ALD ÉRIC JOULIN Michel Bonnefont

Yutao Ma

A NOTE ON SPECTRAL GAP AND WEIGHTED POINCAR É INEQUALITIES FOR SOME ONE-DIMENSIONAL DIFFUSIONS

Keywords: 2010 Mathematics Subject Classification. 60J60, 39B62, 37A30 Spectral gap, Diffusion operator, Weighted Poincaré inequality

We present some classical and weighted Poincaré inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors for a class of spherically symmetric probability measures in dimension larger than 2. Our strategy is based on two main ingredients: on the one hand, the optimal constant in the desired weighted Poincaré inequality has to be rewritten as the spectral gap of a convenient Markovian diffusion operator, and on the other hand we use a recent result given by the two first authors, which allows to estimate precisely this spectral gap. In particular we are able to capture its exact value for some examples.

Introduction

Let M be a smooth connected Riemannian manifold which is complete for the associated Riemannian metric g. Denote µ the probability measure on M with density (with respect to the volume element on M) proportional to e -V , where V is some nice potential, and let σ be some smooth function satisfying convenient assumptions. Let C ∞ (M) be the space of real-valued smooth functions on M and let C ∞ 0 (M) be the subspace of C ∞ (M) consisting of compactly supported functions. We say that µ satisfies a weighted Poincaré inequality with constant λ > 0 and weight σ 2 if for any f ∈ C ∞ 0 (M), the following inequality holds:

λVar µ (f ) ≤ M σ 2 |∇ M f | 2 dµ, (1.1)
where ∇ M is the Riemannian gradient and |•| is the norm with respect to the metric g. Here Var µ (f ) is the variance of f with respect to µ, that is

Var µ (f ) := M f 2 dµ - M f dµ 2 .
Such a functional inequality reduces to the classical Poincaré inequality when there is no weight or, in other words, σ ≡ 1. Recently, the question whether a probability measure satisfies such a functional inequality has attracted a lot of attention, cf. for instance the papers [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF][START_REF] Miclo | Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite?[END_REF][START_REF] Bobkov | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF][START_REF] Cattiaux | Functional inequalities for heavy tailed distributions and application to isoperimetry[END_REF][START_REF] Gozlan | Poincaré inequalities and dimension free concentration of measure[END_REF][START_REF] Bonnefont | Intertwining relations for one-dimensional diffusions and application to functional inequalities[END_REF][START_REF] Bonnefont | Spectral gap for spherically symmetric log-concave probability measures, and beyond[END_REF]. Among the potential applications of these Poincaré type inequalities, one of the most relevant and interesting features is given in terms of measure concentration, large deviations and tail estimates of Lipschitz functionals. The interested reader is referred to the set of notes of Ledoux [START_REF] Ledoux | Concentration of measure and logarithmic Sobolev inequalities[END_REF][START_REF] Ledoux | Spectral gap, logarithmic Sobolev constant, and geometric bounds[END_REF] for an introduction on the rich theory of functional inequalities and their basic (and less basic) properties, and also to the book of Chen [START_REF] Chen | Eigenvalues, inequalities, and ergodic theory[END_REF] in which, among other situations, the one-dimensional case is carefully emphasized. An alternative probabilistic point of view on these functional inequalities consists in regarding the right-hand-side of (1.1) as an energy term derived from a convenient (essentially self-adjoint) Markovian generator. More precisely, the measure µ is invariant for many reversible Markovian dynamics and among them, the diffusion operators we can consider are of the following form: for any

f ∈ C ∞ 0 (M), L σ µ f := σ 2 ∆ M f + ∇ M (σ 2 ) -σ 2 ∇ M V ∇ M f
, where ∆ M is the Laplace-Beltrami operator on M and σ 2 is the weight function appearing in the weighted Poincaré inequality, which now rewrites by integration by parts as the inequality

λVar µ (f ) ≤ - M f L σ µ f dµ.
When σ ≡ 1, the operator reduces to the classical dynamics

L µ f := ∆ M f -∇ M V ∇ M f.
Therefore the best constant in the weighted Poincaré inequality (1.1) is nothing but the spectral gap in L 2 (µ) of the operator -L σ µ (or more rigorously the self-adjoint extension of this operator, still denoted -L σ µ in the remainder of the paper). Recall that the spectral gap of the operator -L σ µ is, when it exists, the largest real λ such that its spectrum lies in {0} ∪ [λ, +∞). We denote it λ 1 (-L σ µ ). It is characterized by the variational identity

λ 1 (-L σ µ ) = inf -M f L σ µ f dµ Var µ (f ) : f ∈ H 1 σ (M, µ), f = const , (1.2)
where H 1 σ (M, µ) is the weighted Sobolev space

H 1 σ (M, µ) := f ∈ L 2 (µ) : M σ 2 |∇ M f | 2 dµ < ∞ ,
which is nothing but the domain of the associated Dirichlet form. According to this point of view, the difficulty to establish a weighted Poincaré inequality with weight σ 2 and to estimate the optimal constant is transferred to the study of the spectral gap of the operator -L σ µ which can be more tractable, in particular in the one-dimensional setting, in view of the recent results emphasized in [START_REF] Bonnefont | Intertwining relations for one-dimensional diffusions and application to functional inequalities[END_REF].

From now on we turn to the one-dimensional framework of the real line, for which the diffusion operator is

L σ µ f := σ 2 f ′′ + (σ 2 ) ′ -σ 2 V ′ f ′ .
To deal with an essentially self-adjoint operator, we require the following set of assumptions on σ, which will be satisfied by our examples all along the paper:

• Smoothness: the function σ ∈ C ∞ (R).

• Ellipticity: the function σ is positive on R.

• Completeness: the metric space (R, d σ ) is complete, where d σ denotes the distance d σ (x, y) = y x du/σ(u) . Beyond the famous Muckenhoupt estimates [START_REF] Muckenhoupt | Hardy's inequality with weights[END_REF] which can be limited in some cases of interest, a practical criterion ensuring a spectral gap for the dynamics has been put forward in [START_REF] Bonnefont | Intertwining relations for one-dimensional diffusions and application to functional inequalities[END_REF] by the two first authors. Using the intertwining approach, one of the main results can be reformulated as follows:

λ 1 (-L σ µ ) ≥ sup f ∈IC ∞ (R) inf V σ f , (1.3)
where IC ∞ (R) is the set of functions in C ∞ (R) with positive derivative and

V σ f := (-L σ µ f ) ′ f ′ .
Moreover the equality holds trivially in (1.3) if λ 1 (-L σ µ ) is an eigenvalue. Such a formula, which is reminiscent of the work of Chen and Wang [START_REF] Chen | Estimation of spectral gap for elliptic operators[END_REF] (see also [START_REF] Chen | Analytic proof of dual variational formula for the first eigenvalue in dimension one[END_REF] and [START_REF] Djellout | Lipschitzian norm estimate of one-dimensional Poisson equations and applications[END_REF] through various alternative forms), provides a convenient criterion ensuring a spectral gap for the dynamics, at least when one has the intuition of the shape of the associated eigenfunctions. For instance if one choose the family of functions f ′ ε = e εV /σ 2 indexed by some convenient real number ε, then after some computations one obtains

L σ µ f = -(1 -ε)V ′ e εV and V σ fε = (1 -ε)σ 2 V ′′ + εV ′ 2 , (1.4)
leading to a potentially interesting lower bound on the spectral gap.

Recently, an estimate of the type (1.3) has been successively used by the authors in [START_REF] Bonnefont | Spectral gap for spherically symmetric log-concave probability measures, and beyond[END_REF] to derive classical and weighted Poincaré inequalities for spherically symmetric probability measures (including the log-concave and heavy tailed cases) in dimension larger than 2, after a reduction of the problem to the one-dimensional setting. However the one-dimensional case, for which an additional analysis is required, has not been treated yet and this is what we intend to present in this short note. In particular we focus our attention on the one-dimensional version of the examples studied in [START_REF] Bonnefont | Spectral gap for spherically symmetric log-concave probability measures, and beyond[END_REF].

This work is presented as follows. In Section 2, we investigate a first family of examples which already appears in [START_REF] Bonnefont | Intertwining relations for one-dimensional diffusions and application to functional inequalities[END_REF] (as an illustration of the intertwining techniques for deriving functional inequalities) and for which we establish classical and weighted Poincaré inequalities. The measure µ is the so-called exponential power distribution of parameter α ∈ [1, ∞), that is to say, the potential V is convex and is given by V (x) = |x| α /α. Using first (1.3) in a convenient way, we are able to derive new estimates for the spectral gap λ 1 (-L µ ) which are different as α ∈ [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF][START_REF] Bobkov | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF] or α > 2, showing the criticality of the Gaussian case α = 2. Next, to illustrate the robustness of formula (1.3), we show in this Gaussian setting how to obtain a family of weighted Poincaré inequalities that refine the classical one, the weight function being σ 2 (x) = 1/(1 + bx 2 ) where b is some positive parameter. In particular we are able to capture the optimal constant λ 1 (-L σ µ ), which is different as b ∈ (0, 1/2) or b ≥ 1/2 but continuous at point b = 1/2. We pursue this approach in Section 3 where we investigate the case of the generalized Cauchy distribution with potential V (x) = β log(1 + x 2 ), β > 1/2. Because of the heavy-tailed phenomenon, such a probability measure does not satisfy the classical Poincaré inequality but a (weaker) weighted Poincaré inequality involving the weight σ 2 (x) = 1 + x 2 . Once again the optimal constant λ 1 (-L σ µ ) is obtained and it reveals to be different as β ∈ (1/2, 3/2] or β > 3/2, with also a continuous transition as point β = 3/2.

Classical and weighted Poincaré inequalities for the exponential power distributions

The first class of examples we consider in this part is of the following form: the potential is V (x) = |x| α /α with α ≥ 1, so that the underlying probability measure, the exponential power distribution of parameter α, has Lebesgue density on R given by

µ(dx) = 1 2α 1/α-1 Γ(1/α) e -|x| α /α dx,
where Γ is the famous Gamma function, Γ(z) = ∞ 0 t z-1 e -t dt, z > 0. Let us start by the case of classical Poincaré inequalities.

Classical Poincaré inequalities.

By the convexity of the potential V , the measure µ satisfies a classical Poincaré or, in other words, the associated classical Markovian operator given for any

f ∈ C ∞ 0 (R) by L µ f (x) = f ′′ (x) -|x| α-2 xf ′ (x),
has a spectral gap. Among the potential cases of interest, some are well-studied in the literature:

• when α = 1, the measure µ is the double-exponential distribution and we have λ 1 (-L µ ) = 1/4; • when α = 2, we consider the standard Gaussian distribution for which we have λ 1 (-L µ ) = 1; • when α = ∞, this is the uniform measure on [-1, 1] and we have λ 1 (-L µ ) = π 2 /4. Actually, it seems difficult to obtain the exact value of the spectral gap beyond these three cases. Below, we propose new estimates which are rather different as α ∈ [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF][START_REF] Bobkov | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF] or α > 2. Such an observation is expected because the Gaussian case α = 2 is critical in terms of convexity of V : the infimum of the second derivative of V is attained at infinity when α ∈ [1, 2) and at the origin when α > 2. In particular, our estimates are sharp as α is close to 1 or 2. However there is still room for a mild improvement in the extremal regime α → ∞ since we do not capture the π 2 /4 (≈ 2.467) by a limiting argument (we only obtain the interval [2, 3] at the limit α → ∞). The result is stated as follows.

Theorem 2.1. If µ stands for the exponential power distribution of parameter α ∈ [1, ∞), then the spectral gap λ 1 (-L µ ) of the classical Markovian operator -L µ satisfies the following estimates:

(i) if 1 ≤ α ≤ 2, we get α 2 4 ≤ λ 1 (-L µ ) ≤ 2 1-2/α ; (ii) if α ≥ 2, then we have 2(1 + α) 1-2/α α ≤ λ 1 (-L µ ) ≤ 3 1-2/α .
Proof. The proof is divided in several parts. Let us first concentrate our attention on the two upper bounds. By the variational formula (1.2), a brief computation with the function f (x) = x entails

λ 1 (-L µ ) ≤ α -2/α Γ(1/α) Γ(3/α) .
Now our objective is to understand the behaviour of the right-hand-side in terms of the parameter α and we thus have to distinguish the two cases:

• On the one hand if α ≥ 2 then our idea is to use the log-convexity of the Gamma function. More precisely, if a, b are two parameters such that a > 0 and b ∈ [0, 1], then the log-convexity of Γ entails the inequality

aΓ(a) = Γ(a + 1) ≤ Γ(a + b) b Γ(a + b + 1) 1-b = (a + b) 1-b Γ(a + b),
which rewrites in a condensed form as

Γ(a) a b Γ(a + b) ≤ a + b a 1-b
.

Using then this estimate with a = 1/α and b = 2/α yields the result.

• On the other hand if α ∈ [1, 2], we proceed differently and use the famous Kershaw inequality [START_REF] Kershaw | Some extensions of W. Gautschi's inequalities for the gamma function[END_REF] controlling the ratio of two Gamma functions: for any (a, b)

∈ (0, ∞) × [1, ∞) such that a ≤ b ≤ a + 1, Γ(a) Γ(b) ≤ a + b -1 2 a-b
.

Then the desired upper estimate holds with the two parameters a = 1 + 1/α and b = 3/α, which verify the required conditions above. The second step of the proof is to establish the lower bound in the case α ∈ [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF][START_REF] Bobkov | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF].

To that aim, we would like to use the inequality (1.3). Note that the measure µ interpolates between the double-exponential and Gaussian distributions. In particular, the spectral gap is attained in the Gaussian case for linear functions whereas for the double-exponential distribution we obtain its exact value by choosing in the variational formula (1.2) the family of functions f a (x) = e a|x| and letting a ↑ 1/2. Hence our idea is to try in (1.3) some functions of the type f ′ ε (x) = e ε|x| α /α , where ε is some constant depending on α and which equals 1/2 when α = 1 and vanishes when α = 2. Note that such a choice of functions f ′ ε is of the form appearing in (1.4) (with σ ≡ 1) and therefore we have 1) , so that the infimum is positive as soon as ε ∈ (0, 1) and thus equals inf

L µ f ε (x) = -(1 -ε)|x| α-2 xe ε|x| α /α , and V fε (x) = (-L µ f ε ) ′ (x) f ′ ε (x) = (1 -ε) (α -1)|x| α-2 + ε|x| 2(α-
V fε = (1 -ε) (α -1) 2 -α 2ε 1-2/α + ε 2 -α 2ε 2-2/α
.

Finally choosing ε = (2 -α)/2, we obtain through (1.3) the announced lower bound

λ 1 (-L µ ) ≥ α 2 /4.
Let us achieve this proof by showing the lower bound when α ≥ 2. As in the previous case, we want to use the estimate (1.3). Since the choice of functions f ′ ε involved in the identity (1.4) does not work (we have no idea of the underlying eigenfunctions), a reasonable strategy is to find a convenient centered function ρ ∈ IC ∞ (R) such that the solution f to the Poisson equation -L µ f = ρ also belongs to the set IC ∞ (R). In other words, the key point is to understand the action of the inverse operator (-L µ ) -1 on a class of smooth increasing functions. Certainly, such an operator is not well-defined on C ∞ (R) but (sometimes) on a convenient subspace of smooth centered Lipschitz functions, the metric under consideration being induced by ρ, cf. [START_REF] Djellout | Lipschitzian norm estimate of one-dimensional Poisson equations and applications[END_REF] for more details. Hence, let us start by the simplest centered function ρ ∈ IC ∞ (R) we may choose, namely the identity function ρ(x) = x. Solving then the Poisson equation (a differential equation of the first order in the derivative f ′ ) yields

f ′ (x) = e |x| α /α ∞ x ye -|y| α /α dy.
Since it belongs to the set IC ∞ (R), it is a good candidate for plugging into the formula (1.3). We have

inf x∈R V f (x) = 1 sup x∈R f ′ (x) = 1 sup x≥0 α 2/α-1 e x ∞
x t 2/α-1 e -t dt , where we used a symmetry argument and a change of variable in the integral to obtain the last equality. Differentiating the denominator with respect to x shows that the supremum over [0, ∞) is attained at the origin since α lies in the region [2, ∞) (it would be infinite otherwise) and we thus get the lower bound

λ 1 (-L µ ) ≥ α 1-2/α Γ(2/α) .
Finally, we apply once again Kershaw's estimate with a = 1 + 1/α and b = 2 to get the required estimate λ 1 (-L µ ) ≥ 2α -1 (1 + α) 1-2/α . The proof is now complete.

2.2.

Weighted Poincaré inequalities for the Gaussian distribution. As mentioned above, we illustrate now how the formula (1.3) might be used to get a weighted Poincaré inequality instead of a classical one, as stated in the last section. For more simplicity, let us focus our attention on the Gaussian setting, that is, the parameter α equals 2, although the forthcoming analysis might be adapted to any α ≥ 1 (the same approach might be applied and one can prove that the weighted Poincaré inequality with (optimal) weight σ 2 (x) = (1 + x 2 ) 1-α holds). The weight function we consider is of the type σ 2 a,b (x) = 1/(a + bx 2 ), with a, b > 0, and the Markovian dynamics is now given by

L σ a,b µ f (x) = σ 2 a,b (x)f ′′ (x) + (σ 2 a,b ) ′ (x) -σ 2 a,b (x)V ′ (x) f ′ (x) = f ′′ (x) a + bx 2 - (2b + a)x + bx 3 (a + bx 2 ) 2 f ′ (x).
We will see later why such a weight is optimal. Since we have clearly λ 1 (-L

σ a,b µ ) = aλ 1 (-L σ 1,b/a µ
), we can assume without loss of generality that the weight is given in the sequel by σ 2 (x) := σ 2 1,b (x) = 1/(1+bx 2 ). Below, we obtain a family (indexed by b) of weighted Poincaré inequalities that strengthen the classical Poincaré inequality for the Gaussian measure. In particular, an interesting feature resides in the continuous transition between two different regimes as b lies in the regions (0, 1/2) or [1/2, ∞), the reason being a lack of integrability in the second case. Such a phenomenon is also shared by an heavy-tailed probability measure such as the generalized Cauchy distribution, as we will observe in Section 3. For the moment, our estimates in the Gaussian setting stand as follows.

Theorem 2.2. If µ denotes the standard Gaussian measure and σ is the weight σ 2 (x) = 1/(1 + bx 2 ), then the spectral gap of the operator -L σ µ is given by

λ 1 (-L σ µ ) = 1 -b if 0 < b < 1/2; 1/4b if b ≥ 1/2.
In other words, the spectral gap given above is the best constant λ > 0 such that we have the following weighted Poincaré inequality: for any f ∈ C ∞ 0 (R),

λVar µ (f ) ≤ R f ′ (x) 2 1 + bx 2 µ(dx).
Proof. If b ∈ (0, 1/2) then with the choice of the centered function f (x) = xe bx 2 /2 , we have L σ µ f = -(1 -b)f and since f is increasing and in L 2 (µ) (because 0 < b < 1/2), the value 1 -b is nothing but the expected spectral gap. Now let us focus our attention on the region b ≥ 1/2 for which the latter argument is no longer available. In contrast to Section 2.1 where there is no weight, the contribution of the weight σ 2 in the generator has to be taken into account and a natural candidate to invoke the formula (1.3) is to pick a family of functions according to the formula (1.4), i.e. of the type f ′ ε (x) = (1 + bx 2 )e εx 2 /2 and then to check the best ε allowing us to use (1.3). We obtain

L σ µ f ε (x) = -(1 -ε)xe εx 2 /2 and V σ fε (x) = (-L σ µ f ε ) ′ (x) f ′ ε (x) = (1 -ε) 1 + εx 2 1 + bx 2 ,
and provided ε ∈ (0, 1) we have

inf V σ fε = ε(1 -ε)/b if 0 < ε ≤ b; (1 -ε) if b ≤ ε < 1.
Finally optimizing in ε ∈ (0, 1) yields the lower bound λ 1 (-L σ µ ) ≥ 1/4b. To establish the upper bound, we choose the function f (x) = xe (1-ε)x 2 /4 with ε ∈ (0, 1) to guarantee the existence of the forthcoming quantities. We have

Var µ (f ) = 1 √ 2π R y 2 e -εy 2 /2 dy = 1 ε 3/2 , and - R f L σ µ f dµ = R σ 2 f ′2 dµ = R 1 + (1 -ε)y 2 2 2 e -εy 2 /2 1 + by 2 dy √ 2π = R (1 -ε) 2 y 4 4 + (1 -ε)y 2 + 1 e -εy 2 /2 1 + by 2 dy √ 2π ≤ (1 -ε) 2 4b R y 2 e -εy 2 /2 dy √ 2π + 1 -ε + b b R e -εy 2 /2 dy √ 2π = (1 -ε) 2 4bε 3/2 + 1 -ε + b b √ ε ,
so that by the variational identity (1.2), we get

λ 1 (-L σ µ ) ≤ -R f L σ µ f dµ Var µ (f ) ≤ (1 -ε) 2 4b + (1 -ε + b)ε b .
Finally letting ε tend to 0 guarantees the desired upper bound. The proof of the equality for b ≥ 1/2 is now achieved.

Note that the lower bound 1/4b in the case b ≥ 1/2 can be derived directly from the case b = 1/2. Indeed using the variational formula (1.2) of the spectral gap shows that we have for any b ≥ 1/2 the comparison λ 1 (-L

σ 1,b µ ) ≥ λ 1 (-L σ 1,1/2 µ
)/2b. As announced at the beginning of this part, the polynomial of degree 2 at the denominator of the weight σ is optimal among all positive polynomials, and also among all powers of x. More precisely, if σ is of the form σ 2 (x) = (1 + x 2 ) -a , where a is some parameter very close to 1 (but greater than 1), then we have λ 1 (-L σ µ ) = 0. Indeed, we have with the functions f ε (x) = e εx 2 /2 , where ε ∈ (0, 1/2),

Var µ (f ε ) = 1 √ 1 -2ε - 1 1 -ε ,
and also

- R f ε L σ µ f ε dµ = 2 2-a ε 2 Γ(3/2 -a) √ 2π(1 -2ε) 3/2-a ,
so that plugging these two quantities into the variational identity (1.2) and taking then the limit as ε ↑ 1/2 yields the conclusion. Before turning to generalized Cauchy distributions in the next section, let us say some words about another functional inequality related to weighted Poincaré inequalities, known as the Brascamp-Lieb inequality [START_REF] Brascamp | On extensions of the Brunn-Minkovski and Prékopa-Leindler theorems, including inequalities for log-concave functions, and with an application to the diffusion equation[END_REF]. For a given measure µ with Lebesgue density proportional to e -V , we assume that the potential V is strictly convex on R and moreover V ′ ∈ L 2 (µ). Such a log-concave framework includes Theorem 3.1. Let µ be the generalized Cauchy distribution of parameter β > 1/2 and let σ be the weight function σ 2 (x) = 1 + x 2 . Then the spectral gap λ 1 (-L σ µ ) of the operator -L σ µ satisfies:

λ 1 (-L σ µ ) = 2(β -1) if β > 3/2; (β -1/2) 2 if 1/2 < β ≤ 3/2.
In other words, the spectral gap given above is the best constant λ > 0 such that we have the following weighted Poincaré inequality: for any f ∈ C ∞ 0 (R),

λVar µ (f ) ≤ R (1 + x 2 )(f ′ (x)) 2 µ(dx).
Proof. As in the Gaussian setting, we start by the case for which the eigenfunction is easily found, that is for β > 3/2. Indeed taking the centered function f (x) = x, we have L σ µ f = -2(β -1)f and since f is increasing and lies in L 2 (µ), the value 2(β -1) is nothing but the spectral gap. When β ∈ (1/2, 3/2], we proceed for the lower bound as in the Gaussian case studied previously and choose according to (1.4) the functions f ′ ε (x) = (1 + x 2 ) ε for some real parameter ε (the present ε corresponds to εβ -1 in the notation of (1.4)). Then we have L σ µ f (x) = -2(β -ε -1)x(1 + x 2 ) ε and also

V fε (x) = (-L σ µ f ε ) ′ (x) f ′ ε (x) = 2(β -ε -1)(1 + (2ε + 1)x 2 ) 1 + x 2 . Now if ε is chosen in the interval (-1/2, β -1) then inf V fε = 2(β -ε -1)(2ε +
1) > 0 and optimizing then in ε (the optimal one is ε = (2β -3)/4 ∈ (-1/2, β -1)) yields through the formula (1.3) the lower estimate λ 1 (-L σ µ ) ≥ (β -1/2) 2 . To obtain the upper bound, take the function f (x) = x(1 + x 2 ) ε with ε < (2β -3)/4. After some careful calculations and using the notation a := β -2ε > 3/2, we have

Var µ (f ) = Γ(a -3/2)Γ(1/2) ZΓ(a -1) 1 - a -3/2 a -1 , and - R f L σ µ f dµ = Γ(a -3/2)Γ(1/2) ZΓ(a -1) (1+2ε) 2 -4ε(1+2ε) a -3/2 a -1 +4ε 2 (a -3/2)(a -1/2) a(a -1) .
Using then the variational formula (1.2) entails Let us conclude this work by two remarks. First, we mention that similarly to the Gaussian case, the weight in the weighted Poincaré inequality is optimal among all powers of x, that is, if the weight function is chosen to be σ 2 (x) = (1+x 2 ) b with some b ∈ (0, 1), then we have λ 1 (-L σ µ ) = 0. Indeed, choose the function f (r) = (1 + r 2 ) ε , with 0 < ε < β/2 -1/4. Then after some tedious computations, we have with the notation a := β -2ε > 1/2, Therefore we obtain by the variational formula (1.2)

λ 1 (-L σ µ ) ≤ 2 (1 + 2ε) 2 (a -1) -4ε(1 + 2ε)(a -3/2) + 4ε 2 (a -3/2)(a -1/2) a = 2 a -1 + 2ε + 3ε 2 a = 2(β -1) + 6ε 2 β -2ε .
λ 1 (-L σ µ ) -1 ≥ Γ(a -1/2) Γ(a) - Γ 2 (β -ε -1/2)Γ(β) Γ(β -1/2)Γ 2 (β -ε) Γ(a + 2 -b) 2ε 2 Γ(a + 1/2 -b)
.

When ε tends to β/2 -1/4, the limit of the right-hand-side of the above inequality is infinite or, in other words, λ 1 (-L σ µ ) = 0. Our second brief observation follows an argument due to Bobkov and Ledoux [START_REF] Bobkov | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF]. Given β > 1/2, we slightly modify the measure µ and replace it by

µ β (dx) = 1 + x 2 2β -1 -β dx Z .
Then it is clear that µ β converges weakly as β tends to infinity to the standard Gaussian measure. Since β has the vocation to be large, we assume β > 3/2 and according to the previous results, we have

λ 1 (-L σ µ β ) = 2(β -1) 2β -1 ,
where σ is the slightly modified weight σ 2 (x) = 1 + x 2 /(2β -1). In other words, we have the optimal weighted Poincaré inequality: for all f ∈ C ∞ 0 (R), 2(β -1) 2β -1

Var

µ β (f ) ≤ R 1 + x 2 2β -1 f ′ (x) 2 µ β (dx).
Passing through the limit as β tends to infinity yields the classical Poincaré inequality for the standard Gaussian distribution, with the optimal constant 1.

Finally letting ε ↑ (2β - 3 ) 2 ,

 32 /4, we getλ 1 (-L σ µ ) ≤ 2(β -1) + 6(2β -3) 2 16(β -2(2β -3)/4) = β -1 2which is the required upper bound. The proof is now complete.

  µ(f ) = Γ(β -ε -1/2)Γ(β) Γ(β -1/2)Γ(β -ε) , µ(f 2 ) = Γ(a -1/2)Γ(β) Γ(β -1/2)Γ(a) ,and-R f L σ µ f dµ = 2ε 2 Γ(a + 1/2 -b)Γ(β) Γ(β -1/2)Γ(a + 2 -b) .

MB is partially supported by the French ANR-12-BS01-0013-02 HAB project. AJ is partially supported by the French ANR-2011-BS01-007-01 GEMECOD and ANR-12-BS01-0019 STAB projects.

YM is partially supported by NSFC 11101040, YETP0264, 985 Projects and the Fundamental Research Funds for the Central Universities.

for instance the case of the exponential power distribution of parameter α, with α in the region [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF][START_REF] Bobkov | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF]. Then the Brascamp-Lieb inequality reads as follows: for any f ∈ C ∞ 0 (R), we have Var µ (f ) ≤ R f ′2 V ′′ dµ, and the constant 1 in front of the variance is optimal. Let us recover briefly this inequality by a simple argument. Note that the Brascamp-Lieb inequality can be rewritten as a weighted Poincaré inequality with the weight function σ 2 = 1/V ′′ . Therefore the best constant in this inequality is nothing but the spectral gap of the Markovian dynamics

Once we have in mind this analogy, we observe that the equality L σ µ V ′ = -V ′ holds, i.e., the function V ′ is always an eigenfunction associated to the eigenvalue 1 (recall that V ′ ∈ L 2 (µ)) and since V ′ is increasing, we deduce that λ 1 (-L σ µ ) = 1. However this Brascamp-Lieb inequality is somewhat limited in the sense that the weight function is fixed in terms of the potential V , excluding de facto a large variety of interesting weighted inequalities (for instance in the Gaussian case we have V ′′ ≡ 1, leading only to the classical Poincaré inequality). Recently, the two first authors obtained a generalization of the Brascamp-Lieb inequality, involving the function V f appearing in the formula (1.3). It is stated as follows. Consider the classical operator

When V is strictly convex, we recover the Brascamp-Lieb inequality above by considering the identity function f (x) = x. However choosing conveniently the function f allows a large variety of weighted Poincaré inequalities. For instance consider in (1.3) the function f ′ = e -W where W has to be carefully chosen. After some brief computations, one has

and rewriting this expression with the new function U := W ′ + V ′ /2, we get a more tractable function

Now the main point is to choose the function U in such way that V f is positive. To that aim, we can consider for instance a polynomial and the easiest form one can think of is U(x) = γx, where γ is some real parameter. The function of interest is then f ′ (x) = e (V (x)-γx 2 )/2 and we get

which can be positive for some given potentials V (provided γ is suitably chosen).

In particular such an argument might be used when the function V ′2 + 2V ′′ is not positive everywhere. In terms of weighted Poincaré inequalities, it leads to the inequality

Coming back to the Gaussian case V (x) = x 2 /2, we have

which is positive for any γ ∈ [0, 1/2]. Therefore we obtain the family (indexed by γ) of inequalities

These inequalities, which reduce as expected to the classical Poincaré inequality when γ = 1/2, enter the general framework of Theorem 2.2 with b = 1/2 -γ ∈ [0, 1/2], for which we recover without any additional effort the optimal constant 1 -b = γ + 1/2.

Weighted Poincaré inequalities for generalized Cauchy distributions

This final part is devoted to study the case of an heavy-tailed distribution, namely the generalized Cauchy distribution of parameter β > 1/2, that is,

where Z is the devoted normalization constant. In the language of the previous sections, the potential V rewrites as V (x) = β log(1 + x 2 ) and therefore is far from being convex at infinity. In particular, the classical Markovian operator L µ f = f ′′ -V ′ f ′ doe not have a spectral gap or, in other words, λ 1 (-L µ ) = 0. In terms of functional inequality, the measure does not satisfy the classical Poincaré inequality, that is, there is no constant λ > 0 such that we have

However, one may hope to get a weighted Poincaré inequality that is weaker than the classical one, and this is indeed the case by choosing the weight function σ 2 (x) = 1 + x 2 , cf. [START_REF] Bobkov | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF][START_REF] Bonnefont | Intertwining relations for one-dimensional diffusions and application to functional inequalities[END_REF]. To that aim, we consider the Markovian dynamics

Below, we give the exact value of the spectral gap λ 1 (-L σ µ ) of the operator -L σ µ . In particular, our results exhibit as in the Gaussian case above a continuous transition at point β = 3/2, which is somewhat expected due to a lack of integrability for β in the region (1/2, 3/2).


Laurent Manivel
email: manivel@math.cnrs.fr

On the asymptotics of Kronecker coefficients

Introduction

Kronecker coefficients are the structure constants for the tensor products of irreducible representations of symmetric groups. Their computation is thus an old and important problem in finite group theory. Through Schur-Weyl duality, they can be interpreted as multiplicities in Schur powers of tensor products and their computation becomes a problem in invariant theory. Applying the Borel-Weil theorem we can see Schur powers of tensor products as spaces of sections of equivariant line bundles on flag varieties, and the problem can be interpreted as a question about Hamiltonian actions on symplectic manifolds. It can also be considered in relation with marginal problems in quantum information theory. Each of these perspectives gives access to interesting pieces of information about Kronecker coefficients, which can be very hard to obtain, or even to guess, from a different perspective. But although we have these several points of views on the Kronecker coefficients, each revealing some part of their properties, they remain quite mysterious, and very basic problems seem completely out of reach at this moment. Let us mention a few important ones.

1. No combinatorial formula is known. We would like the Kronecker coefficients to count some combinatorial objects. Or points in polytopes.

2. A few linear conditions for Kronecker coefficients to be non zero are known. We would like to know all of them. In the terminology of this paper, we would like to know the faces of the Kronecker polyhedra.

3. These polyhedra being described, we would like to know how far we are from being able to decide whether a Kronecker coefficient is zero or not. This is a saturation type problem, related to Mulmuley's conjecture that this decision problem is in P.

4. We would like to understand stretched Kronecker coefficients. They are given by certain quasipolynomials, which we would like to be able to compute efficiently.

The analogue problems for Littlewood-Richardson coefficients have been solved. The Littlewood-Richardson rule gives a combinatorial recipe for their computation, which can be interpreted as a count of integral points in polytopes, once translated in terms of hives or honeycombs, for example. The full list of linear inequalities is known and gives a solution to the famous Horn problem (see eg [START_REF]Eigenvalues, invariant factors, highest weights, and Schubert calculus[END_REF] for a survey). They are enough to decide whether a Littlewood-Richardson coefficients is zero or not, a decision problem which is definitely in P. Finally the stretched versions have been studied and are in fact given by polynomials, not just quasipolynomials [START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF].

This gives some hope for the seemingly similar Kronecker coefficients. There are two different ways to appreciate the connections between these two families of numbers. Littlewood-Richardson coefficients are multiplicities in Schur powers of direct sums rather than tensor products. They are also special Kronecker coefficients: in fact, quite remarkably there is a special facet of the Kronecker polyhedron which is exactly a Littlewood-Richardson polyhedron. The latter is known to be generated by the triple of partitions for which the Littlewood-Richardson coefficient is exactly one, and there is a remarquable equivalence c ν λ,µ = 1 ⇐⇒ c kν kλ,kµ = 1 ∀k ≥ 1.

Such a property will certainly not hold for Kronecker coefficients, but it suggests to introduce the following definition:

Definition. A triple of partitions (α, β, γ) is weakly stable if the Kronecker coefficients g(kα, kβ, kγ) = 1 ∀k ≥ 1.

It is stable if g(α, β, γ) = 0 and for any triple (λ, µ, ν), the sequence of Kronecker coefficients g(λ + kα, µ + kβ, ν + kγ) is bounded (or equivalently, eventually constant).

Stability was introduced in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF] by Stembridge, who proved that it implies weak stability and conjectured that the two notions are in fact equivalent. From the Kronecker polyhedron perspective, stable or weakly stable triples will correspond to very special boundary points.

One of the main objectives of the present paper is to explain how to construct large families of stable triples. Moreover we will be able to describe the Kronecker polyhedron locally around these special points, including by giving explicit equations of the facets they belong to.

Our main tool for studying the Kronecker coefficients will be Taylor expansion. This might look strange at first sight, but recall that using Schur-Weyl duality and the Borel-Weil theorem we can interpret them in terms of spaces of sections of line bundles on flag manifolds. In this context it is very natural to use some basic analytic tools, like Taylor expansion, in order to analyze such sections. We will use these Taylor expansions around certain flag subvarieties, and the fact that these subvarieties allow us to control the Kronecker coefficients will depend on certain combinatorial properties of standard tableaux. The key concept here is that of additivity, studied in detail by Vallejo [START_REF] Vallejo | A stability property for coefficients in Kronecker products of complex S n characters[END_REF][START_REF] Vallejo | Stability of Kronecker coefficients via discrete tomography[END_REF]. In fact this concept already appears in [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF], where very similar ideas are introduced and used to understand certain asymptotic properties of plethysms. Section 3 of [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF] is in fact already devoted to Kronecker coefficients: we explained how the method could be used in this context, what was the role of the additivity property, how we could deduce infinite families of stable triples, and much more. In the present paper we explain our approach in greater detail in the specific context of Kronecker coefficients, and go a bit further than in [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF]. Our results are the following:

1. We show that in a very general setting, stable triples can be obtained through equivariant embeddings of flag manifolds. The asymptotics of the Kronecker coefficients are then governed by the properties of the normal bundle of the embedding, whose weights must be contained in an open half space. When this occurs, a very general stability phenomenon can be observed, and the limit multiplicities can be computed. In particular the local structure of the Kronecker polyhedron can be described (Theorem 2).

2.

The simplest example is a Segre embedding of a product of projective spaces. This takes care of the very first instance of stability, discovered by Murnaghan a very long time ago [START_REF] Murnaghan | The analysis of the Kronecker product of irreducible representations of the symmetric group[END_REF][START_REF] Murnaghan | On the analysis of the Kronecker product of irreducible representations of S n[END_REF], which corresponds to the simplest stable triple (1, 1, 1). This immediately leads to an expression of the limit multiplicities, traditionnally called reduced Kronecker coefficients. Moreover this can readily be generalized: considering the product of a projective space by a Grassmannian we recover a stability property called k-stability by Pak and Panova [START_REF] Pak | Bounds on the Kronecker coefficients[END_REF], and we are able to compute the limit mutiplicities (Proposition 3). Of course this further generalizes to a general product of Grassmannians, which yields a new stability property. In fact, what we show is that (1 ab , a b , b a ) is always a stable triple (Proposition 4). Moreover we can in principle compute the limit multiplicities.

3. Turning to products of flag varieties we explain why the convexity property of the normal bundle of an embedding defined by a standard tableau is in fact equivalent to the property that the tableau is additive (Proposition 6). In fact this is already contained in [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF], even in its version generalized to multitableaux. What we explain in the present paper is how the additive tableaux define minimal regular faces of the Kronecker polyhedron (Proposition 9). Moreover, we describe all the facets containing these minimal faces in terms of special tableaux that we call maximal relaxations (Proposition 10). The corresponding inequalities are completely explicit (Propositions 11 and 12).

4. Each of these special facets consists in stable triples, that we therefore obtain in abundance. Moreover, for each of these triples there is a corresponding notion of reduced Kronecker coefficients. When the stable triple is regular, which is the generic case, we compute this reduced Kronecker coefficient as a number of integral points in an explicit polytope (Proposition 8). As we mentionned above, this is something we would very much like to be able to do for general Kronecker coefficients.

5. The last section of the paper is devoted to certain Kronecker coefficients for partitions of rectangular shape. There are some nice connections with the beautiful theory of θ-groups of Vinberg and Kac (Proposition 13). In particular we prove an identity suggested by Stembridge in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF] by relating it to the affine Dynkin diagram of type E 6 (see Proposition 14, which also contains an interesting identity coming from affine D 4 ).

Acknowledgements. This paper was begun in Berkeley during the semester on Algorithms and Complexity in Algebraic Geometry organized at the Simons Institute for Computing, and completed in Montréal at the Centre de Recherches Mathématiques (Université de Montréal) and the CIRGET (UQAM). The author warmly thanks these institutions for their generous hospitality.

2 From Kronecker to Borel-Weil

Schur-Weyl duality

For any integer n, the irreducible complex representations of the symmetric group S n are naturally parametrized by partitions of n [START_REF] Manivel | Symmetric Functions, Schubert Polynomials and Degeneracy Loci[END_REF]. We denote by [λ] the representation defined by the partition λ of n (we use the notation λ ⊢ n to express that λ is a partition of n, in which case we also say that λ has size n). The Kronecker coefficients can be defined as dimensions of spaces of S n -invariants inside triple tensor products:

g(λ, µ, ν) = dim([λ] ⊗ [µ] ⊗ [ν]) Sn .
Since the irreducible representations of S n are defined over the reals, they are self-dual and therefore, we can also interprete the Kronecker coefficients as multiplicities in tensor products:

[λ] ⊗ [µ] = ν⊢n g(λ, µ, ν)[ν].
Schur-Weyl duality allows to switch from representations of symmetric groups to representations of general linear groups. Recall that irreducible polynomial representations of GL(V ) are parametrized by partitions with at most d = dim V non zero parts. We denote by S λ V the representation defined by the partition λ. Schur-Weyl duality can be stated as an isomorphism

V ⊗n = λ⊢n [λ] ⊗ S λ V between S n × GL(V )-modules.
Here the summation is over all partitions λ of n with at most d non zero parts (the number of non zero parts will be called the length and denoted ℓ(λ)). A straightforward consequence is that

g(λ, µ, ν) = mult(S λ V ⊗ S µ W, S ν (V ⊗ W)),
as soon as the dimensions of V and W are large enough; more precisely, as soons as dim(V ) ≥ ℓ(λ) and dim(W ) ≥ ℓ(µ). Note that consequently, we get the classical result that

g(λ, µ, ν) = 0 if ℓ(ν) > ℓ(λ)ℓ(µ).

The Borel-Weil theorem and its consequences

The GL(U )-representation S λ U is called a Schur module. It is defined more generally for λ an arbitrary non increasing sequence of integers of length equal to the dimension of U . The Borel-Weil theorem asserts that such a Schur module can be realized as a space of sections of a linearized line bundle L λ on the complete flag variety F l(U ). We can therefore understand the Kronecker coefficients as the mutiplicities in the decomposition of

S ν (V ⊗ W ) = H 0 (F l(V ⊗ W ), L ν ) into irreducible GL(V ) × GL(W )-modules.
A direct (well-known) consequence of the Borel-Weil theorem is that the direct sum of all the Schur powers of a given vector space U has a natural algebra structure. This algebra is finitely generated, and therefore if we let U = V ⊗ W , the subalgebra of GL(V ) × GL(W )-covariants is also finitely generated (this is a consequence of the fact that the unipotent radical of a reductive algebraic group is a Grosshans subgroup). This implies the following result. Consider Kron a,b,c := {(λ, µ, ν), ℓ(λ) ≤ a, ℓ(µ) ≤ b, ℓ(ν) ≤ c, g(λ, µ, ν) = 0}.

Proposition 1 (Semigroup property) Kron a,b,c is a finitely generated semigroup.

(The semigroup property is Conjecture 7.1.4 in [START_REF] Klyachko | Quantum Marginal problem and representations of the symmetric group[END_REF], where finite generation is also a conjecture.) Moreover, since the covariant algebra is described in terms of saces of sections of line bundles, it has no zero divisor. This implies the following monotonicity property: Proposition 2 (Monotonicity) If g(λ, µ, ν) = 0, then for any triple (α, β, γ), g(α + λ, β + µ, γ + ν) ≥ g(α, β, γ).

Remark. In fact a stronger property is true: there exists a natural map

([λ] ⊗ [µ] ⊗ [ν]) Sn ⊗ ([α] ⊗ [β] ⊗ [γ]) Sp -→ ([λ + α] ⊗ [µ + β] ⊗ [ν + γ]) S n+p
which is non zero on decomposable tensors. It could be interesting to understand this map better.

The Kronecker polyhedron

The semigroup Kron a,b,c lives inside Z a+b+c , more precisely inside a codimension two sublattice because of the obvious condition |λ| = |µ| = |ν| for a Kronecker coefficient g(λ, µ, ν) to be non zero. Consider the cone generated by Kron a,b,c . The finite generation of the latter implies that this is a rational polyhedral cone P Kron a,b,c , defined by some finite list of linear inequalities.

The interpretation in terms of sections of line bundles on flag manifolds allows to understand this polytope as a moment polytope, and to use the powerful results that has been obtained in this context using the tools of Geometric Invariant Theory, or in an even broader, not necessarily algebraic context, those of Symplectic Geometry.

On the GIT side, there exist general statements that allow, in principle, to determine moment polytopes and in particular the Kronecker polyhedron P Kron a,b,c . For example Klyachko in [START_REF] Klyachko | Quantum Marginal problem and representations of the symmetric group[END_REF] suggested to use the results of Berenstein and Sjamaar, and applied them in small cases. This method has recently been improved in [START_REF] Vergne | Moment cones of representations[END_REF]. This approach has two important limitations. First, it describes the moment polytope by a collection of inequalities which is in general far from being minimal: this means that even if one was able to list these inequalities, many of them would in fact not correspond to any facet of the polytope and would be redundant with the other ones. Second, even if far from being satisfactory, the effective production of this collection of inequalities would require the computation of certain Schubert constants in some complicated homogeneous spaces, which seems combinatorially an extremely hard challenge. More precisely, one would need to decide whether certain Schubert constants are zero or not, which is certainly more accessible than computing them but is certainly far beyond our current level of undertanding.

The first issue has been essentially solved by Ressayre [START_REF] Ressayre | Geometric invariant theory and the generalized eigenvalue problem[END_REF], who devised a way to produce, in principle, a minimal list of inequalities of certain moment polytopes: that is, a list of its facets, or codimension one faces. This approach has been remarkably successful for Littlewood-Richardson coefficients. However, applying them concretely to the Kronecker polyhedron P Kron a,b,c seems completely untractable at the moment.

At least do we know the Kronecker polyhedron P Kron a,b,c for small values of a, b, c. For example ( [START_REF] Klyachko | Quantum Marginal problem and representations of the symmetric group[END_REF][START_REF] Franz | Moment polytopes of projective G-varieties and tensor products of symmetric group representations[END_REF]) P Kron 3,3,3 is defined by the following seven inequallities, and those obtained by permuting the partitions λ, µ, ν:

λ 1 + λ 2 ≤ µ 1 + µ 2 + ν 1 + ν 2 , λ 1 + λ 3 ≤ µ 1 + µ 2 + ν 1 + ν 3 , λ 2 + λ 3 ≤ µ 1 + µ 2 + ν 2 + ν 3 , λ 1 + 2λ 2 ≤ µ 1 + 2µ 2 + ν 1 + 2ν 2 , λ 2 + 2λ 1 ≤ µ 1 + 2µ 2 + ν 2 + 2ν 1 , λ 3 + 2λ 2 ≤ µ 1 + 2µ 2 + ν 3 + 2ν 2 , λ 3 + 2λ 2 ≤ µ 2 + 2µ 1 + ν 2 + 2ν 3 .
Klyachko made extensive computations showing an overwhelming growth of complexity when the parameters increase. For example he claims that P Kron 2,3,6 is defined by 41 inequalities, P Kron 2,4,8 by 234 inequalities and P Kron 3,3,9 by no less than 387 inequalities!

The quasipolynomiality property

The problem of understanding multiplicities in spaces of sections of line bundles is at the core of the study of Hamiltonian actions on symplectic manifolds. The Quantization commutes with Reduction type results have very strong consequences in the context we are interested in. In the algebraic setting, one starts with a smooth projective complex variety M with an action of a reductive group G, and an ample G-linearized line bundle L on M . In this context one considers the virtual G-module RR(M, L) = ⊕ q≥0 (-1) q H q (M, L), whose dimension is given by the Grothendieck-Riemann-Roch formula. It is then a very general statement, due to Meinrenken and Sjamaar [START_REF] Meinrenken | Singular reduction and quantization[END_REF] (see also [START_REF] Vergne | Quantification géométrique et réduction symplectique[END_REF]), that for any dominant weight α, the multiplicity of the irreducible G-module of highest weight kα inside the virtual G-module RR(M, L k ) is given by a quasipolynomial function of k.

Of course we will apply this result to G = GL(U ) × GL(V ) × GL(W ) acting on M = P(U ⊗ V ⊗ W ), and L the hyperplane line bundle on this projective space. Since ample line bundles on projective spaces have no higher cohomology, we are in the favorable situation where

RR(M, L k ) is just the actual G-module H 0 (M, L k ) = S k (U ⊗ V ⊗ W ) * ,
whose multiplicities are precisely the Kronecker coefficients. Applying the full force of Meinrenken and Sjamaar's results, we deduce the following statement:

Theorem 1 The stretched Kronecker coefficient g(kλ, kµ, kν) is a piecewise quasi-polynomial function of (k, λ, µ, ν). More precisely, there is a finite decomposition of the Kronecker polyhedron P Kron a,b,c into closed polyhedral subcones called chambers, and for each chamber C a quasi-polynomial p C (k, λ, µ, ν), such that

g(kλ, kµ, kν) = p C (k, λ, µ, ν) whenever (λ, µ, ν) belongs to C. Corollary 1 For any triple (λ, µ, ν), the stretched Kronecker coefficient g(kλ, kµ, kν) is a quasi- polynomial function of k ≥ 0.
The monotonicity property of Kronecker coefficients easily implies that whenever (λ, µ, ν) is in the interior of the Kronecker polyhedron, the stretched Kronecker coefficient g(kλ, kµ, kν) grows as fast as possible, in the sense that if g(λ, µ, ν) = 0, then g(kλ, kµ, kν) ≃ C(λ, µ, ν)k ngen .

Here we denoted by n gen the generic order of growth, which is also the generic dimension of the GIT-quotient M//G. For P Kron a,b,c we get the value

n gen = abc -a 2 -b 2 -c 2 + 2.
Moreover the coefficient C(λ, µ, ν) can be expressed as the volume of the so-called reduced space M (λ,µ,ν) (see [START_REF] Vergne | Quantification géométrique et réduction symplectique[END_REF], page 21). This volume function is given by the Duistermaat-Heckman measure, which is piecewise polynomial, and not just quasipolynomial.

Conversely, if g(λ, µ, ν) = 0 and g(kλ, kµ, kν) grows like k n for some n < n gen , then the triple (λ, µ, ν) must belong to the boundary of P Kron a,b,c . The extreme case is when n = 0. As observed in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF], this can happen only if g(kλ, kµ, kν) = 1 for all k ≥ 1. Otherwise said, (λ, µ, ν) is a weakly stable triple.

The quasi-polynomiality property has attracted the attention of several authors. Most of them realized that general arguments based on finite generation of covariant rings implied the eventual quasipolynomiality of the stretched Kronecker coefficients, that is, g(kλ, kµ, kν) is given by some quasipolynomial for large enough k. The much stronger property that the stretched Kronecker coefficients are quasipolynomial right from the beginning seem to be much less accessible using algebraic methods only. It was asked as a question in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF]. It is also discussed in [START_REF] Mulmuley | Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry[END_REF] and some explicit computations appear in [START_REF] Briand | Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes 21[END_REF].

Remark 1. One interesting implication of the quasipolynomiality property is that, knowing the Kronecker coefficients asymptotically, in fact we know them completely. Could this be useful to find triples (λ, µ, ν) in the Kronecker polyhedron, such that g(λ, µ, ν) = 0? The study of such holes seems extremely challenging, but is of the greatest importance for Geometric Complexity Theory [START_REF] Buergisser | An overview of mathematical issues arising in the Geometric Complexity Theory approach to VP vs VNP[END_REF].

Remark 2. We have noticed that although the stretched Kronecker coefficient g(kλ, kµ, kν) is a quasi-polynomial function of k ≥ 0, its highest order term is really polynomial. More generally, examples show that the period of the coefficients seem to increase when one consider terms of lower degrees. It would be interesting to prove an explicit statement of this type.

3 Kronecker coefficients and Taylor expansions

The general setup

In [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF] we suggested a method which easily produces lots of stable triples, and yields much more informations about Kronecker coefficients. The idea is very general: we can study a space of sections of any line bundle L on any smooth irreducible variety X by taking the Taylor expansion of these sections along a smooth subvariety Y . To be more precise, we can define a filtration of H 0 (X, L) by the order of vanishing along Y ; more formally, we let

F i = H 0 (X, I i Y ⊗ L) ⊂ H 0 (X, L),
where I Y is the ideal sheaf of Y . Let ι : Y ֒→ X denote the inclusion map. The quotients

I i Y /I i+1 Y = ι * S i N * , for N = ι * T X/T Y the normal vector bundle of Y in X.
Therefore there are natural injective maps

F i /F i+1 ֒→ H 0 (Y, ι * L ⊗ S i N * ).
In words, this map takes a section of L vanishing to order i on Y , to the degree i part of its Taylor expansion in the directions normal to Y . The injectivity is clear: if the degree i part of the Taylor expansion is zero, then the section vanishes on Y at order i + 1.

In general, it will be quite difficult to determine the image of these maps. But the situation improves dramatically if we suppose that L is a sufficiently large tensor power of some given ample line bundle M on X. Indeed, it follows from the general properties of ample line bundles that for any fixed integer i, the map

F i /F i+1 ֒→ H 0 (Y, ι * M k ⊗ S i N * )
must be surjective for k large enough. (This is a straightforward and very classical consequence of Serre's vanishing theorems for ample sheaves.)

Let us suppose moreover that the whole setting is preserved by the action of some reductive group G. Then our filtration splits as a filtration by G-submodules, and a splitting yields an injection of G-modules

H 0 (X, M k ) ֒→ H 0 (Y, ι * M k ⊗ S * N * ).
In fact this statement certainly holds true without any ampleness assumption: it simply asserts that an algebraic section of a line bundle is completely determined by its full Taylor expansion.

In case M is ample, we have a very important extra information: we know that the right hand side is generated by the left hand side up to any given degree, if k is sufficiently large.

Application to Kronecker coefficients

We want to apply the previous ideas in the following situation. The variety X will be a flag manifold of a tensor product V ⊗ W , not necessarily a complete flag; its type (by which we mean the sequence of dimensions of the subspaces in the flag) will be allowed to vary, so we will just denote it by F l * (V ⊗ W ). The line bundle M will be some L λ . If we need it to be ample the jumps in the partition λ will have to be given exactly by the type of the flag manifold, in which case we will say that λ is strict (one should add: relatively to the flag manifold under consideration). The subvariety Y of X will be a product F l * (U ) × F l * (V ) of flag manifolds of certain types, and we will need to construct the embedding ι. Of course our reductive group will be G = GL(V ) × GL(W ).

We have Y = G/P for some parabolic subgroup P of G. It is well known that the category of G-equivariant vector bundles on Y is equivalent to the category of finite dimensional Pmodules. Such modules can be quite complicated since P is not reductive. Let us consider a Levi decomposition P = LU P , where U P denotes the unipotent radical of P and L is a Levi factor, in particular a reductive subgroup of P . We can consider L-modules as P -modules with trivial action of U P , and conversely. In general, any P -module has a canonical filtration by P -modules, such that the associated graded P -module has a trivial action of U P . We can then consider it as an L-module and decompose it as a sum of irreducible modules, as in the usual theory of representations of reductive groups. Equivalentely, any G-homogeneous vector bundle E as an associated graded homogeneous bundle gr(E), which is a direct sum of irreducible ones. Cohomologically speaking, and even at the level of global sections, E and gr(E) can be quite different. In fact E can in principle be reconstructed from gr(E) through a series of extensions than can mix up the cohomology groups in a complicated fashion. Note however that

H 0 (Y, E) is always a G-submodule of H 0 (Y, gr(E)).
Example. On a complete flag variety, the irreducible bundles are exactly the line bundles. On a variety of incomplete flags F l * (U ) this is no longer true, but we can obtain them as follows. Denote by p : F l(U ) → F l * (U ) the natural projection map (which takes a complete flag and simply forgets some of the subspaces in it). Consider a line bundle L λ on F l(U ). Then the pushforward E λ = p * L λ is an irreducible homogeneous vector bundle on F l * (U ) and they are all obtained in this way. (In particular, if λ is strict then E λ is exactly the line bundle L λ on F l * (U ), for which we use the same notation.) Note that the Borel-Weil theorem extends to all the irreducible bundles:

H 0 (F l * (U ), E λ ) = H 0 (F l(U ), L λ ) = S λ U.
Definition. The embedding ι is stabilizing if the normal bundle has the following convexity property: the highest weights of the positive degree part of the symmetric algebra S * (gr(N * )), considered as an L-module, are contained in some open half space of the weight lattice of L.

The line bundle ι * L λ will be of the form L a(λ) ⊗ L b(λ) for some weights a(λ) and b(λ) depending linearly on λ. More generally, given an irreducible homogeneous vector bundle E α on F l * (V ⊗ W ), we will need to understand its pull-back by ι. The resulting homogeneous bundle will scarcely be completely reducible. The associated graded bundle is of the form

gr(ι * E α ) = (ρ,σ)∈T (α) E ρ ⊗ E σ
for some multiset T (α).

Our main result is the following.

Theorem 2 Suppose that the embedding ι is stabilizing. Let λ be any strict partition. Then 1. The Kronecker coefficient g(kλ, ka(λ), kb(λ)) = 1 for any k ≥ 0.

2. For any triple (α, β, γ) the Kronecker coefficient g(α + kλ, β + ka(λ), γ + kb(λ)) is a non decreasing, bounded function of k, hence eventually constant. Otherwise said, the triple(λ, a(λ), b(λ)) is stable.

3. If moreover λ is ample, then the limit Kronecker coefficient is given by the multiplicity of the weight (β, γ) inside gr(ι * E α ) ⊗ S * (gr(N * )).

4. In particular, if α is strict, this is the multiplicty of (β -a(α), γ -b(α)) inside S * (gr(N * )).

Proof. The Kronecker coefficients g(kλ, ka(λ), kb(λ)) are positive since the restriction map

S kλ (V ⊗ W ) = H 0 (F l * (U ), L k λ ) -→ H 0 (F l * (V ) × F l * (W ), ι * L k λ ) = S ka(λ) V ⊗ S kb(λ) W
is surjective. Indeed this map is non zero since the line bundle L λ is generated by global sections, which means that it has a section that does not vanish at any given point. The surjectivity then follows from Schur's lemma since the right hand side is irreducible. Moreover, we have seen that Taylor expansions induces a G-embedding

S kλ (V ⊗ W ) ֒→ H 0 (F l * (V ) × F l * (W ), ι * L k λ ⊗ S * (N * )).
As we also noticed, replacing N * by gr(N * ) can only result in making the right hand side larger, so we even have an inclusion

S kλ (V ⊗ W ) ֒→ H 0 (F l * (V ) × F l * (W ), ι * L k λ ⊗ S * (gr(N * ))).
By the Borel-Weil theorem, the right hand side is now a sum of irreducible GL(V ) × GL(W )modules whose highest weights are of the form (ka(λ), kb(λ)) plus a highest weight of S * (gr(N * )).

But since these weights are supposed to belong to a strictly convex cone, none can give a positive contribution of highest weight (ka(λ), kb(λ)) except the weight zero in degree zero. This proves that g(kλ, ka(λ), kb(λ)) = 1 for any k ≥ 0. Now consider an arbitrary triple (α, β, γ). By the Borel-Weil theorem again, we have

S α+kλ (V ⊗ W ) = H 0 (F l * (U ), E α ⊗ L k λ ).
This immediately implies that g(α + kλ, β + ka(λ), γ + kb(λ)) is a non decreasing function of k, by considering the map

H 0 (F l * (U ), E α ⊗ L k λ ) ⊗ H 0 (F l * (U ), L λ ) -→ H 0 (F l * (U ), E α ⊗ L k+1 λ )
and its restriction through ι. Moreover, the same approach using Taylor expansions can then be used without much change, in particular we get an injection

S α+kλ (V ⊗ W ) ֒→ H 0 (F l * (V ) × F l * (W ), gr(ι * E α ) ⊗ ι * L k λ ⊗ S * (gr(N * ))).
The same argument as before then implies that g(α+kλ, β+ka(λ)) is bounded by the multiplicity of the representation of highest weight (β, γ) inside gr(ι * E α ) ⊗ S * (gr(N * )), which is finite by the convexity hypothesis. What remains to check is that for k large enough, we have in fact equality. But being finite, the multiplicity (β, γ) inside gr(ι * E α ) ⊗ S * (gr(N * )) only comes from some finite part of the symmetric algebra of the conormal bundle. Then it follows formally from the properties of ample line bundles that if k is large enough, all the maps involved in the Taylor expansions are surjective in boundeed degrees, and it makes no difference to replace all the homogeneous bundles involved by the associated graded bundles. This concludes the proof. 2 In particular, if they are non zero, then they remain nonzero after stretching. Kirillov [START_REF] Kirillov | An invitation to the generalized saturation conjecture[END_REF] and Klyachko [START_REF] Klyachko | Quantum Marginal problem and representations of the symmetric group[END_REF] conjectured in certain special cases that the converse property, that is saturation, should hold. We could ask more generally if our limit (or reduced) Kronecker coefficients should have the saturation property, but we actually have only very limited evidence for that.

Remarks.

1. We have not tried to bound the minimal value of k starting from which the Kronecker coefficient g(α+kλ, β +ka(λ), γ +kb(λ)) does stabilize, but it would in principle be possible to give an effective version of the previous theorem. Indeed an irreducible homogeneous bundle which has non zero sections has no higher cohomology by Bott's theorem, and this would be sufficient to ensure for example that replacing the conormal bundle by its graded associated bundle makes no difference at the level of global sections. Nevertheless the resulting statements would probably be rather heavy, and presumably not even close to be sharp.

2. We have expressed the limit multiplicities in a rather compact form, but we will see that these expressions are in fact very complicated. Moreover they usually involve other Kronecker coefficients, as well as Littlewood-Richardson coefficients and other interesting variants. It seems we are playing with Russian dolls, but with growing complexity when we open a doll to find a smaller, more mysterious one...

Tangent and normal bundles

In order to apply the previous theorem, we need to be able to understand the normal bundle of our embedding ι and the associated graded bundle. The starting point for that is of course to understand the tangent bundle to a flag variety; this is classical and goes as follows.

Suppose we consider a variety F l * (V ) of flags (0

= V 0 ⊂ V 1 ⊂ • • • ⊂ V r-1 ⊂ V r = V )
where V i has dimension d i . Each of these spaces defines a homogeneous vector bundle on F l * (V ), but not irreducible in general. The irreducible homogeneous bundles are obtained by considering the quotient bundles

Q i = V i /V i-1 , for 1 ≤ i ≤ r.
Each of these bundles is irreducible, as well as each of their Schur powers. More generally, each irreducible vector bundle on F l * (V ) is of the form

E λ = S α 1 Q 1 ⊗ • • • ⊗ S αr Q r
for some non increasing sequences (α 1 , . . . , α r ) of relative integers. Each homogeneous vector bundle F can then be constructed from such irreducible bundles by means of suitable extensions; the set of irreducible bundles involved does not depends on the process and their sum is gr(F ). For example we have non trivial exact sequences 0 → V i-1 → V i → Q i → 0, and by induction we deduce that gr(

V i ) = Q 1 ⊕ • • • ⊕ Q i .
The tangent bundle to F l * (V ) at the flag

V • = (0 = V 0 ⊂ V 1 ⊂ • • • ⊂ V r-1 ⊂ V r = V
) can be naturally identified with the quotient of End(V ) by the subspace of endomorphisms preserving V • . Taking the orthogonal with respect to the Killing form we get a natural identification of the cotangent bundle:

T * F l * (V ) ≃ {X ∈ End(V ), X(V i ) ⊂ V i-1 , 1 ≤ i ≤ r}.
From this we can easily deduce the following statement:

1 The associated bundle of the tangent bundle of F l * (V ) is

gr(T F l * (V ) ) = ⊕ 1≤i<j≤r Hom(Q i , Q j ).
In order to describe the normal bundle of an embedding ι : F l * (V )×F l * (W ) ֒→ F L * (V ⊗W ), we will have to take the quotients of two such bundles and it is clear that the result will be quite complicated in general. Rather than writing down general formulas, we will compute these quotients in certain specific situations; mainly, when there are only few terms (typically for Grassmannians, whose tangent bundles are irreducible) or for complete flag varieties (whose irreducible bundles are line bundles, hence much easier to handle).

First examples

We now have all the necessary ingredients in hand in order to apply Theorem 2. What remains to be done is to construct suitable embeddings between flag varieties and determine whether they are stabilizing. We begin with a few simple examples.

Murnaghan's stability

As an appetizer we begin with a well-known instance of stability, first discovered by Murnaghan [START_REF] Murnaghan | The analysis of the Kronecker product of irreducible representations of the symmetric group[END_REF][START_REF] Murnaghan | On the analysis of the Kronecker product of irreducible representations of S n[END_REF]. This is the statement that for any triple of partitions (α, β, γ), the Kronecker coefficient g(α + (k), β + (k), γ + (k)) is eventually constant for large k (a sharp bound on k has been given in [START_REF] Briand | The stability of the Kronecker product of Schur functions[END_REF]). The limit value is called a reduced Kronecker coefficient and is denoted ḡ(α ′ , β ′ , γ ′ ), where α ′ is deduced from α by suppressing the first part.

To interpret this in our setting, consider the Segre embedding ι : P(V ) × P(W ) ֒→ P(V ⊗ W ).

If L denotes the tautological line bundle on projective space, then with some abuse of notations, ι * L = L ⊗ L. From Lemma 1 we easily derive that N = Hom(L, V /L) ⊗ Hom(L, W/L).

In particular this embedding is clearly stabilizing since any component of S * N will have negative degree on both copies of L. This implies Murnaghan's stability right away. Moreover, if Q is the quotient bundle on P(V ⊗ W ), then the associated bundle of

ι * Q is gr(ι * Q) = L ⊗ W/L ⊕ V /L ⊗ L ⊕ V /L ⊗ W/L.
Applying Theorem 2, we deduce that ḡ(α

′ , β ′ , γ ′ ) is the multiplicity of S β ′ A ⊗ S γ ′ B inside S α ′ (A ⊗ B ⊕ A ⊕ B) ⊗ S * (A ⊗ B).
This is equivalent to Lemma 2.1 in [START_REF] Briand | The stability of the Kronecker product of Schur functions[END_REF]. The monotonicity property that is a special case of Corollary 3 has been observed in [START_REF] Christandl | Nonzero Kronecker coefficients and what they tell us about spectra[END_REF].

k-stability

A generalisation of Murnaghan's stability has been considered recently by Vallejo [START_REF] Vallejo | A diagrammatic approach to Kronecker squares[END_REF] and Pak and Panova [START_REF] Pak | Bounds on the Kronecker coefficients[END_REF] This k-stability, as coined by the latter authors, corresponds to the generalized Segre embedding ι :

P(V ) × Gr(k, W ) ֒→ Gr(k, V ⊗ W ),
sending a pair (L, M ) of a one dimensional subspace L ⊂ V and a k dimensional subspace M ⊂ W , to the k dimensional subspace L ⊗ M of V ⊗ W . If L denotes the tautological line bundle on the Grassmann variety, then ι * L = L k ⊗ L. Extending the computation we made for projective spaces we get that grN = Hom(L, V /L) ⊗ End 0 (M ) ⊕ Hom(L, V /L) ⊗ Hom(M, W/M ).

For essentially the same reasons as in the previous case, this embedding is manifestly stabilizing. This implies Theorem 1.1 in [START_REF] Pak | Bounds on the Kronecker coefficients[END_REF] without further ado. Theorem 10.2 in [START_REF] Vallejo | A diagrammatic approach to Kronecker squares[END_REF] gives an effective version.

Moreover we have access to the limit multiplicities, called the k-reduced Kronecker coefficients. For this we need to consider a vector bundle E λ on Gr(k, V ⊗ W ) and pull it back by ι. Since E λ is a Schur power of a the quotient bundle Q and

gr(ι * Q) = L ⊗ W/M ⊕ V /L ⊗ W ⊕ V /L ⊗ W/M,
we get the following result.

Proposition 3 The k-reduced Kronecker coefficient ḡk (λ, µ, ν) is equal to the multiplicity of the product A µ - -⊗ S µ + A + ⊗ S ν -B -⊗ S ν + B + inside S λ (A -⊗ B + ⊕ A + ⊗ B -⊕ A + ⊗ B + ) ⊗ Sym(A * -⊗ A + ⊗ End 0 (B -)) ⊗ Sym(A * -⊗ A + ⊗ B * -⊗ B + ).
For Geometric Complexity Theory the most relevant Kronecker coefficients are those indexed by two partitions with equal rectangular shapes [START_REF] Buergisser | An overview of mathematical issues arising in the Geometric Complexity Theory approach to VP vs VNP[END_REF]. This corresponds to taking λ and ν equal to the empty partition. To get a contribution from the previous formula we must avoid all the terms contributing positively to B + , which kills the second symmetric algebra. The first will contribute through S µ A + ⊗ S µ (End 0 (B -)) (because of the Cauchy formula), and extracting the term with ν -= 0 means that we take the GL(B -)-invariants inside S µ (End 0 (B -)). Since End 0 (B -) is just sl k , we get: Corollary 4 The k-reduced Kronecker coefficient ḡk (0, µ, 0) is equal to the dimension of the GL k -invariant subspace of S µ (sl k ).

An effective version of this result was derived in [15] using a completely different method. It seems interesting to notice that Schur powers of sl(V ) = End 0 (V ) are often involved in the stable Kronecker coefficients, along with the Kronecker coefficients themselves and the Littlewood-Richardson coefficients. These multiplicities are also of interest for themselves; more generally, many interesting phenomena appear when one considers Schur powers of the adjoint representation of a simple complex Lie algebra, see eg [START_REF] Kostant | Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, and the gmodule structure of Λg[END_REF]. Our methods can be applied to the study of the asymptotics of these coefficients; we hope to come back to this question in a future paper.

Grassmannian stability

We can obviously generalize the Segre embedding to any product of Grassmann varieties. For any positive integers a, b consider the natural embedding ι : Gr(a, V ) × Gr(b, W ) ֒→ Gr(ab, V ⊗ W ).

If L denotes the tautological line bundle on the Grassmann variety, then ι * L = L b ⊗ L a . Extending the computation we made for projective spaces we get that grN = End 0 (A)⊗Hom(B, W/B)⊕Hom(A, V /A)⊗End 0 (B)⊕Hom(A, V /A)⊗Hom(B, W/B).

This embedding is again stabilizing and we get right away the following generalization of kstability.

4 Let a, b be positive integers. For any triple (λ, µ, ν), the Kronecker coefficient

g(λ + (t) ab , µ + (at) b , ν + (bt) a )
is a non decreasing, eventually constant, function of t.

We could call the limits (a, b)-reduced Kronecker coefficients, and provide an expression generalizing Theorem 3. We leave this to the interested reader. Note that in this statement (λ, µ, ν) are not just partitions but integer sequences for which the arguments of the Kronecker coefficient are partitions, at least for large enought t. For example λ = (λ + , λ -) where λ + is a non decreasing sequence of length ab (with possibly negative entries), and λ -is a partition (of arbitrary length).

Note that when λ -is the empty partition, drastic simplifications occur. This follows from the usual formula S α+(t

) d V = S α V ⊗ (det V ) t when d is the dimension of V . Since det(A ⊗ B) = (det A) dim B ⊗ (det B) dim A , we deduce that g(λ + (t) ab , µ + (at) b , ν + (bt) a ) = g(λ, µ, ν).
This is Theorem 3.1 in [START_REF] Vallejo | A stability property for coefficients in Kronecker products of complex S n characters[END_REF].

Stability and standard tableaux 5.1 Classification of embeddings

In this section we focus on the equivariant embeddings of complete flag varieties Let T be such a standard tableau of rectangular shape a × b. The corresponding embedding ι T is defined as follows. Let A • and B • be two complete flags in A and B respectively. Choose adapted basis u 1 , . . . , u a of A and v 1 , . . . , v b of B, that is, such that A i = u 1 , . . . , u i and

B j = v 1 , . . . , v j . Then define the complete flag C • in A ⊗ B by C k = u i ⊗ v j , T (i, j) ≤ k .
Here T (i, j) denotes the entry of T in the box (i, j). Clearly this flag does not depend on the adapted basis but only on the flags, and we can let ι

T (A • , B • ) = C • .
Proposition 6 The embedding ι T : F l(A) × F l(B) ֒→ F l(A ⊗ B) is stabilizing if and only if the standard tableau T is additive.

Here we use the terminology of Vallejo, the additivity condition being used in [START_REF] Vallejo | Stability of Kronecker coefficients via discrete tomography[END_REF]. In our paper [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF] no special terminology was introduced for this additivity property, which means the following: there exist increasing sequences

x 1 < • • • < x a and y 1 < • • • < y b such that T (i, j) < T (k, l) ⇐⇒ x i + y j < x k + y l .
Remark. There is a huge number embeddings ι T : recall that by the hook length formula, the number of standard tableaux of shape a × b is ST (a, b) = (ab)!/h(a, b), where h(a, b) = (a + b -1)!!/(a -1)!!(b -1)!!. This grows at least exponentially with a and b. Among these, the proportion of additive tableaux probably tends to zero when a and b grow, but their number should still grow exponentially. Note that each of this additive tableau corresponds to a certain chamber in the complement of the arrangement of hyperplanes H ijkl defined by the equalities x i + y j = x k + y l in R a+b-2 + (we may suppose that x 1 = y 1 = 0). This looks very much like the hyperplance arrangement associated to a root system.

The partitions a T (λ) and b T (λ) such that ι * T L λ = L a T (λ) ⊗ L b T (λ) are easily described; one just needs to read the entries in each line or column of T and sum the corresponding parts of λ:

a T (λ) i = b j=1 λ T (i,j) , b T (λ) j = a i=1
λ T (i,j) .

(T, λ)-reduced Kronecker coefficients

Applying our general statements of Theorem 2, we get:

Proposition 7 Let T be any additive tableau. For any partition λ, the triple (λ, a T (λ), b T (λ)) is stable.

Although this statement does not appear explicitely in [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF], it is discussed p.735, in the paragraph just before the Example. It was recently rediscovered by Vallejo [START_REF] Vallejo | Stability of Kronecker coefficients via discrete tomography[END_REF], inspired by the work of Stembridge [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF], and following a completely different approach. (As a matter of fact there is a slight difference between our definition of additivity and that of Vallejo. When the tableau T is additive, as we intend it, then for any partition λ of length at most ab, the matrix A with entries a ij = λ T (i,j) is additive in the sense of Vallejo, and all such matrices are obtained that way. As we just stated it, this implies that the triple (λ, a T (λ), b T (λ)) is stable. If moreover λ is strict, then we will get more information, in particular we will be able to compute the stable Kronecker coefficients. )

Note that the additivity property is introduced in section 3.1.2 of [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF] and explained to be equivalent to the convexity property of the normal bundle that implies stability. At that time we were mainly interested in plethysm and we treated the case of Kronecker coefficients rather quickly, giving details only for a sample of the results that were amplified in the strongly similar case of plethysm. Also we treated directly the multiKronecker coefficients

g(µ 1 , . . . , µ r ) = dim([µ 1 ] ⊗ • • • ⊗ [µ r ]) Sn ,
for which the method applies with essentially no difference.

The fact that additivity is equivalent to stability is easy to understand. Recall, as a special case of Lemma 1, that the tangent bundle of a complete flag manifold F l(U ) has associated graded bundle

gr(T F l(U ) ) = ⊕ i<j Hom(Q i , Q j ),
where the quotient bundles are now line bundles. Applying this to U = V ⊗ W and pulling-back by ι T we will get a formula in terms of the quotient line bundles on F l(V ) and F l(W ), that we will denote by E i and F j . The formula reads

gr(ι * T T F l(V ⊗W ) ) = ⊕ T (i,j)<T (k,l) Hom(E i ⊗ F j , E k ⊗ F l ).
Let us denote by e 1 , . . . , e a and f 1 , . . . , f b natural basis of the weight lattices of GL(V ) and GL(W ), respectively. The formula shows that the weights of the restricted tangent bundle * T (T F l(V ⊗W ) ) are the e k + f l -e i -f j for T (i, j) < T (k, l). We claim that the weights of the normal bundle are exactly the same. Indeed, since the tableau T is increasing on rows and columns, the weights e k -e i and f l -f j appear for k > i and l > j with multiplicity b and a respectively, in particular greater than one. Since they are the weights of the tangent bundles of F l(V ) and F l(W ), we see that when we go from the restricted tangent bundle to the normal bundle the list of weights will not change, only certain multiplicities will decrease, but remaining positive. In particular the generated cone will not be affected. Finally, recall that the additivity property asks for the existence of sequences x 1 < • • • < x a and y 1 < • • • < y b such that T (i, j) < T (k, l) if and only if x i + y j < x k + y l , otherwise written as x k -x i + y l -y j > 0. This precisely means that the corresponding linear form is positive on each of the weights e k -e i + f l -f j with T (i, j) < T (k, l).

Remark. Of course it is not necessary, in order to determine the cone generated by the weights of gr(N * ), to compile the full list of M = (a -1)(b -1)(ab + a + b)/2 weights. The ab -1 of them obtained by reading the successive entries of the tableau T from 1 to ab will suffice, since all the other weights will obviously be sums of these.

Note that from Proposition 7, we can deduce right away the following special case, which is a generalization of Proposition 4.

Corollary 5

For any partition µ of size m, the triple (1 m , µ, µ * ) is stable.

Proof. Consider a rectangle a × b in which the diagram of µ can be inscribed. Consider µ and the conjugate partition µ * as integer sequences of lengths a and b respectively, by adding zeroes if necessary. Consider the increasing sequences α 1 , . . . , α a and β 1 , . . . , β b defined by α i = i -µ i -1 and β j = j -µ * j . Then α i + β j = -h ij , the opposite of the hook length of µ for the box (i, j). In particular α i + β j is negative exactly on the support of µ. Let T be the corresponding additive tableau, and let λ = 1 m . Then a T (λ) = µ and b T (λ) = µ * . Hence (1 m , µ, µ * ) is stable. 2

Let us denote the value of g(α + kλ, β + ka T (λ), γ + kb T (λ)), for k very large, by g T,λ (α, β, γ), and call it a (T, λ)-reduced Kronecker coefficient. If λ is strictly decreasing of length ab, or ab -1, so that the corresponding line bundle on the flag manifold is very ample, we have seen that this (T, λ)-reduced Kronecker coefficient can be computed as the multiplicity of the weight (β -a T (α), γ -b T (α)) inside the symmetric algebra S * (gr(N * )). The weights of gr(N * ) can readily be read off the tableau T . Let us denote them by (u i , v i ), for 1 ≤ i ≤ M = (a -1)(b -1)(ab + a + b)/2. Denote by P T,λ (µ, ν) the polytope defined as the intersection of the quadrant t 1 , . . . , t M ≥ 0 in R M with the affine linear space defined by the condition that

N i=1 t i (u i , v i ) = (µ, ν).
Proposition 8 The (T, λ)-reduced Kronecker coefficient g T,λ (α, β, γ) is equal to the number of integral points in the polytope P T,λ (β -a T (α), γ -b T (α)).

Of course we would then be tempted to stretch the triple (α, β, γ). We would then obtain the stretched (T, λ)-reduced Kronecker coefficient g T,λ (kα, kβ, kγ) as given by the Ehrhart quasi-polynomial of the rational polytope P T,λ (β -a T (α), γ -b T (α)). This suggests interesting behaviours for multistretched Kronecker coefficients, but we will not pursue on this route.

Let us simply notice an obvious consequence for (T, λ)-reduced Kronecker coefficients, the following translation invariance property:

g T,λ (α, β, γ) = g T,λ (α + δ, β + a T (δ), γ + b T (δ))
for any partition δ.

Faces of the Kronecker polytope

we observed, the convexity property of the embeddings defined by additive tableaux has very interesting consequences for the Kronecker polytope. Let us first reformulate Corollary 2.

Proposition 9 Each additive tableau T defines a regular face f T of the Kronecker polyhedron P Kron a,b,ab , of minimal dimension.

Regular means that the face meets the interior of the Weyl chamber, which is the set of strictly decreasing partitions. Ressayre proved in [START_REF] Ressayre | Geometric invariant theory and the generalized eigenvalue problem[END_REF], in a more general setting, that the maximal codimension of a regular face is the rank of the group which in our case is GL(V ) × GL(W ). This exactly matches with the codimension of f T .

Around this minimal face f T , we know that the local structure of P Kron a,b,ab is described by the convex polyhedron generated by the weights of gr(N * ).

A face of our polytope will be defined by a linear function which is non negative on all these vectors, and vanishes on a subset of them that generate a hyperplane. Such a linear function will be defined by sequences x 1 ≤ • • • ≤ x a and y 1 ≤ • • • ≤ y b such that x i + y j ≤ x k + y l when T (i, j) < T (k, l). The different values of x i + y j define a partition of the rectangle a × b into disjoint regions, and this partition is a relaxation of T , in the sense that each region is numbered by consecutive values of T . The hyperplane condition can be interpreted as the fact that the vectors e k -e i + f l -f j , for (i, j) and (k, l) belonging to the same region, generate a hyperplane in the weight space. This is also a maximality condition: we cannot relax any further while keeping the compatibility condition with T . We deduce: Proposition 10 The facets F R of the Kronecker polyhedral cone Kron a,b,ab containing the minimal face f T are in bijective correspondence with the maximal relaxations R of the tableau T .

Example. Recall that for a = b = 3 there are 42 standard tableaux fitting in a square of size three, among which 36 are additive. The total number of maximal relaxations of these additive tableaux is 17. Up to diagonal symmetry they are as follows, where each square is filled by the entry x i + y j in box (i, j) for some sequences (x 1 = 0, x 2 , x 3 ) and (y 1 = 0, y 2 , y 3 ). 0 0 0 0 0 0 1 1 1

0 0 0 1 1 1 1 1 1 0 0 1 1 1 2 2 2 3 0 1 1 1 2 2 2 3 3 0 1 2 2 3 4 3 4 5 0 0 1 0 0 1 1 1 2 0 0 1 1 1 2 2 2 3 0 1 1 1 2 2 1 2 2 0 1 2 1 2 3 2 3 4
(x i + y j )α T (i,j) ,
T is any standard tableau compatible with R.

We mean that R is a relaxation of T . It is clear then that the inequality does not depend on T , since taking another compatible tableau T ′ amounts to switching some entries only inside the regions defined by R, on each of which the sum x i + y j is constant.

From rectangles to arbitrary tableaux

If we want to restrict to the Kronecker cone Kron a,b,c for some c < ab, we just need to intersect Kron a,b,ab with a linear space L c of codimension ab -c, which meets each of our minimal faces f T . But this raises two issues. First, two distinct minimal faces can have the same intersection with L c . Second, it is not clear whether the intersection of a facet F R of Kron a,b,ab with L c will still be a facet of Kron a,b,c .

The first issue is easy to address: two tableaux T and T ′ will give the same minimal face of Kron a,b,c if and only if they coincide up to c, that is, their entries smaller of equal to c appear in the same boxes. We thus get minimal faces of Kron a,b,c parametrized by standard tableaux S of size c inside the rectangle a × b, which are additive in the same sense as before.

To address the second issue we can modify our embeddings ι T accordingly. A standard tableau S of size c inside the rectangle a × b defines an embedding

ι S : F L(V ) × F l(W ) ֒→ F L c (V ⊗ W ),
where we denote by F L c (U ) the partial flag manifold of U parametrizing flags of the form

(0 = U 0 ⊂ U 1 ⊂ • • • ⊂ U c ⊂ U )
, where U i has dimension i. We will require that S fits exactly in the rectangle, not in any smaller one. Denote the quotient bundles on F l(U ) by Q 1 , . . . , Q c , Q c+1 , they are all line bundles except the last one. We have ι * S Q k = E i ⊗ F j whenever S(i, j) = k ≤ c, while ι * S Q c+1 is not irreducible, but has associated graded bundle

gr(ι * S Q c+1 ) = ⊕ (i,j) / ∈S E i ⊗ F j .
The pull-back of the tangent bundle is then given by the same formula as before,

gr(ι * S T F l(V ⊗W ) ) = ⊕ S(i,j)<S(k,l) Hom(E i ⊗ F j , E k ⊗ F l ),
except that for this to be correct, we need to consider S as a rectangular tableau of size a × b, in which the boxes (i, j) that do not belong to S are all numbered by the same arbitrarily large number, say S(i, j) = ∞. Exactly as before we deduce that the weights of the normal bundle are the e k -e i + f l -f j with S(i, j) < S(k, l), and the convexity condition translates into the same additive property, that we can summarize by saying that S must be a piece of a rectangular additive standard tableau. The discussion above then goes through exactly as in the rectangular case, except that we don't need to care about the boxes of the rectangle that are not supported by S. We get the following slight extension of our previous results:

x i β i + b j=1 y j γ j ≥ (i,j)∈S (x i + y j )α S(i,j) .
Remark. More variants could be explored. F l * (V ) × F l * (W ) ֒→ F l * (V ⊗ W ) where arbitrary types appear at the source are easily constructed in terms of tableaux, and their stability could be analyze. This will be more complicated in general since the quotient bundles will have ranks bigger than one. Moreover it will only give access to stable triples on the boundary of the Weyl chamber.

Of course we could aso readily extend the discussion to an arbitrary number of vector spaces, describe embeddings of arbitrary products of flag manifolds in terms of multidimensional tableaux, observe that the convexity condition on the weights of the normal bundle is again an additivity condition, and deduce stability properties for multiKronecker coefficients. This was partly done in [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF].

Rectangles: stability and beyond

There exist stable triples which do not come from additive tableaux, and it would be nice to understand them. Stembridge in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF] observed that [START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF][START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF][START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF] is stable, and it is certainly not additive. (Nevertheless it is highly degenerate, in the sense that it belongs to a very small face of the dominant Weyl chamber. In particular this example leaves open the question of the existence of non additive stable triples in the interior of the Weyl chamber.) In this section we make a connection with Cayley's hyperdeterminant and the Dynkin diagram D 4 , and we explain another observation by Stembridge in terms of affine E 6 .

Finite cases

The rectangular Kronecker coefficients, ie those involving partitions of rectangular shape, are of special interest because of their direct relation with invariant theory. For three factors, g(p a , q b , r c ) = dim(S k (A ⊗ B ⊗ C)) SL(A)×SL(B)×SL(C) when k = pa = qb = rc and a, b, c are the dimensions of A, B, C. Of course this connection also holds for a larger number of factors.

There seems to exist only few results on these quotients. The cases for which there are only finitely many orbits of GL(A)×GL(B)×GL(C) inside A⊗B ⊗C have been completely classified in connection with Dynkin diagrams(see eg [START_REF] Manivel | Prehomogeneous spaces and projective geometry[END_REF] and references therein). One can deduce all the possible dimensions a, b, c for which there exists a dense orbit, through the combinatorial process called castling transforms. In this situation there exists one invariant for SL(A)×SL(B)×SL(C) at most, depending on the codimension of the complement of the dense open orbit, which can be one (in which case its equation is an invariant) or greater than one (in which case there is no invariant). The former case gives a weakly stable triple.

In this classification through Dynkim diagrams, the triple tensor products correspond to triple nodes, so there are only few cases, coming from diagrams of type D or E. The first interesting case is D 4 , corresponding to a = b = c = 2. The invariant is the famous hyperdeterminant first discovered by Cayley, which has degree four. This implies that g(n 2 , n 2 , n 2 ) = 1 when n is even and g(n 2 , n 2 , n 2 ) = 0 when n is odd. In particular, [START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF][START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF][START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF] is a weakly stable triple, and even a stable triple, as shown by Stembridge [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF].

The next two diagrams, D 5 and D 6 , give (a, b, c) = (2, 2, 3) and (2, 2, 4) respectively. The corresponding invariants have degree 6 and 4. They correspond to the weakly stable triples (33, 33, 222) and [START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF][START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF]1111) (which we already met among stable triples). For D n , n ≥ 7, we get (a, b, c) = (2, 2, n -2) but there is no non trivial invariant anymore. Finally the triple nodes of E 6 , E 7 , E 8 yield the triples (a, b, c) = (2, 3, 3), [START_REF] Briand | Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture, with an appendix by K. Mulmuley[END_REF][START_REF] Briand | The stability of the Kronecker product of Schur functions[END_REF][START_REF] Buergisser | An overview of mathematical issues arising in the Geometric Complexity Theory approach to VP vs VNP[END_REF], [START_REF] Briand | Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture, with an appendix by K. Mulmuley[END_REF][START_REF] Briand | The stability of the Kronecker product of Schur functions[END_REF][START_REF] Christandl | Nonzero Kronecker coefficients and what they tell us about spectra[END_REF]. There is no invariant for the latter case, but an invariant of degree 12 in the two previous ones, yielding the weakly stable triples (66, 444, 444) and (66, 444, 3333). Let us summarize our discussion:

13 The triple nodes of the Dynkin diagrams of types D 4 , D 5 , E 6 , E 7 yield the non additive weakly stable triples [START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF][START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF][START_REF] Rassart | A polynomiality property for Littlewood-Richardson coefficients[END_REF], (33,33,222), (66, 444, 444), (66, 444, 3333).

Affine cases

This discussion can be upgraded from Dynkin to affine Dynkin diagrams. Indeed it is a theorem of Kac [8] that when we consider a representation associated to a node of an affine Dynkin diagram, the invariant algebra is free, or otherwise said, is a polynomial algebra. (If the chosen node is the one that has been attached to the usual Dynkin diagram, the associated representation is just the adjoint one, so the theorem generalizes the well-known result that the invariant algebra of the adjoint representation is free.)

There will be four cases related to Kronecker coefficients, corresponding to the affine Dynkin diagrams with a unique multiple node. Let us introduce the following notation: g D4 (n) = g(n 2 , n 2 , n 2 , n 2 ), g Ê6 (n) = g(n 3 , n 3 , n 3 ), g Ê7 (n) = g((2n) 2 , n 4 , n 4 ), g Ê8 (n) = g((3n) 2 , (2n) 3 , n 6 ). Applying Kac's results we immediately identify the generating series of these rectangular Kronecker coefficients.

Proposition 14

The generating series of the rectangular Kronecker coefficients g D4 (n), g Ê6 (n), g Ê7 (n) and g Ê8 (n) are the following: n≥0 g D4 (n)q n = 1 (1 -q)(1 -q 2 ) 2 (1 -q 3 ) , n≥0 g Ê6 (n)q n = 1 (1 -q 2 )(1 -q 3 )(1 -q 4 ) , n≥0 g Ê7 (n)q n = 1 (1 -q 2 )(1 -q 3 ) , n≥0 g Ê8 (n)q n = 1 1 -q .

The last of these identities simply expresses the fact that (33, 222, 1 6 ) is a weakly stable triple, as we already know. The previous one can be rewritten as g((2n) 2 , n 4 , n 4 ) = n + π 6 (n) 6

where π 6 is the 6-periodic function with first 6 values (6, -1, 4, 3, 2, 1). The identity for g Ê6 (n) has been suggested by Stembridge ([24], Appendix). It can also be rewritten as a quasipolynomial identity: Remark. Let us mention that there should exist lots of non additive stable triples. For example, [15] implies that if λ is a partition of size 2n, then (n 2 , n 2 , λ) is weakly stable when λ is even and of length at most four, as well as (n 4 , (2n) 2 , 2λ) when λ has length at most three and λ 1 ≤ λ 2 + λ 3 . It would be interesting to prove that these weakly stable triples are in fact stable, and to get more examples, or more general procedures to construct (weakly) stable triples. Theorem 6.1 in [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF] gives a criterion for stability that covers the additive triples, but not only those. It would be interesting to understand these non additive triples more explicitely, find a geometric interpretation, compute the stable Kronecker coefficients, and decide to which extent they could help to describe the Kronecker polyhedra.

g(n

Corollary 2 Corollary 3

 23 The set of triples a(λ), b(λ)) is a face of the Kronecker polyhedron. Moreover the local structure of the polyhedron around this face is given by the cone generated by the weights of S * (gr(N * )). Limit Kronecker coefficients have the monotonicity property.

ιProposition 5

 5 : F l(A) × F l(B) ֒→ F l(A ⊗ B), the combinatorics being more transparent in that case. Denote by a and b the respective dimensions of A and B. The following two statements appear in[START_REF] Manivel | Applications de Gauss et pléthysme[END_REF]: The equivariant embeddings ι : F l(A) × F l(B) ֒→ F l(A ⊗ B) are classified by standard tableaux of rectangular shape a × b.

  It splits the square into six regions, three of size one and three of size two. There are therefore eight compatible standard tableaux T , which are all additive. This gives eight minimal faces f T incident to the facet F R .Proposition 11The defining inequalities of the facet F R associated to a maximal relaxation R defined by sequences (x i , y j ) are of the form a i=1 x i β i + b j=1 y j γ j ≥ a i=1 b j=1

Proposition 12

 12 Let S by a standard tableau of height a, width b, size c. Suppose that S is additive. Then the set of stable triples of the form (λ, a S (λ), b S (λ)) defines a minimal regular face f S of the Kronecker polytope P Kron a,b,c . Moreover the facets of the polytope containing this minimal face f S are in bijection with the minimal relaxations R of S. If R is defined by non decreasing sequences (x i , y j ), the equation of the facet F R is a i=1

  π 6 is 6-periodic with (35, -8, 27, 8, 19, 0) and π 2 is 2-periodic with period[START_REF] Murnaghan | The analysis of the Kronecker product of irreducible representations of the symmetric group[END_REF][START_REF] Klyachko | Quantum Marginal problem and representations of the symmetric group[END_REF].

	3 , n 3 , n 3 ) =	(n + 1)(n + 2) 48	+	n + 1 16	π 2 (n) +	1 48	π 12 (n),
	n 3 + 12n 2 + 29n + 18 72	+	n + 1 72	π 2 (n) +	1 72	π 6 (n),

where π 12 is 12-periodic with period (37, -12, 9, 16, 21, -48, 25, 0, 21, 4, 9, 0) and π 2 is 2-periodic with period

[START_REF] Briand | The stability of the Kronecker product of Schur functions[END_REF][START_REF] Briand | Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes 21[END_REF]

. Finally the quadruple Kronecker coefficient g(n 2 , n 2 , n 2 , n 2 ) = where