1 -te -α = β∈Q + M t (β) α∈S e -β 1 -te -α .
Proposition 2.5 suggests to consider (13) as a t-analogue of (11) when λ → +∞. Observe that the cardinality of B(λ, h) is then independent of λ, and if wt(h) = µ we have ( 15)

w + µ = b∈B(λ,h)
e wt(b) .

If B(λ) has an atomic decomposition, then clearly χ + λ has the atomic decomposition [START_REF] Kashiwara | Crystal bases of modified quantized enveloping algebra[END_REF], where A λ,µ is the number of vertices of weight µ in H(λ).

Definition 2.7. The crystal B(λ) admits a t-atomic decomposition if it admits an atomic decomposition and there exists a statistic c : H(λ) → Z ≥0 such that the polynomials

(16) A λ,µ (t) = h∈H(λ) wt(h)=µ t c(h) satisfy (12).
As we can see, the t-atomic decomposition property of χ + λ (t) is part of Definition 2.7. Assuming that B(λ) has a t-atomic decomposition, one can extend the statistic c to B(λ) + by setting [START_REF] Lecouvey | Kostka-Foulkes polynomials, cyclage graphs and charge statistic for the root system Cn[END_REF] c(b

) := c(h) + wt(h) -wt(b), ρ ∨ , for any b ∈ B(λ, h) .
The t-analogues of the combinatorial formulas ( 15) and ( 7) immediately follow from Definition 2.7:

w + µ (t) = b∈B(λ,h) t c(b)-c(h) e wt(b) , (18)
χ + λ (t) = b∈B(λ) + t c(b) e wt(b) . (19)
Moreover, by comparing ( 19) with [START_REF] Dhillon | The Weyl-Kac weight formula[END_REF], we obtain the following combinatorial formula for Kostka-Foulkes polynomials:

(20) K λ,µ (t) = b∈B(λ) wt(b)=µ t c(b) .
To summarize, the existence of a t-atomic decomposition of a crystal is highly desirable because: (i) it implies the t-atomic decomposition of χ + λ (t) and of the Kostka-Foulkes polynomials K λ,µ (t), which are now realized combinatorially; (ii) it leads to combinatorial formulas for both K λ,µ (t) and the atomic polynomials A λ,µ (t), namely [START_REF] Lecouvey | Combinatorics of generalized exponents[END_REF] and ( 16), respectively.

Atomic decomposition of the crystal B(∞)

In this section we assume that the Lie algebra g is of type A n-1 , B n , C n , D n , and G 2 . Let r be the rank of g.

3.1.

Marginally large tableaux. Marginally large tableaux were introduced by Hong and Lee [START_REF] Hong | Young tableaux and crystal B(∞) for finite simple Lie algebras[END_REF] in order to describe the crystal B(∞) associated to g. Recall that they can be regarded as g-tableaux (that is, of type

A n-1 , B n , C n , D n , or G 2 ) with d rows such that d = n in types B n , C n , d = n -1 in types A n-1 , D n , and d = 2 in type G 2 ;
furthermore, for any i = 1, . . . , d -1, the number of boxes in row i containing i is equal to 1 plus the number of boxes in row i + 1 (see the example below). Write T (∞) for the set of marginally large tableaux associated to g. Since marginally large tableaux are special cases of tableaux for each type considered, the set T (∞) comes with a crystal action which is essentially the same as in the finite crystal up to renormalization of rows, in order to insure that the obtained tableau is marginally large. This renormalization is defined as follows. Consider a marginally large g-tableau T , and let T be a g-tableau obtained from T by modifying a letter in row i. If T is not marginally large, this means that we have modified the rightmost letter i in row i. Then the renormalization of T is the marginally large tableau obtained from T by adding a letter k in each row k between 1 and i, the others rows remaining unchanged. One can then define crystal operators F1 , . . . , Fr on T (∞) by setting

Fi (T ) = fi (T ) ,
where fi is the ordinary Kashiwara crystal operator on the g-tableau T . It was established in [START_REF] Hong | Young tableaux and crystal B(∞) for finite simple Lie algebras[END_REF] that the crystal structure on T (∞) obtained in this way is isomorphic to B(∞).

Example 3.1. If g is of type A 3 , then T = 1 1 1 1 1 1 1 3 5 5 2 2 2 3 3 4 3 5
is a marginally large tableau, and

f2 (T ) = 1 1 1 1 1 1 1 1 3 5 5 2 2 2 3 3 3 4 3 5
.

We define a multisegment as a multiset of positive roots. We can write a multisegment m as

m = α∈R + m α α ,
which means that the multiset m contains m α times the positive root α. Let M be the set of g-multisegments. In [22, Proposition 3.7], a bijection Ξ1 is given from T (∞) to M. This bijection depends on the type considered. For example, in type A n-1 , for any positive root α ij = ε i -ε j with 1 ≤ i < j ≤ n and any marginally large tableau T , the integer m α i,j is equal to the number of letters j in the i-th row of T . For any

T ∈ T (∞) such that Ξ(T ) = α∈R + m α α, write |T | = α∈R + m α .
Example 3.2. Resuming the previous example, we get

Ξ(T ) = α 1,3 + 2α 1,5 + 2α 2,3 + α 2,4 + α 3,5
and |T | = 7.

Modified crystal operators and atomic decomposition of B(∞)

. We refer to [START_REF] Lee | Tableaux, crystal bases and the Gindikin-Karpelevitch formula[END_REF] for a complete description of the bijection Ξ in each case. In the sequel, we only need the following two properties of the map Ξ.

(1) Starting from any marginally large tableau T and any simple root α i , i = 1, . . . , r, there is a unique marginally large tableau T such that

Ξ(T ) = Ξ(T ) + α i .
We then set T = F i (T ). ( 2) For any marginally large tableau T and any i = 1, . . . , n such that Ξ(T ) = α∈R + m α α with m α i > 0, there exists a (unique) marginally large tableau T such that T = F i (T ).

More precisely, the construction of T from T is as follows.

• For any i = 1, . . . , r -1, T is obtained from T by replacing the rightmost letter i located in row i by a letter i + 1 and then by renormalizing if needed.

• For i = 2 in type G 2 ,
T is obtained from T by replacing the rightmost letter 2 located in row 2 by a letter 3.

• For i = n in type C n ,
T is obtained from T by replacing the rightmost letter n located in row n by a letter n. • For i = n in type D n , T is obtained from T by replacing the rightmost letter n -1 located in row n -1 by a letter n. • For i = n in type B n , the rows of T can only contain one letter 0. Then we have:

-if 0 belongs to row n, T is obtained from T by replacing this letter 0 by a letter n; -if 0 does not belong to row n, T is obtained from T by replacing the rightmost letter n located in row n by a letter 0.

Observe that F i (T ) is a marginally large tableau for any i = 1, . . . , n and any T ∈ T (∞). We also define the operators E i , i = 1, . . . , n, such that E i (T 1 ) = T 2 if there exists T 2 ∈ T (∞) satisfying F i (T 2 ) = T 1 , and E i (T 1 ) = 0 otherwise. The operators F i and E i , i = 1, . . . , n, are the modified crystal operators. We have F i (T ) = Fi (T ) and E i (T ) = Fi (T ) in general. Furthermore, by property (2), we have E i (T ) = 0 for any T such that m α i > 0. Now we can endow T (∞) with a new colored directed graph structure B(∞) such that T i T if and only if T = F i (T ). A source vertex for this structure is a marginally large tableau T such that E i (T ) = 0 for any i = 1, . . . , r. Let us denote by ST (∞) the set of source vertices in T (∞).

Theorem 3.3.

(

) We have S ∈ ST (∞) if and only if Ξ(T ) = α∈R + \S m α α, that is, when m α i = 0 for any i = 1, . . . , r. 1
T ∈B(S)

t |T | e wt(T ) = t |S| e wt(S) α∈S (1 -te -α )
.

Proof. The first part follows from the fact that

Ξ(E i (T )) = Ξ(T ) -α i ,
for any i = 1, . . . , r. For the second part, we observe that, for any marginally large tableau T and for any i = j in {1, . . . , r}, we have E i E j (T ) = 0 and E i E j (T ) = 0 if and only if m α i > 0 and

m α j > 0; in this case, we have E i E j (T ) = E i E j (T ). Thus S = r i=1 E mα i i (T )
is the unique source vertex of the connected component corresponding to T , and does not depend on the order in which the operators E i are applied in the right hand side. For the third part, observe we have

B(S) = r i=1 F mα i i (S) | m α i ∈ Z ≥0 ,
where the operators F i , i = 1, . . . , r commute. Since we have wt

r i=1 F mα i i (S) = wt(S) - r i=1 m α i α i ,
the weights of the vertices in B(S) are all distinct. This also yields the desired equality [START_REF] Lecouvey | Lusztig's q-analogue of weight multiplicity and one-dimensional sums for affine root systems[END_REF].

Corollary 3.4. The partition

B(∞) = S∈ST (∞) B(S)
is a t-atomic decomposition of B(∞); in other words, we have

α∈R + 1 1 -te -α = β∈Q + M t (β) α∈S e -β 1 -te -α , where M t (β) = S∈ST (∞) wt(S)=β t |S| .

The partial order on dominant weights

Before we consider the atomic decomposition of finite crystals, we need some information about the partial order on dominant weights that was defined in Section 2.1. In full generality, this poset was first studied in [START_REF] Stembridge | The partial order of dominant weights[END_REF], so we will recall some results from this paper.

The components of the dominant weight poset are lattices. Each cocover is of the form µ µ -α, where α is a positive root, so we can represent it as a downward edge in the Hasse diagram labeled by α. The cocovers were completely described in [START_REF] Stembridge | The partial order of dominant weights[END_REF]Theorem 2.8]. Fixing a dominant weight λ, we will be interested in the lower order ideal determined by λ. This is known to be an interval [ 0, λ], with 0 a minimal element of the dominant weight poset.

Type

A n-1 . Now λ is a partition (λ 1 ≥ . . . ≥ λ n-1 ≥ 0)
, and let N = |λ| := i λ i . We identify partitions with their Young diagrams, and we denote a partition with p parts a, q parts b (a ≥ b) etc. by (a p b q . . .). As explained below, the interval [ 0, λ] mentioned above can be identified with the similar interval in the poset of partitions of N with the dominance order; the latter is defined by (µ 1 , µ 2 , . . .) ≤ (ν 1 , ν 2 , . . .) if and only if j i=1 µ i ≤ j i=1 ν j , for any j. As a partition of N , the element 0 is the partition with N/n columns of height n, and the last column of height p := N mod n; as an element of the dominant weight poset, 0 is the fundamental weight ω p . We identify a partition of N with the partition obtained from it by removing all columns of height n (if a partition of N is greater or equal to 0, then it has no columns of height larger than n).

The dominance order on partitions of N was first studied in [START_REF] Brylawski | The lattice of integer partitions[END_REF], so we recall some results in this paper. Conjugation of partitions is an antiautomorphism. The cocovers µ µ -α ij , where α ij = ε i -ε j is a positive root (i.e., i < j), are labeled by (i, j). It turns out that there are only two types of cocovers, namely: [START_REF] Lee | Tableaux, crystal bases and the Gindikin-Karpelevitch formula[END_REF] (. . . ab . . .) (. . . (a -1)(b + 1) . . .) , and (. . . (a + 1)a p (a -1) . . .) (. . . a p+2 . . .) ,

where in the first case a ≥ b + 2 and the cocover is labeled by a simple root. These types are referred to as (*) and (**), respectively, while a cocover of type (**) which is not of type (*) is called proper.

An important result in [START_REF] Brylawski | The lattice of integer partitions[END_REF] concerns the structure of short intervals in the dominance order. To state it, we need some more definitions. Consider two distinct cocovers µ ν and µ π of a partition µ, which are labeled (i, j) and (k, l), where we assume i < k. These cocovers can only have one of the following relative positions (in terms of their labels): . . . (a -1)c(e + 1) . . .

(i) nonoverlapping if j < k; (ii) partially overlapping if j = k; (iii) fully overlapping if k = j -1.
In subcase (c), we have b = a -1, c = b -1, d = c -1, p, q ≥ 1
, while i is the position of a, j = i + p + 1 is the position of the first c, and k = j + q is the position of d.

. . . ab p c q d . . .

(i,j) u u (j-1,k) A A u u u u A A A A . . . b p+2 c q-1 d . . .
= a -1, c = b -1, d = c -1, e = d -1, p, q ≥ 1
, while i is the position of a in the partition µ, j = i + p + 1 is the position of c, and k = j + q + 1 is the position of e.

. . . ab p cd q e . . .

(j-1,j+1) (i,j) t t (j,k) B B t t t t B B B B
. . . b p+2 d q e . . . . . . ab p-1 c 3 d q-1 e . . . Due to the conjugation automorphism of the dominance order, given two distinct covers µ ν and µ π of µ, the isomorphism type of the interval [µ, ν ∨ π] is always given by one of the above graphs turned upside down. So we have the corresponding Cases A1 -A3 . In fact, the structure of the intervals in Cases A1 and A2 is identical with that in Cases A1 and A2, respectively; however, Case A3 leads to the different structure shown below; the two covers above are again shown in bold.

Case A3 . Here a ≥ c + 1, d = c -1, d ≥ f + 1, while i is the position of a in the partition µ at the bottom.

. . . (a + 1)cd(f -1) . . .

(i+2,i+3) t t (i,i+1) B B . . . (a + 1)c(d -1)f . . . (i+1,i+2) (i,i+1) B B . . . a(c + 1)d(f -1) . . . (i+2,i+3) t t (i+1,i+2) . . . (a + 1)d 2 f . . . (i,i+1) B B B B B B . . . a(c + 1)(d -1)f . . . (i+1,i+2) . . . ac 2 (f -1) . . . (i+2,i+3) t t t t t t . . . acdf . . . 4.2. Type C n . Now λ is a partition (λ 1 ≥ . . . ≥ λ n ≥ 0)
, and we use the same notation as in Section 4.1. It is easy to see that the minimal element 0 mentioned above (i.e., the unique minimal element below λ) is either 0 or ω 1 = (10 n-1 ), depending on |λ| being even or odd, respectively. By [34, Theorem 2.8], a cocover in the corresponding partial order on dominant weights is either of the same form [START_REF] Lee | Tableaux, crystal bases and the Gindikin-Karpelevitch formula[END_REF] as in type A, or has one of the following three forms: [START_REF] Lusztig | Singularities, character formulas, and a q-analog of weight multiplicities. Analyse et topologie sur les espaces singuliers (II-III)[END_REF] (. . .

1 2 0 n-k-1 ) (. . . 0 n-k+1 ) , (. . . 21) (. . . 10) , (. . . (a + 2)) (. . . a) ,
where 1 ≤ k ≤ n -1; these three cocovers are labeled by the roots ε k + ε k+1 , ε n-1 + ε n , and 2ε n , respectively. For simplicity, we denote a root ε i + ε j by α i or (i, ). In the sequel, we will see that it is desirable for the last cover in (23) never to appear; the reason is that α n is a long root, and hence does not appear in the W -orbit of α 1 . In fact, there is an easy condition which simplifies the setup even more. From now on, we work under the assumption of Proposition 4.1, and we call this the type C stable range.

To the authors' knowledge, an analogue of the classification of short intervals that was described above is not available beyond type A. In order to address this problem in type C n , in the stable range, we focus on the new cases involving a pair of cocovers and covers.

Given distinct cocovers µ ν and µ π of a partition µ, we can assume that they are labeled (i, j) and (k, k + 1), where necessarily i < k. It is easy to see that these cocovers can only have one of the following relative positions (in terms of their labels): (i) nonoverlapping if j < k; (ii) partially overlapping if j = k; (iii) fully overlapping if j = k + 2. We are led to Cases C1-C3 below; the two cocovers above are shown in the diagrams below in bold, and for simplicity we omit the trailing 0's in a partition. Thus, we proved the following result. Proposition 4.2. A lower bound of ν and π is always obtained as in one of the Cases C1-C3. Furthermore, we claim that the structure of the interval [ν ∧ π, µ] is always given by one of the diagrams below. The proof is completely similar to that of [4, Proposition 3.2], which was mentioned above. However, Proposition 4.2 suffices for our purposes. . . .

1 p+2 (j-1,) 7 7
. . . 21 p-2

(i,j-2) x x . . . 1 p
Case C2: partially overlapping cocovers, with (i, j) of type (*), so j = i + 1. We have a ≥ 2, while i is the position of a in the partition µ.

. . . a1 2 (i,i+1) w w (i+1,i+2) ) ) w w w w ) ) ) ) . . . (a -1)21 (i+1,i+3) . . . a (i,i+1) Ò Ò . . . (a -1)1 3 (i+2,i+3) 9 9
. . . (a -1)1

Case C3: partially overlapping cocovers, with (i, j) proper of type (**). Here p ≥ 1, while i is the position of (the shown) 3 in the partition µ, and j = i + p + 1 is the position of the first 1.

. . . 32 p 1 2 (j-1,j+2) (i,j) u u (j,j+1) A A u u u u A A A A . . . 2 p+2 1 (j,j+2) . . . 32 p-1 1 4 (i,j-1) u u (j+1,j+2) A A . . . 32 p (j-1,j) . . . 2 p+1 1 3 (j+1,j+2) A A . . . 32 p-1 1 2 (i,j-1) u u . . . 2 p+1 1
Now consider two distinct covers µ ν and µ π of µ, labeled (i, j) and (k, k + 1), respectively.

Proposition 4.3. An upper bound of ν and π is always obtained as in one of the Cases C1-C2.

Proof. We have µ k = µ k+1 = 0 and π k = π k+1 = 1. Clearly, we must have j < k and µ k-1 > 0. If j < k -1, or j = k -1 and µ k-1 > 1, we are in Case C1 (a). Otherwise, when j = k -1 and µ k-1 = 1, we have either i < j -1 or i = j -1; but these are precisely Cases C1 (b) and C2, respectively.

Furthermore, we claim that the structure of the interval [µ, ν ∨π] is always given by the diagrams in Cases C1 or C2. The proof is again completely similar to that of [4, Proposition 3.2]. However, Proposition 4.3 suffices for our purposes.

Type

D n . Now λ is a sequence (λ 1 ≥ . . . ≥ λ n ) with λ i ∈ 1
2 Z, all congruent mod Z, such that λ n-1 + λ n ≥ 0. By [34, Theorem 2.8], a cocover in the corresponding partial order on dominant weights is either of the same form [START_REF] Lee | Tableaux, crystal bases and the Gindikin-Karpelevitch formula[END_REF] as in type A (with entries in 1 2 Z now allowed), or has one of the following three forms:

(. . . 1 2 0 n-k-1 ) (. . . 0 n-k+1 ) , (. . . (a + 1)a n-l-1 (-a + 1)) (. . . a n-l (-a)) , (24)
(. . . (a + 1)(b + 1)) (. . . ab) , where 1 ≤ k, l ≤ n -1 and a ≥ 1 2 ; these cocovers are labeled by the roots ε k + ε k+1 , ε l + ε n , and ε n-1 + ε n , respectively.

We will now assume that λ i ∈ Z. This implies that the interval [ 0, λ] only contains weights µ = (µ 1 ≥ . . . ≥ µ n ) with µ i ∈ Z. Note that, in this case, there are the same possibilities for the minimal element 0 as in type C, see Section 4.2. Proof. Assume that the statement fails. Then there is µ ≤ λ with µ n-2 ≥ 1, µ 1 ≥ 2, and µ n-1 +µ n ≥ 1. Combining this with the fact that |λ| ≥ |µ|, we obtain |λ| ≥ n. But this contradicts the condition in the proposition.

From now on, we work under the assumption of Proposition 4.4, and we call this the type D stable range. Clearly, all the results about the type C stable range in Section 4.2 apply to the type D one.

Modified crystal operators on finite crystals

We now assume that g is a simple Lie algebra over C of rank r which satisfies the following condition: we can (and will) label the Dynkin diagram of g with the integers 1, . . . , r such that the parabolic Dynkin diagram obtained by removing the node r is of type A r-1 . In other words, the only excluded Lie type is the one of type F 4 . 5.1. Definition of the modified crystal operators. For any w ∈ W , define the modified crystal operator k w on B(λ) as the conjugation k w := w f1 w -1 of the ordinary crystal operator f1 by the Kashiwara action of w on B(λ) [START_REF] Kashiwara | Crystal bases of modified quantized enveloping algebra[END_REF] 2 . This means that k w (b) = 0 precisely when f1 applied to w -1 (b) is 0. For any b ∈ B(λ), we have

wt(k w (b)) = wt(b) -α ,
where α = w(α 1 ) .

2 These operators were considered in [START_REF] Kashiwara | Crystal bases of modified quantized enveloping algebra[END_REF]Remark 7.4.2], and it was observed there that they do not coincide with the ordinary ones, but to the authors' knowledge they were not studied further.

When α is a positive root, we set f w := k w . When w ∈ W is such that w (α 1 ) = -α is a negative root, set w := w s 1 , so that we have w(α 1 ) = α and ( 25)

k w = ws 1 f1 s 1 w -1 = wẽ 1 w -1 .
It is then convenient to write e w := k w . In the sequel, we will only consider operators f w and e w with w(α 1 ) ∈ R + . Assume moreover that w 1 and w 2 are two elements of W such that w 1 (α 1 ) = w 2 (α 1 ) = α ∈ R + . Then w -1 2 w 1 (α 1 ) = α 1 , and therefore w -1 2 w 1 belongs to the parabolic subgroup of W generated by the simple reflections s 3 , . . . , s n . It follows that the actions of f1 and w -1 2 w 1 (respectively, ẽ1 and w -1 2 w 1 ) commute. We thus have f w 1 = f w 2 and e w 1 = e w 2 . Therefore, given any positive root α in the orbit of α 1 3 , it makes sense to define f α := f w and e α := e w , where w is any element in W such that w(α 1 ) = α. When α = α i is a simple root, we simply write f i := f α i and e i := e α i . We clearly have f 1 = f1 , but f i = fi in general, and similarly for e 1 .

We endow the vertices of B(λ) with the structure of a colored directed graph B(λ) with edges b α b when b = f α (b). As noted above, the graph B(λ) is different from the Kashiwara crystal B(λ), and in fact, unlike the latter, the former is not connected in general. Proof. Indeed, let w ∈ W be such that w(α 1 ) = α. The first equality is equivalent to

w • (b ) = w • w f1 w -1 w -1 • (w • (b)
) . Now observe that w • w(α 1 ) = -α * and thus, by [START_REF] Mirković | Geometric Langlands duality and representations of algebraic groups over commutative rings[END_REF], the right-hand side of the above equality is e α * (w • (b)), as desired. (

1) If wt(b), α > 0, then f α (b) = 0. In particular, if wt(b) -α is dominant, where α ∈ R + , then f α (b) = 0. (2) If wt(b), α < 0, then e α (b) = 0. Proof. Choose w ∈ W such that w(α 1 ) = α. If wt(b), α > 0, then w -1 wt(b), w -1 (α) = wt(w -1 (b)), α 1 > 0, which implies that f1 (w -1 (b)) = 0. But this is equivalent to f α (b) = 0.
For the second part of (1), let µ := wt(b), and observe that µ, α ∨ = µ -α, α ∨ + 2 ≥ 2; so we can apply the first part. Then we deduce (2) from (1) by using Lemma 5.1. (1)

If α + β ∈ R + , then f α f β (b) = f β f α (b) = f α+β (b) = 0 . (2) If α, β = 0, then f α f β (b) = f β f α (b) = 0 .
We first reduce our theorem to an equivalent simpler statement. Its two parts will be proved in Sections 5.4 and 5.5, respectively. Lemma 5.4. Theorem 5.3 is true if and only the following two statements hold. 3 Observe that the orbit W α1 coincides with the set of all roots only for simply laced root systems.

(1) For any b such that wt(b), α 1 > 0 and wt(b), α 2 > 0, we have

f 1 f 2 (b) = f 2 f 1 (b) = f α 1 +α 2 (b) = 0 .
(2) For any b such that wt(b), α 1 > 0 and wt(b), α 3 > 0, we have

f 1 f 3 (b) = f 3 f 1 (b) = 0 .
Proof. The group W acts on the set D = {{α, β} | α = β, α, β ∈ W α 1 } by w(α, β) = (w(α), w(β)) for any w ∈ W . When α, β = 0, we must have r ≥ 3 and (w(α), w(β) = 0 for any w ∈ W . There thus exists w ∈ W such that w(α) = α 1 and w(α) = α 3 . We then obtain

f α f β (b) = f β f α (b) ⇐⇒ (wf α w -1 )(wf β w -1 )(w(b)) = (wf β w -1 )(wf α w -1 )(w(b)) ⇐⇒ f 1 f 3 (w(b)) = f 3 f 1 (w(b)) .
When α + β > 0, we must have r ≥ 3 in types B r ,and C r 4

. By using that α and β belong to W α 1 , one checks by considering the set of positive roots in each type A r , B r , C r and D r that there always exists w in the Weyl group W such that w(α) = α 1 and w(β) = α 2 , or w(α) = α 2 and w(β) = α 1 . In the first case, one gets

f α f β (b) = f β f α (b) = f α+β (b) ⇐⇒ (wf α w -1 )(wf β w -1 )(w(b)) = (wf β w -1 )(wf α w -1 )(w(b)) = wf α+β w -1 (w(b)) ⇐⇒ f 1 f 2 (w(b)) = f 2 f 1 (w(b)) = f α 1 +α 2 (w(b)) .
The second case is similar.

We have an analogous result to Theorem 5.3 for the e • operators. By analogy with Theorem 5.3, the above result is proved based on the following reduction. In turn, the two parts of Lemma 5.6 are proved in the same way as those of Lemma 5.4, also in Sections 5.4 and 5.5, respectively. Lemma 5.6. Theorem 5.5 is true if and only the following two statements hold. Proof. The proof of Assertion 1 is similar to that of Lemma 5.4. To prove Assertion 2, we can assume with no loss of generality that α = ε i -ε j and β = ε k -tε l with i < j < k < l and t ∈ {±1}.

Let w ∈ W be such that w(α) = α 1 and w(β) = α 3 . We have As mentioned, we start with a direct proof of the first parts of Lemmas 5.4 and 5.6; these are based on the description of the actions of f 2 and f α 1 +α 2 on semistandard tableaux. We then prove the second parts by using the cyclage of Lascoux and Schützenberger (which can also be used to reprove the first parts). 5.3. Cyclage, charge and the modified crystal operators. Given a semistandard tableau T , write C(T ) = x → T where x → T is the semistandard tableau obtained after the row-insertion of the south-west letter x of T in the tableau T \ {x}. The combinatorial procedure T → C(T ) is called the cyclage of the tableau T . It is known (see [START_REF] Lascoux | Crystal graphs and q-analogue of weight multiplicities for the root system An[END_REF]) that the sequence of cyclages applied to T will eventually lead to the unique row-tableau R µ where µ = wt(T ) is the weight of T . The number co(T ) of cyclage operations used in this sequence is called the cocharge of T . The charge of T is then defined as c(T ) = µ -co(T ) where µ = n-1 i=1 (i -1)µ i .

w -1 (i) = 1, w -1 (2) = j, w -1 (3) = k and w -1 (4) = tl. Then γ = w -1 (α 2 ) = ε j -ε k is
Example 5.7.

For T = 1 1 4 2 2 3
we get

T 1 = C(T ) = 1 1 3 2 2 4 , T 2 = C 2 (T ) = 1 1 2 2 3 4 , T 3 = C 3 (T ) = 1 1 2 4 2 3
T 4 = C 4 (T ) = 1 1 2 2 3 4 , T 5 = C 5 (T ) = 1 1 2 2 3 4 T 6 = C 6 (T ) = 1 1 2 2 3 4 .
Therefore co(T ) = 6 and c(T ) = 7 -6 = 1.

We can endow the set Tab µ of semistandard tableaux of weight µ with the structure of an oriented graph with an oriented edge T T when T = C(T ). We then have a unique sink vertex in Tab µ which is the row tableau of evaluation µ. Given µ and ν two weights, we write ν ≤ µ when µ -ν can be written as a linear combination of simple roots ε i -ε i+1 , i = 1, . . . , n -1 with nonnegative integral coefficients. We refer to [START_REF] Lascoux | Crystal graphs and q-analogue of weight multiplicities for the root system An[END_REF] for a proof of the following theorem. It is well-know that the charge statistic yields a combinatorial description of the Kostka polynomials in type A.

Theorem 5.10. For any partitions λ and µ we have

K λ,µ (t) = T ∈Tab(λ)µ t c(T )
where Tab(λ) µ is the set of semistandard tableaux of shape λ and evaluation µ.

The following proposition is a consequence of Theorems 5.8 and 5.10. It shows how the charge and cocharge statistics are modified when a modified crystal operator is applied to a tableau T of shape λ (regarded as a vertex of the crystal B(λ)). For any positive root α, let |α| be the height of α, that is the number of simple roots appearing in the decomposition of α as a sum of simple roots. Proof. Since f α is obtained by conjugation of the action of f1 , we get the equality co(f α (T )) = co(T ) from Assertion 1 and 2 of Theorem 5.8. Write α

= ε i -ε i with 1 < i < j < n. Then f α (T ) has weight µ -α. Since co(f α (T )) = co(T ), we have c(f α (T )) -c(T ) = µ -α -µ = (i -1)((µ i -1) -µ i ) + (j -1)((µ j + 1) -µ j ) = j -i = |α| .
5.4. Proof of Lemmas 5.4 (1) and 5.6 [START_REF] Braden | From moment graphs to intersection cohomology[END_REF]. Observe first that it suffices to prove Lemmas 5.4 (1) and 5.6 (1) in type A 2 . To do this, we will use the tableau realization of the crystal B(λ). As we restrict to type A 2 , we view λ as a partition λ = (λ 1 ≥ λ 2 ≥ 0), and we index the vertices of B(λ) by semistandard tableaux of shape λ with the alphabet {1 < 2 < 3}. We adopt the English notation for semistandard tableaux. The row reading of a semistandard tableau T is the word w(T ) obtained by reading its rows from right to left and top to bottom. The actions of f 1 , s α 1 , and s α 2 on B(λ) are computed by using the following well-know procedures.

For any i = 1, 2, write w i for the subword of w(T ) formed by the letters in {i, i + 1}. Let w red i = (i + 1) r i s be the subword of w i obtained by recursively deleting factors i(i + 1). Now consider w red i as a subword of w(T ). If r > s, then s α i is obtained by replacing in w(T ) the r -s rightmost letters i + 1 of w red i with i. Otherwise, s α i is obtained by replacing in w(T ) the s -r leftmost letters i of w red i with i + 1. Observe here that for each factorization w i = (i + 1) a u i i a , we have

s i (w i ) = (i + 1) a s i (u i )i a .
The action of f 1 on a tableau T is straightforward. If T does not contain any letter 1, then f 1 (T ) = 0. Otherwise f 1 (T ) is obtained by replacing its rightmost letter 1 (always in its first row) with 2.

Let us now describe the actions of f α 1 +α 2 and f 2 on a tableau T . Let m

(k) i = m (k)
i (T ) be the number of letters i in the k-th row of T . Let µ = (µ 1 , µ 2 , µ 3 ) be the content of T , i.e., µ i is the number of letters i in T . Lemma 5.12. Assume µ 1 > µ 3 .

(1) If T does not contains any letter 2, then f α 1 +α 2 (T ) is obtained by changing in T the rightmost letter 1 into a letter 3. (2) Otherwise f α 1 +α 2 (T ) is obtained by changing in T the rightmost letter 1 into a letter 2 and

• the rightmost letter 2 in its first row into a letter 3 when m

3 , • the rightmost letter 2 in its second row into a letter 3 when m

(2) 2 > m (1) 3 . Proof. Recall that f α 1 +α 2 (T ) = s α 2 f 1 s α 2 (T ). Let w(T ) = 3 a 2 b 1 c 3 d 2 e .
The assumption µ 1 > µ 3 is written c > a+d ≥ 0. Let m = min(a, e). By the definition of the actions of s α 2 and f 1 , the contributions of the m leftmost letters 2 in the second row and that of the m rightmost letters 3 in the first row cancel. So, without loss of generality, we have the following two cases, in which we show the successive actions of s α 2 , f 1 , and s α 2 ; the paired entries when applying s α 2 are underlined.

Case 1: a ≥ e = 0.

3 a 2 b 1 c 3 d = 3 a 2 max(b-d,0) 2 min(b,d) 1 c 3 min(b,d) 3 max(d-b,0) sα 2 --→ sα 2 --→ 3 max(b-d,0) 2 a 2 min(b,d) 1 c 3 min(b,d) 2 max(d-b,0) f 1 -→ f 1 -→ 3 max(b-d,0) 2 a+1 2 min(b,d) 1 c-1 3 min(b,d) 2 max(d-b,0) sα 2 --→ sα 2 --→ 3 a+1 2 max(b-d,0) 2 min(b,d) 1 c-1 3 min(b,d) 3 max(d-b,0) = 3 a+1 2 b 1 c-1 3 d .
Case 2: e > a = 0.

2 b 1 c 3 d 2 e = 2 max(b-d,0) 2 min(b,d) 1 c 3 min(b,d) 3 max(d-b,0) 2 e sα 2 --→ sα 2 --→ 3 max(b-d,0) 2 min(b,d) 1 c 3 min(b,d) 3 e 2 max(d-b,0) f 1 -→ f 1 -→ 3 max(b-d,0) 2 min(b,d)+1 1 c-1 3 min(b,d)+1 3 e-1 2 max(d-b,0) sα 2 --→ sα 2 --→ 2 max(b-d,0) 2 min(b,d)+1 1 c-1 3 min(b,d)+1 3 max(d-b,0) 2 e-1 = 2 b+1 1 c-1 3 d+1 2 e-1 . Lemma 5.13. Assume µ 2 > µ 3 . If m (2) 2 ≤ m (1) 3 (respectively m (2) 2 > m (1)
3 ), the tableau f 2 (T ) is obtained by changing in T the rightmost letter 2 in the first (respectively second) row into a letter 3.

Proof. Recall that f 2 (T ) = s 1 f α 1 +α 2 s 1 (T ).

Let w(T ) = 3 a 2 b 1 c 3 d 2 e , so we have c ≥ e. The assumption µ 2 > µ 3 is written b + e > a + d. Based on Lemma 5.12, we consider the following two cases, in which we show the successive actions of s α 1 , f α 1 +α 2 , and s α 1 ; the paired entries when applying s α 1 are underlined.

Case 1: a ≥ e. We have b + e > a + d ≥ d + e, so b > d ≥ 0. 3 a 2 b 1 c 3 d 2 e = 3 a 2 b 1 c-e 1 e 3 d 2 e sα 1 --→ 3 a 2 c-e 1 b+e 3 d 2 e f α 1 +α 2 ----→ 3 a+1 2 c-e 1 b+e-1 3 d 2 e = = 3 a+1 2 c-e 1 b-1 1 e 3 d 2 e sα 1 --→ 3 a+1 2 b-1 1 c 3 d 2 e .
Case 2: e > a ≥ 0.

3 a 2 b 1 c 3 d 2 e sα 1 --→ 3 a 2 c-e 1 b+e 3 d 2 e f α 1 +α 2 ----→ 3 a 2 c-e+1 1 b+e-1 3 d+1 2 e-1 = = 3 a 2 c-e+1 1 b 1 e-1 3 d+1 2 e-1 sα 1 --→ 3 a 2 b 1 c 3 d+1 2 e-1 .
Corollary 5.14. Consider a tableau T of weight (µ 1 , µ 2 , µ 3 ) such that µ 2 ≥ µ 3 . Then e 2 (T ) = 0 if and only if m

(2) 2 ≥ m (1)
3 and m

(2)

3 = 0. Proof. Assume m (2) 2 < m (1)
3 . Since m

3 > 0, we can consider the tableau T obtained by changing the leftmost letter 3 in 2 in the first row of T . By the previous lemma, we have then e 2 (T ) = T = 0. Similarly, when m 3 > 0, we also get e 2 (T ) = T where T is then obtained from hypothesis, a > µ 4 + b, thus a -1 ≥ µ 4 and m

(2) 2 (U ) ≥ µ 4 = m (1)
3 (U ). We yet get the contradiction e 2 (U ) = 0 by Corollary 5.14.

We can now prove Lemma 5.18 by induction on the cocharge.

Proof. Observe first Lemma 5.18 is clearly true for row tableaux, that is for tableaux of cocharge 0. Assume it holds for any tableau of cocharge k and the tableau T considered as cocharge k + 1. If T has two rows we are done by the previous lemma. So we can assume that T has at least 3 rows. Set T = e 3 (T ). By Corollary 5.9, we can apply our induction hypothesis to the tableau C(T ), we must have e 1 (C(T )) = 0. We claim this implies that e 1 (T ) = 0. Indeed, the conditions e 1 (C(T )) = 0 but e 1 (T ) = 0 would imply that the letter of T which is used in the cyclage operation is a 2 (any other letter would not modify the locations of the letters 1 and 2 in T ). But by definition of the cyclage, 2 is then the leftmost letter of the shortest row of T , which is only possible if T has two rows contrary to the assumption we made. To terminate the proof it suffices to apply Lemma 5.17 to T = e 1 (T ) = e 1 e 3 (T ).

Remarks 5.20. (1) One can use a similar method to give alternative proofs of Lemmas 5.15 and 5.16, that is, without making explicit the action of f 2 and f α 1 +α 2 . Note that only the basic properties of cyclage mentioned in Section 5.3 are used to carry out the induction step, and no related combinatorics.

(2) On another hand, it would be interesting to prove Lemmas 5.17 and 5.18 in a similar way to Lemmas 5.15 and 5.16, that is, by making explicit the action of f 3 . While such a proof would only use the crystal structure (without referring to cyclage), we found it challenging because the Weyl group action is not easy to express in any of the combinatorial models we considered, so we were led to an unmanageable number of cases. Nevertheless, the special case considered in the proof of Lemma 5.19 is manageable with the tableau model. The main goal is to identify situations in which the components of the poset B(λ) + define an atomic, respectively t-atomic decomposition, cf. Definitions 2.6 and 2.7. Remark 6.1. It is important to use the setup mentioned above, as we found that several other variations fail, as explained below.

Proof of the atomic decomposition

• If we consider all vertices of B(λ), rather than just those of dominant weight, then Lemma 5.17 It suffices to check that these diagrams commute, which follows by using Theorem 5.3 (1)-( 2) repeatedly. In fact, we apply this theorem to the corresponding triangles and diamonds, after we verify its hypotheses by inspecting the diagrams. In the case of the pentagons, for instance in Case A2 (a), we let f ij := f α ij , and calculate: Proof. The notation and related conditions are the same as in Section 4.1. Like in the proof of Lemma 6.3, the goal is to lift the diagrams in Cases A1 -A3 from the dominant weight poset to B(λ) + . In fact, we will perform the lift along the edges of slightly modified diagrams (by starting from the bottom) in the cases indicated below, while we use the same diagrams as in Section 4.1 in Cases A1 (a) and A1 (b).

f i+2,j f i+1,i+2 (f i,i+1 (b)) = f i+1,j f i,i+1 (b) = f i,i+1 f i+1,j (b)
Case A1 (c).

. . . ab p c q d . . . . . . ab p-1 cbc q-1 d . . . As we saw, all of its minimal vertices have weight 0. We will first prove the uniqueness of a minimal vertex. Assuming the contrary, let C be the subposet of C consisting of vertices connected via directed paths to more than one minimal vertex. Find an undirected path in C connecting two distinct minimal vertices, and consider its local maxima and minima which are not endpoints. As noted above, there are directed paths from the local minima to minimal vertices. By considering these paths, we can see that some local maximum must be in C, so C = ∅; see It is now clear that the components of B(λ) + define an atomic decomposition, where in each component we choose its maximum as the distinguished vertex h ∈ H(λ).

To get the t-atomic decomposition, we fist need, according to Definition 2.1, a statistic on H(λ). One can use the realization of B(λ) in terms of semistandard tableaux. Obviously, the natural candidate is the charge statisitcs c. For any dominant weight µ (i.e. for any partition with at most n parts), we set

A λ,µ = h∈H(λ) wt(h)=µ t c(h) .
We then have for any dominant weight ν

K λ,ν (t) = T ∈B(λ)ν t c(T ) = ν≤µ≤λ h∈H(λ) wt(h)=µ T ∈B(λ)ν ∩B(λ,h) t c(T ) .
Now by Proposition 5.11, we obtain the equality c(T ) = c(h) + µ -ν, ρ ∨ for any T ∈ B(λ) ν ∩ B(λ, h). Indeed we have wt(h) -wt(T ) = µ -ν and T can be obtained from h by applying modified crystal operators f α , each of them increasing the charge by α, ρ ∨ . Also, the set B(λ) ν ∩ B(λ, h) is reduced to a singleton because the connected component B(λ, h) of B(λ) + contains exactly one vertex of weight ν ≤ λ. Thus we can write

1 1 1 2 2 3 (1,3) (2,4) 1 1 1 1 3 2 2 3 (3,4) 1 1 1 2 4 3 (1,2) 1 1 4 2 2 3 1 1 2 2 3 3 (3,4) 1 1 1 1 1 2 3 4 (1,2)
K λ,ν (t) = ν≤µ≤λ t µ-ν,ρ ∨ h∈H(λ) wt(h)=µ t c(h) = ν≤µ≤λ t µ-ν,ρ ∨ A λ,µ (t) ,
which is equivalent to the t-atomic decomposition by Proposition 2.2.

Example 6.6. Consider λ = (3, 2, 1) in type A 3 . The modified crystal graph B(λ) + is shown in Figure 3. Its vertices are labeled by semistandard Young tableaux whose content is a partition, and its edges are labeled as above. In particular, this graph gives the following atomic decomposition of the character: Proof. In addition to the cases considered in the proofs of Lemmas 6.3 and 6.4, we need to consider the cases involving the new type of cover in the corresponding dominant weight poset, namely Cases C1-C3 (for extending the first lemma, by Proposition 4.2) and Cases C1 -C2 (for extending the second one, by Proposition 4.3). The goal is the same: to lift the corresponding diagrams from the dominant weight poset to B(λ) + . We use the same reasoning as in the mentioned lemmas, and give more details below.

χ λ = w (3,
For instance, in order to prove Lemma 6.3 in Case C2, we let f i := f α i , and calculate:

f i+2,i+3 f i+1,i+3 (f i,i+1 (b)) = f i+1,i+2 f i,i+1 (b) = f i,i+1 f i+1,i+2 (b)
; indeed, each of the two equalities follows from Theorem 5.3 (1) after we verify its hypothesis by inspecting the corresponding diagram. (2,3) is also a dominant weight. Otherwise, there is another root α which is dominant, and we have 0 ≤ α ≤ α. More precisely, both α and α -α are short roots of R + . In the crystal B(α), there are vertices b α of weight α, one for each root α of g and r vertices of weight 0.

1 3 2 3 (1,2) 1 3 2 3 (1,2) 2 2 3 3 1 1 3 3 (1,2) 2 3 3 2 1 1 3 3 1 2 3 3 (1,2)
Lemma 7.1. For any simple α i and any index j ∈ {1, . . . , r}, we have fj (b α i ) = 0 if and only if i = j.

Proof. Since α i = -α the lowest weight in B(α), there is at least an index j such that f j (b

α i ) = 0. If j = i, the vertex f j (b α i ) has weight α i -α j = α ∈ R. When α ∈ R + (resp. when -α ∈ R + ), we get a contradiction because α i = α j + α is not simple (resp. α j = α i + (-α) is not simple).
It follows from the lemma that the vertices b i = fα i (b α i ), i = 1, . . . , r, are the zero weight vertices in B(α). Recall also that

K α,0 (t) = r i=1 t m i ,
where m 1 , . . . , m r are the classical exponents of g. We can choose m 1 = |α| since |α| is the greatest exponent.

Assume first that α 1 is short, that is, the root system is not of type B r or F 4 . In the simply laced case, the highest root α is in the orbit of α 1 and, since α, α > 0, we derive by Lemma 5.2 that f α( α) = 0 is a vertex of zero weight in B(α). In fact, the previous lemma also implies that f α( α) = b 1 , because f α (b i ) = e α (b i ) = 0 for any i = 1 and any positive root α. Indeed, each vertex b i has zero weight and thus is fixed under the action of the Weyl group, while f1 (b i ) = 0 if and only if i = 1. In the non-simply laced case (that is, in types C r and G 2 under our assumption), we get similarly f α-α( α) = b α (because α, α -α > 0) and f α( α) = b 1 . In all cases, the previous actions of the modified operators correspond to coverings in the dominant weight poset. Therefore, with the notation of Section 2. We then get

A α, α(t) = h∈H( α) wt(h)= α t c(h) = 1 and A α,0 (t) = h∈H( α) wt(h)=0 t c(h) = t m 2 + • • • + t mr .
For types C r and G 2 , we also have

A α, α(t) = h∈H( α) wt(h)= α t c(h) = 0 .
Finally, by Proposition 2.2, we get the desired t-atomic decomposition K α,0 (t) = t 0 A α,0 (t) + t | α| A α, α(t) in the simply laced case t 0 A α,0 (t) + t | α|-| α| A α, α(t) + t | α| A α, α(t) for types C r and G 2 .

Now it remains to consider types B r and F 4 where α 1 is a long root. Recall Stembridge's result [START_REF] Stembridge | The partial order of dominant weights[END_REF] stating that any simple root gives a cover in dominant weight poset. Thus, we cannot define the modified crystal operators based on f1 , because this would not permit to get the covering relations corresponding to short roots. Now, if we define the modified crystal operators by Weyl group conjugation of an ordinary crystal operator fi with α i short root, this will not match the coverings corresponding to long roots. Nevertheless, for the adjoint representation in type B r and F 4 , the coverings we need only make the short roots appear. So we can choose any short simple root α i 0 (there is one in type B r and two in type F 4 ) and define, for any short root α ∈ R + , the modified operators f α = u α fi 0 u -1 α , where u α ∈ W is such that u(α i 0 ) = α. By using the same arguments as above, we then also get a t-atomic decomposition for K α,0 (t), this time with B(α, b i ) = {b i } for any i = i 0 and B(α, α) : b α α-α b α α b i 0 .

Remark 7.2. The example of the adjoint representation illustrates the restrictions we imposed in this paper: the atomic decompositions we obtained are for types A r , as well as for C r and D r when the rank is sufficiently large, depending on the considered partition. This is because, in these cases, the covering relations only make appear roots in the orbit of α 1 , and we can use the modified operators defined based on f1 ; for them, we were able to prove simple commutation relations in Theorems 5.3 and 5.5. We expect that these operators suffice to derive an atomic decomposition of crystals in the simply laced cases with mild assumptions on the highest weight considered (as in Example 2.4). In the non-simply laced cases, the situation is more involved; namely, atomic decompositions of crystals will certainly require one to consider modified crystal operators defined by the conjugation of two Kashiwara operators, one for each root length, while new commutation relations will be needed. 7.2. More about the charge. In Section 2.3, we defined the t-atomic decomposition of B(λ) from a charge statistic on H(λ) which propagates on the vertices b of each component B(λ, h), h ∈ H(λ), by the formula [START_REF] Lecouvey | Kostka-Foulkes polynomials, cyclage graphs and charge statistic for the root system Cn[END_REF]. When such an atomic decomposition exists, it yields a combinatorial description of K λ,µ (t) by [START_REF] Lecouvey | Combinatorics of generalized exponents[END_REF]. In type A, Theorem 6.5 gives a t-atomic decomposition of crystals, where c is the Lascoux-Schützenberger charge statistic.

Conversely, if we fix ν ∈ P + and assume that we have both a combinatorial description of the Kostka polynomials K λ,µ (t) with λ ≤ ν due to a statistic c defined on the crystals B(λ), λ ≤ ν, and an atomic decomposition of each B(λ) in which ( 17) holds, we will obtain a t-atomic decomposition of B(λ) based on c, exactly as in the proof of Theorem 6.5. Now assume that ν is a partition and g is of type C r with r sufficiently large. In [START_REF] Lecouvey | Kostka-Foulkes polynomials, cyclage graphs and charge statistic for the root system Cn[END_REF] a statistic was defined on Kashiwara-Nakashima tableaux which conjecturally gives a combinatorial description of the Kostka-Foulkes polynomials. The definition of this statistic is involved, but we expect that it satisfies [START_REF] Lecouvey | Kostka-Foulkes polynomials, cyclage graphs and charge statistic for the root system Cn[END_REF] when the atomic decomposition of B(λ) based on the modified crystal operators is used. In [START_REF] Lecouvey | Combinatorics of generalized exponents[END_REF], we also defined a statistic on King tableaux of zero weight, and proved that it yields a combinatorial description of the generalized exponents K λ,0 (t). It is then tempting to combine the latter construction with the Sheats bijection between King and Kashiwara-Nakashima tableaux [START_REF] Sheats | A symplectic jeu de taquin bijection between the tableaux of King and De Concini[END_REF], the atomic decomposition of the crystal B(λ) with λ a partition of even rank, and (17) in order to define another statistic on the whole B(λ) + . Two interesting questions would then arise: does the new statistic coincide with that of [START_REF] Lecouvey | Kostka-Foulkes polynomials, cyclage graphs and charge statistic for the root system Cn[END_REF], and does it give a combinatorial description of the Lusztig t-analogue? If the answers are affirmative, we would get, in type C: (1) a generalization of the Lascoux-Schützenberger charge; (2) an efficient algorithm, based on crystal combinatorics, for calculating K λ,µ (t) starting from K λ,0 (t).

Geometric interpretation

We give such an interpretation in terms of the geometric Satake correspondence. For a reductive group G, this important theory exhibits a geometric realization of the irreducible representation V (λ) of highest weight λ of the Langlands dual group, as the intersection cohomology IH * (Gr λ ) of the Schubert variety denoted Gr λ in the affine Grassmannian Gr G for G; there is also a geometric basis of MV-cycles [START_REF] Mirković | Geometric Langlands duality and representations of algebraic groups over commutative rings[END_REF]. However, it is hard to give concrete formulas for the MV cycles and the action. We will show how one can understand the combinatorics of the geometric Satake correspondence via our combinatorial atomic decomposition.

The Schubert variety Gr λ in Gr G has a Bott-Samelson desingularization Σ → Gr λ → Gr G . Thus, we have cohomology maps [START_REF] Nakayashiki | Kostka-Foulkes polynomials and energy function in sovable lattice models[END_REF] H * (Gr G ) → H * (Gr λ ) → H * ( Σ) IH * (Gr λ ) ⊕ other summands .

The direct sum decomposition, as H * (Gr G )-modules, is given by the Decomposition Theorem, see e.g. [START_REF] De Cataldo | The decomposition theorem, perverse sheaves and the topology of algebraic maps[END_REF].

IH * (Gr λ ) has the truncation filtration (or standard Grothendieck filtration), which gives the Kostka-Foulkes polynomials when restricted to the weight spaces [START_REF] Ginzburg | Perverse sheaves on a Loop group and Langlands' duality[END_REF]. The degree 0 piece in this filtration is the cohomology of the constant sheaf, i.e., H * (Gr λ ). This has the basis of classes of Schubert varieties inside Gr λ , which are indexed by the weights of V (λ) considered without multiplicity, as recorded in the layer sum polynomials. In this language, the atomic decomposition decomposition in Definition 2.1 is expressing the fact that there is a refinement of the truncation filtration (with the H * (Gr G )-action), whose successive quotients are isomorphic to H * (Gr µ ) for µ ∈ P + (λ). These quotients correspond precisely to the blocks of the partition in the combinatorial atomic decomposition, see Definition 2.6.

2. 3 .Definition 2 . 6 .

 326 Atomic decomposition of finite crystals. Let B(λ) + be the subset of B(λ) of vertices with dominant weights. The crystal B(λ) has an atomic decomposition if there exists a subset H(λ) ⊂ B(λ) + and a partition[START_REF] Lascoux | Sur une conjecture de H. O. Foulkes[END_REF] B(λ) + = h∈H(λ)B(λ, h)such that h ∈ B(λ, h) is a distinguished vertex, and each component B(λ, h) consists of exactly one vertex of weight ν for each ν ≤ wt(h).

)

  Each connected component B of the graph B(∞) contains a unique source vertex S. We then write B = B(S), and say that B(S) is an atom of B(∞). (3) For any S ∈ ST (∞), the vertices of B(S) have different weights, and

  By [4, Proposition 3.2], the interval [ν ∧ π, µ] can only have one of the following structures; the two cocovers above are shown in the diagrams below in bold. Case A1: cocovers which are (a) nonoverlapping; (b) partially overlapping and both of type (*); (c) fully overlapping and both proper of type (**). As subcase (a) is easy, only subcases (b) and (c) are represented in the diagrams below. In subcase (b), we have a ≥ c + 2 and c ≥ e + 2, while i is the position of a in the partition µ. . . . ace . . .

  t t t B B B B . . . (a -1)(c + 1)e . . . (i+1,i+2) B B . . . a(c -1)(e + 1) . . .

  ab p-1 c q+2 . . .

  b p+1 c q+1 . . . Case A2: partially overlapping cocovers, where (a) the first is of type (*) and the second proper of type (**); (b) vice versa. In subcase (a), we have a ≥ c + 2, d = c -1, e = d -1, p ≥ 1, while i is the position of a in the partition µ and j = i + p + 2 is the position of e. . . . acd p e . . .

5 t t t t 5 5 5 5 .

 55 . . (a -1)(c + 1)d p e . . . (i+1,i+2) . . . ad p+2 . . .(i,i+1) | | . . . (a -1)c 2 d p-1 e . . . (i+2,j) B B . . . (a -1)cd p+1 . . . In subcase (b), we have b = a -1, c = b -1, c ≥ e + 2, p ≥ 1, while i is the position of a and j = i + p + 1 is the position of c. . . . ab p ce . . .

B

  B B B . . . ab p (c -1)(e + 1) . . .

4 .

 4 . . ab p-1 c 2 (e + 1) . . .

  (i,j-1) t t . . . b p+1 c(e + 1) . . . Case A3: partially overlapping cocovers, both proper of type (**). Here b

  ab p d q+2 . . .

(j- 1

 1 ,j) . . . b p+1 c 2 d q-1 e . . . (j+1,k) B B . . . ab p-1 c 2 d q+1 . . . (i,j-1) t t . . . b p+1 cd q+1 . . .

Proposition 4 . 1 .

 41 If n > (|λ| + 1)/2, then the first cocover in[START_REF] Lusztig | Singularities, character formulas, and a q-analog of weight multiplicities. Analyse et topologie sur les espaces singuliers (II-III)[END_REF] is the only one which can appear in the Hasse diagram of the interval [ 0, λ] beside the type A cocovers in[START_REF] Lee | Tableaux, crystal bases and the Gindikin-Karpelevitch formula[END_REF].Proof. Assume that a partition µ ≤ λ has the form (. . . 21) or (. . . 2). Then we must have µ 1 = . . . = µ n-1 ≥ 2. Combining this with the fact that |λ| ≥ |µ|, we obtain |λ| ≥ 2n -1 or |λ| ≥ 2n, respectively. But this contradicts the condition in the proposition.

Case C1 :

 C1 cocovers which are (a) nonoverlapping; (b) fully overlapping, with (i, j) proper of type (**). As subcase (a) is easy, only subcase (b) is represented in the diagram below.In subcase (b), we have p ≥ 2, while i is the position of (the shown) 2 in the partition µ, and j = i + p + 1 is the position of the first 0.

Proposition 4 . 4 .

 44 If n > |λ|, then the first cocover in[START_REF] Macdonald | Spherical functions on a group of p-adic type[END_REF] is the only one which can appear in the Hasse diagram of the interval [ 0, λ] beside the type A cocovers in[START_REF] Lee | Tableaux, crystal bases and the Gindikin-Karpelevitch formula[END_REF].

5. 2 .Lemma 5 . 1 .

 251 First properties of the modified crystal operators. Let w • be the maximal length element of the Weyl group W . Recall that w • induces an involution * on the set of positive roots of g such that w • (α) = -α * . For any b, b ∈ B(λ), we have b = f α (b) ⇐⇒ w • (b ) = e α * (w • (b)) .

Lemma 5 . 2 .

 52 Consider b ∈ B(λ) and a positive root α ∈ W α 1 .

Theorem 5 . 3 .

 53 Assume the root system considered is of classical type. Consider two positive roots α and β in W α 1 and a vertex b in B(λ) such that wt(b), α > 0 and wt(b), β > 0.

Theorem 5 . 5 .( 1 ) 2 )

 5512 Assume the root system considered is of classical type. Consider two positive roots α and β in W α 1 and a vertex b in B(λ) such that wt(b), α ≥ 0 and wt(b), β ≥ 0. Assume also that e α (b) = 0 and e β (b) = 0. If α + β ∈ R + , then e α e β (b) = e β e α (b) = e α+β (b) = 0 . (Assume that α, β = 0, and that there is a positive root γ such that the triple (α, γ, β) belongs to the W -orbit of the triple of simple roots (α 1 , α 2 , α 3 ) in a subsystem of type A 3 . If we also have wt(b), γ > 0, then e α e β (b) = e β e α (b) = 0.

( 1 )( 2 )

 12 For any b such that wt(b), α 1 ≥ 0, wt(b), α 2 ≥ 0, e 1 (b) = 0, and e 2 (b) = 0, we have e 1 e 2 (b) = e 2 e 1 (b) = e α 1 +α 2 (b) = 0 . For any b such that wt(b), α 1 ≥ 0, wt(b), α 2 > 0, wt(b), α 3 ≥ 0, e 1 (b) = 0 and e 3 (b) = 0, we have e 1 e 3 (b) = e 3 e 1 (b) = 0 .

  the unique positive root such that α, γ = γ, β = -1. We get wt(b ), α 1 ≥ 0, wt(b ), α 2 > 0, wt(b ), α 3 ≥ 0, e 1 (b ) = 0 and e 3 (b ) = 0 with b = w(b).

Theorem 5. 8 .( 1 ) 2 )Corollary 5 . 9 .

 81259 For any σ ∈ S n the Kashiwara action T -→ σ(T ) gives an isomorphism of oriented graphs from Tab µ to Tab σ(µ) . (Assume µ 1 > µ 2 . Then the action of the Kashiwara crystal operator f1 yields an embedding of oriented graphs from Tab µ to Tab µ-α 1 . (3) If C(T ) = C(T ) where T and T are two tableaux with the same shape, then T = T . Thus the embedding in (2) is the unique which preserves the shape of the tableaux. Consider a tableau T and a positive root α such that f α (T ) = 0. Then f α (C(T )) = C(f α (T )) = 0. 5

Proposition 5 . 11 .

 511 Let T be a vertex of B(λ) of weight µ and α a positive root such that f α (T ) = 0. Then co(f α (T )) = co(T ) and c(f α (T )) = c(T ) + |α| .

Fix a dominant

  weight λ for a classical Lie algebra. Consider the subgraph of B(λ) consisting of the vertices of dominant weight, and the edges b α f α (b) for which wt(b) wt(f α (b)) is a cocoverin the dominant weight poset. This new colored directed graph on the vertices of B(λ) + will be denoted by B(λ) + . It can also be viewed as a poset (with cocovers given by the above edges), and the weight function is a poset projection to the interval [ 0, λ] in the dominant weight poset. The two points of view will be used interchangeably.

Lemma 6 . 2 . 6 . 1 . 1 . 6 . 3 .

 6261163 fails, for instance, in type A 3 , for λ = (4, 1, 1). • In type A n-1 , we obtain the same results by defining the modified crystal operators based on f n-1 rather than f 1 (see Section 5.1), due to the symmetry of the Dynkin diagram. However, if any other node of the Dynkin diagram is used, the connected components of the corresponding B(λ) do not satisfy the properties in Theorem 6.5, which are needed for the atomic decomposition. An example is λ = (4, 3) in type A 3 . • The same complication arises if we do not remove the modified crystal edges which do not correspond to cocovers in the dominant weight poset. An example is λ = (3, 1, 1) in type C 3 . Lemma 5.2 (1) immediately gives the following result, which is a converse of the fact that every edge in B(λ) + projects to a cocover in the dominant weight poset (by definition). Given a vertex b in B(λ) + of weight µ and a cocover µ µ -α in the dominant weight poset (with α ∈ R + ), we have an edge b α f α (b) in B(λ) + . Type A n-1 . We refer freely to Section 4.Lemma Consider two distinct edges b b and b b in B(λ) + . The vertices b and b have a lower bound in this poset. Proof. Let µ := wt(b), ν := wt(b ), and π := wt(b ). We have cocovers µ ν and µ π, so the interval [ν ∧ π, µ] has one of the structures in Cases A1-A3. Starting from the crystal vertex b, we can apply the f • operators indexed by the labels in the corresponding diagrams, by Lemma 6.2.

Lemma 6 . 4 .

 64 ; indeed, each of the two equalities follows from Theorem 5.3 (1). Consider two distinct edges b b and b b in B(λ) + . The vertices b and b have an upper bound in this poset.

  (i,j-1)s s (j,k) C C . . . b p+2 c q-1 d . . .

  (j,k) C C C C C C . . . ab p-1 c q+2 . . .

0 0 1 1 Figure 1 .Figure 2 .

 112 Figure 1. Proof of Theorem 6.5: C = ∅.

Figure 1 .

 1 Fixing a minimal vertex b in C, we must have distinct edges b b and b b . By Lemma 6.3, b and b have a lower bound b in C. By considering a directed path from b to a minimal vertex, we can see that b or b are in C, which contradicts the minimality of b; see Figure 2. Therefore, C has a minimum b min . The existence of a maximum b max is proved in a completely similar way, by using Lemma 6.4 instead. We then need to show that there are no two vertices of the same weight in a component. Assume for contradiction that b and b are such vertices, with wt(b) = wt(b ) = µ. Then we can find a saturated decreasing chain from µ to 0 in the dominant weight poset. According to Lemma 6.2, by applying to b and b the f • operators corresponding to the labels on the mentioned chain, we obtain directed paths from these vertices to the minimum b min in the considered component. However, using the reverse sequence of labels and starting from b min , it is clearly impossible to reach two different vertices via the e • operators.

Figure 3 .

 3 Figure 3. The modified crystal graph B(λ) + in Example 6.6.

Figure 4 .

 4 Figure 4. The modified crystal graph B(λ) + in Example 6.9.

  3, we have H(α) = {b α, b 2 , . . . , b r }, with B(α, b i ) = {b i } for any i = 2, . . . , r, and B(α, α) :    b α α b 1 in the simply laced cases b α α-α b α α b 1 in types C r and G 2 . Now define a statistic c on H(α) such that c(b α) = 0 and {c(b 2 ), . . . , c(b r )} = {m 2 , . . . , m r }. We can then extend it to B(α) + by setting c(b 1 ) = c(b α) + α, ρ ∨ = |α| , and c(b α) = c(b α) + α -α, ρ ∨ = |α| -|α| .

  Theorem 2.2]. The smallest counterexample we found is in type D 4 , and is given in Example 2.4 below. However, a slight increase in rank corrects this problem, and in fact we will see that this is a general phenomenon. Example 2.4. Consider λ := 2ω 1 + 2ω 2 in type D 4 . For simplicity, we let w abcd := w µ for µ := aω 1 + bω 2 + cω 3 + dω 4 . With this notation, we have: χ λ = w λ + w 4000 + w 1111 + w 2002 + w 0022 + w 2020 + 2w 2100 + w 0200 + + 4w 1011 + 5w 0002 + 5w 0020 + 11w 2000 -3w 0100 + 17w 0000 .

  Types C n and D n . This section refers to the stable ranges in types C n and D n , namely to a corresponding graph/poset B(λ) + . We refer freely to Sections 4.2 and 4.3, and also to the results in Section 6.1. Lemma 6.7. Lemmas 6.3 and 6.4 hold in types C n and D n , in the corresponding stable ranges.

2,1) + w (2,2,2) + w (3,1,1,1) + w (2,2,1,1) .

6.2.

As far as we are aware, such a bijection is not known in types E6, E7, E8, and F4.

In type G2, there are two short positive roots α and β such that α + β is short. Thus the lemma does not hold since α2 is a long root.

Nevertheless, there are tableaux T such that fα(c(T )) = 0 but fα(T ) = 0.

T by changing the leftmost letter 3 in 2 in the second row. Conversely, when m

3 and m

(2) 3 = 0, the tableau T obtained from T by changing the leftmost letter 3 in 2 in the first row (if any) does not satisfy f 2 (T ) = T . Thus, we have e 2 (T ) = 0.

The following lemma is a rephrasing of the statement in Lemma 5.4 [START_REF] Braden | From moment graphs to intersection cohomology[END_REF]. It is an easy consequence of Lemmas 5.12 and 5.13. Lemma 5.15. If the tableau T is such that µ 1 > µ 2 > µ 3 , then we have

Lemma 5.6 (1) is rephrased as follows.

Lemma 5. [START_REF] Lascoux | Crystal graphs and q-analogue of weight multiplicities for the root system An[END_REF]. If the tableau T is such that µ 1 ≥ µ 2 ≥ µ 3 , e 1 (T ) = 0, and e 2 (T ) = 0, then we have e 1 e 2 (T ) = e 2 e 1 (T ) = e α 1 +α 2 (T ) = 0 .

Proof. Let T := e 2 (T ). We have wt(T ), α 2 > 0. Therefore, the action of f 2 on T , which produces T , is described by Lemma 5.13. Thus, we have the following two cases, where we use the notation m

3 (T ). In this case, we have m

3 (T ). In this case, we have m

2 (T ). In the same way as in Case 1, we deduce T := e 1 (T ) = 0.

In both cases, we have wt(T ) = wt(T ) + α 1 + α 2 . Therefore, the hypothesis of Lemma 5.15 is satisfied for T , and the proof is completed by applying this lemma. 5.5. Proof of Lemmas 5.4 (2) and 5.6 [START_REF] Brion | Points entiers dans les polyèdres convexes[END_REF]. The difficulty is that the second parts of Lemmas 5.4 and 5.6 reduce to type A 3 , rather than type A 2 , like the first parts. The action of the modified crystal operator f 3 can also be described in the same spirit as in Lemma 5.13, but this requires the enumeration of numerous configurations. Fortunately, in order to prove Lemmas 5.4 (2) and 5.6 [START_REF] Brion | Points entiers dans les polyèdres convexes[END_REF], this can be avoided by using the "canonical" cyclage on semistandard tableaux. Lemma 5.4 (2) is now rephrased as follows.

Lemma 5.17. If the tableau T is such that µ 1 > µ 2 and µ 3 > µ 4 , then we have

Proof. By Lemma 5.2, we have f 1 f 3 (T ) = 0 and f 3 f 1 (T ) = 0. We argue by induction on the cocharge. When co(T ) = 0, the tableau T has row shape, and the equality is clear since

is the unique row of weight wt(T ) -α 1 -α 3 . Assume that the lemma holds for any tableau with cocharge k -1, and consider T such that co(T ) = k. Set f 1 f 3 (T ) = U and f 3 f 1 (T ) = U . By Theorem 5.8 (2), we get f 1 (C(T )) = C(f 1 (T )). Then f 3 f 1 (C(T )) = f 3 C(f 1 (T )). But the cyclage operation also commutes with f 3 (in fact with any modified crystal operator), because it commutes with the actions of W and f 1 , by Theorem 5.8 (1), [START_REF] Brion | Points entiers dans les polyèdres convexes[END_REF]. So we have f

We obtain similarly the equality f 1 f 3 (C(T )) = C(f 1 f 3 (T )) = C(U ). By our induction hypothesis, we thus deduce that C(U ) = C(U ). But since C(U ) and C(U ) have the same shape (i.e., that of C(T )), Theorem 5.8 (3) implies the desired equality U = U . Lemma 5.6 (2) is rephrased as follows.

Lemma 5.18. If the tableau T is such that µ 1 ≥ µ 2 > µ 3 ≥ µ 4 , e 1 (T ) = 0 and e 3 (T ) = 0, then we have e 1 e 3 (T ) = e 3 e 1 (T ) = 0 .

We first prove a weaker version for two-row tableaux.

Lemma 5.19. Lemma 5.18 is true for any two-row tableau T .

Proof. Observe first we must have µ 3 > 0. Otherwise µ 3 = µ 4 = 0 and we cannot have e 3 (T ) = 0 for this would give a tableau of evaluation (µ 1 , µ 2 , 1, -1). Set T = e 1 (T ). When e 3 (T ) = 0, the tableau T = e 3 (T ) has weight (µ 1 + 1, µ 2 -1, µ 3 + 1, µ 4 -1) and by applying Lemma 5.17 to T we get

which is equivalent to the desired equality.

We claim we cannot have e 3 (T ) = 0. Indeed, assume e 3 (T ) = 0. Recall we have e 3 = s 2 s 3 e 2 s 3 s 2 . We obtain e 2 s 3 s 2 (T ) = 0 and e 2 s 3 s 2 (T ) = 0 because e 3 (T ) = 0. Set U = s 3 s 2 (T ) and U = s 3 s 2 (T ). Then e 2 (U ) = 0 and e 2 (U ) = 0. Since T = f 1 (T ) we can write

We shall need the two integers a = m

3 (s 3 (U )) (otherwise µ 3 ≥ a ≥ µ 2 ). By Lemma 5.12, f ε 1 -ε 3 (s 3 (U )) is thus obtained by changing a letter 1 in a letter 3 in the first row of s 3 (U ). We get U by applying s 3 to f ε 1 -ε 3 (s 3 (U )), that is by changing µ 2 -µ 4 letters 3 in letters 4 in its first row. We see that U is obtained from U by changing a letter 1 in a letter 4 in its first row (up to reordering). Therefore U has no letter 2 in its second row and m 3 (s 3 (U )) = µ 4 + b. By Lemma 5.12, f ε 1 -ε 3 (s 3 (U )) is then obtained from s 3 (U ) by changing a letter 1 in a letter 2 in its first row and a letter 2 in a letter 3 in its second row. We get U by applying s 3 to f ε 1 -ε 3 (s 3 (U )), that is by changing in letters 4 all the letters 3 unpaired with the letters 4 located in the second row. We see that U is obtained from U by changing a letter 1 in a letter 2 in its first row and a letter 2 in a letter 4 in its second row. Thus, there is no letter 3 in the second row of U . Moreover m Thus, it suffices to focus on the three diagrams above. Their distinctive feature is that the weight in the middle is not dominant, but a single pair of consecutive entries is in increasing order. In each of the three cases, we start by applying Theorem 5.5 [START_REF] Brion | Points entiers dans les polyèdres convexes[END_REF] to the diamond at the bottom (the extra condition in the theorem is part of the assumptions corresponding to the mentioned cases). Then we can apply the corresponding e • operator to the determined vertex of nondominant weight, by Lemma 5.2 (2). Finally, starting from the determined top vertex, we can apply the corresponding f • operators along all the remaining edges of the modified diagrams, by Lemma 6.2. The commutativity of the remaining triangles and diamonds is checked by Theorem 5.3 (1); indeed, in each case we verify the hypothesis by inspecting the corresponding diagram. This concludes the proof.

Theorem 6.5. The components of B(λ) + define a t-atomic decomposition. Moreover, these components are isomorphic to intervals of the form [ 0, µ] in the dominant weight poset via the weight projection, and the distinguished vertex h ∈ H(λ) in each of them is chosen to be the corresponding maximum.

Proof. By Lemma 6.2, the weight projection of a lower order ideal determined by a vertex of weight µ is the interval [ 0, µ].

We now turn to Lemma 6.4. Like in the proof of its type A version, we will perform the lift along the edges of slightly modified diagrams (by starting from the bottom) in the cases indicated below, while we use the same diagram as in Section 4.2 in Case C1 (a). The reasoning is completely similar, based on repeatedly applying Theorems 5.5 (2) and 5.3 [START_REF] Braden | From moment graphs to intersection cohomology[END_REF], after carefully verifying their hypotheses each time. Note that the special condition in Theorem 5.5 [START_REF] Brion | Points entiers dans les polyèdres convexes[END_REF] requires n > 4 in type D n , but this is clearly true in the stable range. s s

. . . (a -1)21

. . . Proof. We use the same reasoning as in the first part of the proof of Theorem 6.5 (the one referring to the atomic decomposition, as opposed to the t-atomic decomposition). The proof is based on Lemma 6.7 instead.

Example 6.9. Consider λ = (2, 1, 1) in type C 3 . The modified crystal graph B(λ) + is shown in Figure 4. Its vertices are labeled by Kashiwara-Nakashima tableaux of dominant weight, and its edges are labeled as above. In particular, this graph gives the following atomic decomposition of the character: χ λ = w (2,1,1) + 2w (1,1,0) + w (0,0,0) .

Additional facts and perspectives

7.1. The t-atomic decomposition for the adjoint representation. Let α be the highest root of the Lie algebra g. When the root system of g is simply laced, α is the unique nonzero root which