Gonzalo Canduela-Rodriguez

Maarten K Sabbe

Jean.-François Joly

Marie.-Françoise Reyniers
email: mariefrancoise.reyniers@ugent.be

Guy B Marin

Thermodynamic study of benzene and hydrogen coadsorption on Pd(111)

Supporting information

Thermodynamic calculations: Three different approaches have been used to evaluate surface free energies. The first one is defined as PW91-G, in which the Gibbs free energy for the adsorbate system (G ads ) and bulk (G bulk ) are calculated as shown in eq. ( 4), from the enthalpy and entropy terms, which include vibrational contributions and consider immobile species. In this equation, the enthalpy consists of the sum of electronic energy E el , the zero point vibrational energy (ZPVE), and the thermal correction to the enthalpy U vib : ( ) ( )

vib vib el S T U ZPVE E S T H G ⋅ - + + = ⋅ - = (1)
The molar gas phase Gibbs free energy, G gas , accounts for the same terms as the ones in eq. ( 4), but also considers the free energy contributions from the translational and rotational gas phase modes, U trans+rot and S trans+rot , which are also calculated from statistical thermodynamics. Additionally, the molar gas phase Gibbs free energy includes the term that accounts for the gas phase pressure, in which k B is the Boltzmann constant, T the temperature (K), p i the pressure of the gas phase species i (bar), and p o the standard pressure of 1 bar, as shown in eq. ( 5).

p p T R S S T U U ZPVE E S T H G ln , , ⋅ ⋅ + + ⋅ - + + + = ⋅ - = + + (2)
The unit cell used in the calculation has two surfaces because a vacuum layer is included between periodic images to avoid interactions in the direction perpendicular to the slab. The Pd atoms in the upper surface are relaxed during the geometry optimization ("top surface"), on which the corresponding coverage of species i is adsorbed. However, the Pd atoms that compose the lower ("bottom") surface are constrained to the bulk geometry, and on this surface no species are adsorbed (i.e. mixed slab in Figure 1). The electronic energy obtained from the periodic DFT calculations corresponds to the whole slab, which includes these two surfaces. Therefore, the surface free energy that can be derived from this electronic energy corresponds to an average value of these two surfaces, while only the upper one is of interest.

In order to calculate an approximate value for a completely optimized surface (Γ el,relaxed ), the following was proposed by Chizallet et al. [10] and is done in this work; the average surface energy of the mixed-surface unit cell is counted two times (mixed slab), and the surface energy of a completely non-optimized unit cell (fixed slab) is subtracted, i.e. energy of the unit cell with all atoms in the bulk structure and without adsorbate (see eq. ( 6) and Figure 1). Substituting eq. ( 4), eq. ( 5) and eq. ( 6) into eq. ( 3), the surface free energy in case1: PW91-G is obtained as shown in eq. ( 7). The adsorbate, bulk and gas phase contributions are multiplied by two because of the definition of the electronic energy term in eq. ( 6). Furthermore, the Γ (T,p) can be also written in terms of the contribution from the electronic energy (Γ el,relaxed ), enthalpy (ZPVE + U vib terms) (Γ H ) and entropy (Γ S ), together with the gas pressure term, as shown in eq. ( 8). Mind that the electronic energy is not included in the enthalpy term, and that the temperature is included in the entropy term. These terms are useful for comparing the cases that include or not vibrational contributions by defining the free energy contribution from the thermochemistry, sum of enthalpy and entropy terms or Γ thermochem =Γ H +Γ S . Each enthalpy and entropy term is, in turn, calculated as the difference between the system and bulk + gas phase terms.

A E n E A E n E n E         ⋅ ⋅ - -           ⋅ ⋅ - ⋅ - ⋅ = Γ ∑ 2 2 2 , , , , , , , (3)

Optimized atom Fixed atom

Mixed slab Mixed slab

Fixed slab Relaxed slab

( ) ( ) ( ) ( ) ( ) ( )                 ⋅                 ⋅ ⋅ + + ⋅ - + ⋅ - - ⋅ ⋅ - + ⋅ ⋅ - ⋅ ⋅ - + ⋅ + Γ = Γ ∑ + A p p T R S S T U ZPVE n A S T U ZPVE n A S T U ZPVE i i
( ) ( )                       ⋅ ⋅ ⋅ - Γ + Γ + Γ = =                       ⋅ ⋅ ⋅ - Γ - Γ - Γ + Γ - Γ - Γ + Γ = Γ A p p T R n A p p T R n o i i S H (4)
In literature, a common approach is to approximate the Gibbs free energy to the electronic energy from DFT, thus neglecting vibrational contributions in the enthalpy and entropy, because the calculation of vibrational frequencies is not straightforward for many systems and time consuming. This case is defined in this work as PW91-E el . Although this approach neglects vibrational contributions for gas phase species, it considers free energy contributions from the translational and rotational gas phase modes (H trans+rot and S trans+rot ). The surface free energy as a function of the conditions is calculated in PW91-E el following eq. ( 10), which in turn is also writen, as for PW91-G approach in eq. ( 8), as the sum of the different terms in eq. ( 9).

( ) ( ) [ ] ( ) ( )                       ⋅ ⋅ ⋅ - Γ + Γ + Γ = =                                 ⋅ ⋅ + ⋅ - ⋅ - Γ = Γ ∑ + + A p p T R n A p p T R T S T T H o i i S H

Figure 1 :

 1 Figure 1: Scheme illustrating the calculation of the surface free energy of a fully optimized surface, obtained from the mixed and fixed slabs. To avoid the use of an average surface free energy between an optimized and non-optimized surface, the electronic energy of the regular unit cell (with one optimized and another non-optimized surfaces) is accounted two times, and the electronic energy of a non-optimized unit cell is subtracted.

Figure 2 :

 2 Figure 2: Thermodynamic phase diagram of benzene adsorption at the bridge(30) site on Pd(111) as a function of temperature (K) and benzene pressure (bar) for standard pressure of p o = 1 bar. The electronic energies have been obtained with the PW91 functional, and free energy vibrational contributions are not included.

  Geometrical parameters of benzene hollow-hcp(0) and hydrogen coadsorbed on Pd(111) for hydrogen coverages between 0 (clean slab) and θ H = 1.89. Average C-C bond length (pm), angle of C-H bond relative to the C atom at the opposite side of the ring (º), distance between the benzene centre of mass and the Pd top layer (pm), between the two first Pd layers (pm), and finally average bond distance (pm) between all hydrogen atoms, considering that fcc sites form three bonds with the Pd atoms

Table S1 : Adsorption energies in kJ mol -1 of hydrogen adsorbed at different coverages and sites up to saturation coverage. The values obtained with the vdW-DF functional for the hollow fcc site are shown between brackets.

 S1

	Results
						Site			0.25 ML			ΔE ads (kJ mol H 0.5 ML 0.75 ML -1 )	1 ML
						fcc			-54.2 (-43.4) -51.5 (-40.6) -49.5 (-38.3) -47.9 (-36.6)
						hcp			-49.3			-46.6			-44.5	-42.9
						bridge			-39.2			-35.9			-32.6	-29.1
						top			-1.6			1.1			3.5			7.0
						octasub		-17.0			-15.6			-15.2	-14.5
						tetrasub31		-17.6			-17.6			-18.4	-19.6
						tetrasub13		-13.5			-13.6			-13.8	-14.4
	T	,	p		el	,	relax	2	i	trans	rot	gas	,	i	trans 2	rot	gas	,	i	ln	gas o	,	i
																							(6)
											ln
			el	,	relax				2		2

Table S2 : Adsorption energy (kJ mol -1 ) per mol of hydrogen for the adsorption of 0.25 ≤ Θ H ≤ 1 hydrogen coverage on a covered slab with Θ H,initial = 1, for different combinations. The values obtained with the vdW-DF functional for the combination of Θ H =1 hydrogen fcc and addition of 0.25 ≤ Θ H ≤ 1 tetrasub13 hydrogen is shown between brackets.

 S2

	Θ H,initial = 1 0.25 ≤ Θ H ≤ 1	ΔE ads (kJ mol H Θ H,total =1.25 Θ H,total =1.5 Θ H,total =1.75 -1 )	Θ H,total =2
		tetrasub31	-38.6	-32.5	-27.6	-24.2
	fcc	octasub tetrasub13	-38.9 -39.8 (-28.3) -34.3 (-22.6) -30.1 (-18.4) -26.8 (-15.1) -32.7 -28.0 -23.3
		hcp	-33.1	-20.0	-0.3	19.1
		octasub	-36.3	-31.7	-28.0	-24.2
	hcp	tetrasub31 tetrasub13	-30.5 -36.1	-23.2 -31.5	-18.0 -27.9	-14.9 -25.2
		fcc	-30.3	-20.0	0.0	19.1
		octasub	5.0	3.2	1.6	0.4
	top	tetrasub31	7.3	6.4	4.7	2.5
		tetrasub13	5.9	4.5	2.7	1.1
		fcc	-18.3	-20.8	-22.8	-23.3
		hcp	-18.4	-21.6	-23.7	-24.2
	octasub	top	-8.4	-4.4	-1.0	0.4
		tetrasub31	-5.0	-2.1	0.9	1.3
		tetrasub13	-6.7	-3.2	-1.8	-0.6
		fcc	-23.2	-25.3	-25.7	-24.2
		hcp	-18.9	-18.5	-17.1	-14.9
	tetrasub31	top	-12.8	-7.6	-2.4	2.7
		octasub	-12.7	-7.3	-1.7	1.3
		tetrasub13	-14.5	-10.4	-6.8	-2.5
		fcc	-20.1	-23.6	-25.8	-26.8
	tetrasub13	hcp	-19.6	-22.6	-24.4	-25.2
		top	-10.1	-5.9	-2.2	1.1

Table S 3: Benzene adsorption energies in kJ mol -1 on a pre-covered surface with hydrogen at surface and/or subsurface sites, and benzene adsorbing on bridge(30) or hollow-hcp(0) sites.

 S

	Table S 4:
			E ads,benzene (kJ/mol)
	Benzene + Surface hydrogen	Benzene + Subsurface hydrogen
	Hydrogen coverage	Bridge(30) Hollow-hcp(0) Hydrogen coverage		Bridge(30) Hollow-hcp(0)
	0	-114.9	-100.0	---		-114.9	-100.0
	0.11 fcc	-106.0	-89.9	0.11 octasub		-105.7	-91.8
	0.11 fcc + 0.11 hcp	-91.9	-78.8	0.22 tetrasub		-100.1	-83.7
	0.22 fcc + 0.11 hcp	-77.0	-70.8	0.22 octasub + 0.11 tetrasub31	-88.9	-76.6
	0.33 fcc + 0.11 hcp	-62.8	-64.4	0.44 tetrasub31		-87.1	-71.1
	0.56 fcc	-42.5	-58.1	0.56 tetrasub31		-71.0	-59.3
	0.67 fcc	-11.5	-27.5	0.67 tetrasub31		-70.7	-56.9
	0.78 fcc	29.9	-15.1	0.78 tetrasub31		-62.1	-54.4
	0.89 fcc	36.4	-13.3	0.89 tetrasub31		-62.0	-51.0
	0.89 fcc+ 0.11 tetrasub31	35.0	-12.5	1 tetrasub31		-61.2	-60.8
		Benzene + Surface (θ surface = 0.89) + Subsurface (0.22 ≤ θ subsurface ≤ 1)
	Hydrogen coverage (θ surface + θ subsurface )	Bridge(30)	Hollow-hcp(0)
	0.89 fcc + 0.22 tetrasub13		33.7			-14.0
	0.89 fcc + 0.33 tetrasub13		34.5			-15.7
	0.89 fcc + 0.44 tetrasub13		27.6			-13.4
	0.89 fcc + 0.56 tetrasub13		30.5			-13.3
	0.89 fcc + 0.67 tetrasub13		32.8			-11.3
	0.89 fcc + 0.78 tetrasub13		22.7			-10.3
	0.89 fcc + 0.89 tetrasub13		24.6			-10.0
	0.89 fcc + 1 tetrasub13		23.8			-11.5


Laurent Manivel
email: manivel@math.cnrs.fr

On the asymptotics of Kronecker coefficients, 2

Keywords: Symmetric group, Kronecker coefficient, stability, Schur-Weyl duality, Borel-Weil theorem, face, facet, simplicial

Kronecker coefficients encode the tensor products of complex irreducible representations of symmetric groups. Their stability properties have been considered recently by several authors (Vallejo, Pak and Panova, Stembridge). In [3] we described a geometric method, based on Schur-Weyl duality, that allows to produce huge series of instances of this phenomenon. In this note we show how to go beyond these so-called additive triples. We show that the set of stable triples defines a union of faces of the moment polytope. Moreover these faces may have different dimensions, and many of them have codimension one.

Introduction

The complex representation theory of symmetric groups is well understood: the irreducible representations, usually called Specht modules, are indexed by partitions and their dimensions are given by the famous hook length formula. But the multiplicative structure of the representation ring has always remained elusive. The multiplicities in tensor products of Specht modules are called Kronecker coefficients. They are poorly undestood and notoriously hard to compute. We refer to the introduction to [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF] for a discussion of some of the most basic questions about Kronecker coefficients whose answers remain out of reach.

That Kronecker coefficients enjoy certain stability properties has been observed by Murnaghan in 1938 [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]. Such properties are extremely surprising in that they involve representations of differents groups, but they become less mysterious once translated in terms of representations of general linear groups, thanks to Schur-Weyl duality. More stability phenomena have been discovered during the last twenty years, and the wealth of examples we are now aware of makes more urgent the need to understand and organize them better. That is one of the goals of this note.

We use the following terminology, taken from [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF] and [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF]. We denote by [λ] the Specht module associated with the partition λ. This is an irreducible representation of the symmetric group S n , if λ is a partition of n. Kronecker coefficients are defined by the identity

[λ] ⊗ [µ] = ⊕ ν g(λ, µ, ν)[ν].
They are symmetic in λ, µ, ν and of course, non negative.

Definition. A triple of partitions (λ, µ, ν) is weakly stable if the Kronecker coefficients g(kλ, kµ, kν) = 1 ∀k ≥ 1.

It is stable if g(λ, µ, ν) = 0 and for any triple (α, β, γ), the sequence of Kronecker coefficients g(α + kλ, β + kµ, γ + kν) is bounded, or equivalently, eventually constant. We call the asymptotic value of this coefficient a stable Kronecker coefficient.

Stability implies weak stabilty. The converse implication is not known. Conjecturally, the two notions should be equivalent.

In order to get nice finiteness properties we restrict to partitions whose length, rather that size, are bounded; the length ℓ(λ) of a partition λ is the number of non zero parts. We would then like to understand stability phenomena in relation with the Kronecker semigroup and the Kronecker polytohedron. The former is Kron a,b,c := {(λ, µ, ν), ℓ(λ) ≤ a, ℓ(µ) ≤ b, ℓ(ν) ≤ c, g(λ, µ, ν) = 0}. This is a finitely generated semigroup. A more precise version of the semigroup property is the elementary, but useful monotonicity property: if g(λ, µ, ν) = 0, then for any triple (α, β, γ), g(α + λ, β + µ, γ + ν) ≥ g(α, β, γ).

The semigroup Kron a,b,c lives inside a codimension two sublattice of Z a+b+c , because of the obvious condition |λ| = |µ| = |ν| for a Kronecker coefficient g(λ, µ, ν) to be non zero. We call this lattice the weight lattice. The cone generated by Kron a,b,c is a rational polyhedral cone P Kron a,b,c , that we call the Kronecker polyhedron. it is defined by some finite list of linear inequalities, giving the equations of its facets (the maximal faces, of codimension one). The number of facets is huge already for small values of the parameters, and certainly grows exponentially with a, b, c (see [START_REF] Klyachko | Quantum Marginal problem and representations of the symmetric group[END_REF] and [START_REF] Vergne | Moment cones of representations[END_REF]).

In [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF][START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF], we showed that certain minimal faces of the Kronecker polyhedron are made of stable triples. These minimal faces were defined in terms of certain standard tableaux with the additivity property. Let us suppose for simplicity that c = ab (this is not a restriction, since it is well known that for a Kronecker coefficient g(λ, µ, ν) to be non zero, the condition ℓ(λ) ≤ ℓ(µ)ℓ(ν) on the lengths is required). Consider a standard tableau T of rectangular shape a × b. Such a tableau is additive if there exist increasing sequences

x 1 < • • • < x a and y 1 < • • • < y b of real (or rational) numbers such that T (i, j) < T (k, l) ⇐⇒ x i + y j < x k + y l .
The main stability result in [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF] was the following:

Proposition 1 Let T be any additive standard tableau of rectangular shape a × b. For any partition λ = (λ 1 , . . . , λ ab ), define two partitions a T (λ), b T (λ) by

a T (λ) i = b j=1 λ T (i,j) , b T (λ) j = a i=1 λ T (i,j) . Then (λ, a T (λ), b T (λ)) is a stable triple.
Moreover the set of these additive triples, for a fixed T , is exactly the set of lattice points inside a minimal face f T of P Kron a,b,ab defined by this standard tableau.

We want to stress here that the fact that stable triples can be related to faces of the Kronecker polyhedron is by no means a surprise. A general statement is the following. More generally, one can associate to any face of the Kronecker polyhedron a positive integer, which gives the order of growth of the stretched Kronecker coefficients on the interior of the face. This will be discussed in the last section of this paper.

Concentrating on (weakly) stable triples, it is natural to try to describe the faces of P Kron a,b,c that are maximal in SKron a,b,c . We will call the faces of P Kron a,b,c whose intersection with the weight lattice are contained in SKron a,b,c the stable faces, and those that are maximal in SKron a,b,c , the maximal stable faces (which we don't expect a priori to be maximal faces in P Kron a,b,c , or facets). Among many other questions, we can ask: what is the maximal dimension of such a face? what can be their dimensions? could they all be of the same dimension? can the additive stable faces be maximal in SKron a,b,c ? more generally, what are the stable faces containing a given additive stable face?

The main goal of this note is to answer some of these questions, in particular the last one, and draw some unexpected consequences. In [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF] we explained how to describe the local structure of the Kronecker polyhedron around an additive face. Among the faces that contain such an additive face, we will distinguish those that have a property that we will call strong simpliciality. We will prove:

Theorem 1 Among the faces of P Kron a,b,c that contain an additive face, the stable ones are exactly those that are strongly simplicial.

A priori, we would have expected the stable faces to be very special, in particular to have high codimension. Surprisingly, our Theorem has the following consequence: Corollary 1 The polyhedral cone P Kron a,b,c always contains stable facets. This means that there exist families of stable triples of the largest possible dimension. It would be extremely interesting to have a full classification. We can give many explicit examples of strongly simplicial facets and show that there always exist many of them (Proposition 4). We can also describe their structure, which is that of a cone over hypercube (Proposition 3). The vertices of this hypercube are in bijection with the additive faces contained in the facet.

Another striking phenomenon is the following. Consider an additive face, and the maximal stable faces that contain it. It may very well happen that these maximal faces have different dimensions! In fact it seems quite plausible that the maximal stable faces can have all the possible dimensions between the smallest and maximal possible dimensions. In particular the set of stable triples seems to have a very intricate structure in general.

Strongly simplicial faces 2.1 The geometric method

Let us briefly recall the main features of the geometric method used in [START_REF] Manivel | Applications de Gauss et pléthysme[END_REF][START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF] in order to approach Kronecker coefficients. Let A, B be complex vector spaces of finite dimensions a, b. By Schur-Weyl duality, Kronecker coefficients are the multiplicities of the Schur powers S λ (A ⊗ B), when decomposed into irreducible representations for GL(A) × GL(B). By the Borel-Weil theorem,

S λ (A ⊗ B) = H 0 (F l(A ⊗ B), L λ )
for a suitable linearized line bundle L λ on the variety F l(A ⊗ B) of complete flags in A ⊗ B. A standard tableau T defines an embedding

ι T : F l(A) × F l(B) ֒→ F l(A ⊗ B).
The induced map on equivariant Picard groups

ι * T : P ic(F l(A ⊗ B)) ≃ Z ab → P ic(F l(A) × F l(B)) ≃ Z a × Z b is precisely our map λ → (a T (λ), b T (λ))
when expressed in natural basis. In particular, restriction gives a nonzero map

H 0 (F l(A ⊗ B), L λ ) -→ H 0 (F l(A) × F l(B), L a T (λ) ⊗ L b T (λ) ) = S a T (λ) A ⊗ S b T (λ) B, (1)
implying that the Kronecker coefficient g(λ, a T (λ), b T (λ)) is positive. Then we can define a filtration of H 0 (F l(A ⊗ B), L λ ) by the order of vanishing on F l(A) × F l(B). This allows to define an injective map

H 0 (F l(A ⊗ B), L λ ) ֒→ H 0 (F l(A) × F l(B), L a T (λ) ⊗ L b T (λ) ⊗ S * N * ), (2)
where N denotes the normal bundle of the embedding ι T , and S * N * is the symmetric algebra of the dual bundle, the conormal bundle. This map must be thought of as taking a section of L λ , to its Taylor expansion in the normal directions to F l(A) × F l(B). Moreover, if λ is strictly decreasing, the line bundle L λ is very ample. By the usual properties of ample bundles, the previous map becomes surjective onto every finite part of S * N * if L λ is sufficiently ample (that is, if the differences λ i -λ i+1 are large enough). This shows that the multiplicities in S λ (A ⊗ B) = H 0 (F l(A ⊗ B), L λ ), otherwise said, the Kronecker coefficients, are somehow controlled by the normal bundle. This works particularly well when the embedding ι T is convex, in the sense that the weights of the normal bundle are contained in a strictly convex cone. Combinatorially, this exactly means that the tableau T is additive. Then the Kronecker coefficient g(α

+ kλ, β + ka T (λ), γ + kb T (λ)) is bounded by the multiplicity of L β-a T (α) ⊗ L γ-b T (α) inside S * N * ,
and the latter is finite by convexity. This implies that we can focus on a finite part of this algebra, independantly of k. But then, if λ is strict and k is large enough, the surjectivity of (2) in finite degrees implies that we have equality between the latter multiplicity, and the Kronecker coefficient. In particular this coefficient does not depend on k, when big enough: this is the stability phenomenon. But of course we get much more information since we are in principle able to compute the stable Kronecker coefficients, directly from the normal bundle.

Combinatorially, the weights of the conormal bundle are determined as follows. Denote by e 1 , . . . , e a and f 1 , . . . , f b basis of the character lattices of maximal tori in GL(A) and GL(B).

If T (i, j) = k (ie the box (i, j) is numbered k in T ), let g k = e i + f j .
Then the weights of the normal bundle are the differences g ℓ -g k for ℓ > k. Among those weights, the horizontal and vertical ones are those of the form e p -e q and f p -f q . They will appear repeatedly, in fact in the conormal bundle their multiplicities are b -1 for the horizontal ones, a -1 for the vertical ones. In particular these multiplicities will be bigger than one as soon as we suppose that a, b > 2. All the other weights have multiplicity one. Of course, the multiplicity of L β-a T (α) ⊗ L γ-b T (α) inside S * N * , which gives the stable Kronecker coefficient, can be obtained as the number of ways to express the weight (β -a T (α), γ -b T (α)) as a non negative linear combination of the weights of the conormal bundle, considered with their multiplicities.

Of course this is possible only when (β -a T (α), γ -b T (α)) belongs to the cone generated by those weights, which we call the conormal cone. This cone gives a local picture of the Kronecker polyhedron locally around f T . In particular any face of the latter containing f T , can be identified with a face of the conormal cone, and conversely.

Perturbations of additive triples

Now consider a triple of the form (λ, a T (λ) + σ, b T (λ) + θ), where σ and θ are not necessarily partitions, but sequences (or weights) such that a T (λ) + σ and b T (λ) + θ are partitions.

By the injectivity of (2), the Kronecker coefficient g(kλ, k(a T (λ)+σ), k(b T (λ)+θ)) is bounded by the multiplicity of (kσ, kθ) as a weight of S * N * . If we suppose that the line generated by (σ, θ) belongs to the conormal cone, this multiplicity will eventually become positive, and we expect it to grow to infinity with k. But this is not necessarily the case: the multiplicity will remain bounded if (σ, θ) belongs to a face of the cone which is strongly simplicial. By this we mean that the weights of the conormal bundle contained in the face, considered with their multiplicities, define a basis of the linear space generated by the face. Then the multiplicity will be 0 or 1, the second possibility occuring exactly when (kσ, kθ) belongs to the lattice generated by the latter weights.

Let us insist on the definition of strongly simplicial faces.

Definition. A face F of the Kronecker polytope is strongly simplicial if it contains an additive face f T such that locally around f T , the face F corresponds to a face in the cone generated by the conormal bundle which is strongly simplicial in the sense that:

1. it is a face of dimension d generated by d vectors g k 1 +1 -g k 1 , . . . , g k d +1 -g k d ,
2. none of these vectors is horizontal (unless b = 2) or vertical (unless a = 2), 3. no other vector of the form g p -g q belongs to the face, 4. in particular the pairs

(k 1 , k 1 + 1), . . . , (k d , k d + 1) do not intersect.
The structure of a strongly simplicial face is not difficult to describe. Recall that an additive tableau T is defined by parameters

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 .
Of course these parameters are not unique. In fact the tableau T really corresponds to a connected component C T of the complement of the collection of hyperplanes defined by the equations x i + y j = x k + y l inside the parameter space.

Locally around the additive face f T , the Kronecker polyhedron is, by hypothesis, the simplicial cone over the vectors g k 1 +1 -g k 1 , . . . , g k d +1 -g k d . Let us choose one of them, say g ks+1 -g ks . Since it is neither horizontal nor vertical, we can exchange the entries k s and k s + 1 in T and obtain another standard tableau T s . We claim that T s is again additive. Indeed, if the entries k s and k s + 1 of T appear in boxes (i, j) and (k, l), the fact that g ks+1 -g ks is an extremal vector of the cone implies that the hyperplane x i + y j = x k + y l is really a facet of C T . Crossing this facet we get into a component corresponding to T s , which is therefore additive.

Iterating the process, we deduce that the 2 d standard tableaux obtained by considering all the possibilities to exchange the entries (k 1 , k 1 + 1) . . . (k d , k d + 1), are all additive. Moreover the Kronecker polyhedron, around each of the corresponding additive faces, is described by the same cone, up to a change of signs for the generators. This implies that our strongly simplicial face is contained in the set of triples

(λ, µ, ν) = (λ, a T (λ), b T (λ)) + d i=1 u s (g ks+1 -g ks ), (3)
with 0 ≤ u s ≤ λ ks -λ ks+1 . Note that the coefficients u 1 , . . . , u d need to be integers. But it is a priori possible that we get a triple of partitions (λ, µ, ν) given by the same expression but with rational coefficients u 1 , . . . , u d , not all integral. In this case, the Kronecker coefficient g(λ, µ, ν) would certainly be zero.

Otherwise said, the identity (3) defines a lattice L F , which could be a proper sublattice of the intersection of the weight lattice with the linear span of F . In this lattice, F is simply defined by the inequalities 0 ≤ u s ≤ λ ks -λ ks+1 for 1 ≤ s ≤ d. Recall that d is the number of generators of the face in the normal directions of an additive face it contains. In particular the codimension of F is the codimension of an additive face (that is, a + b -2) minus d. We get the following description of strongly simplicial faces.

Proposition 3 A strongly simplicial face F of codimension δ in the Kronecker polyhedron is a cone over a hypercube of dimension a + b -2 -δ.

The main result of this paper is the following.

Theorem 2 A strongly simplicial face F of the Kronecker polyhedron is stable. More precisely, any point in F is a stable triple if it belongs to L F , and the corresponding Kronecker coefficient is zero otherwise.

Proof. Consider a triple of the form (kλ + α, k(a T (λ) + σ) + β, k(b T (λ) + θ)) + γ), where as before (σ, θ) belongs to the simplicial face corresponding to F in the conormal cone of the additive face f T . As we have seen, the corresponding Kronecker coefficient is bounded by the multiplicity of (kσ + β -a T (α), kθ + γ -b T (α)) as a weight of S * N * . Suppose that we have expressed this weight as a non negative integer linear combination t 1 η 1 + • • • + t N η N of the weights η 1 , . . . , η N of the conormal bundle, considered with their multiplicities. Suppose these weights are indexed in such a way that the first d generate our simplicial face. By projection along the direction of this face, we get a relation of the form

p F (β -a T (α), γ -b T (α)) = t d+1 p F (η d+1 ) + • • • + t N p F (η N ), (4)
where p F denotes the projection. But the projected weights p F (η d+1 ), . . . , p F (η N ) generate a strictly convex cone, so the latter equation has only finitely many non negative integer solutions (t d+1 , . . . , t N ). These solutions do not depend on k, and for each of these, the original equation has at most one solution in (t 1 , . . . , t d ), since it can be considered as an equation in the simplicial face F . This proves that the Kronecker coefficient g(kλ

+ α, k(a T (λ) + σ) + β, k(b T (λ) + θ)) + γ), is bounded independently of k.
This is precisely the definition of stability, up to the fact that the Kronecker coefficient g(λ, a T (λ) + σ, b T (λ) + θ) must be equal to one. Recall that by [START_REF] Stembridge | Generalized stability of Kronecker coefficients[END_REF], Propoition 3.2, the only alternative is that it is equal to zero. So what remains to prove is that if (λ, µ, ν) is a point of F that also belongs to the lattice L F , the Kronecker coefficient g(λ, µ, ν) cannot be zero.

To check this we will use that (λ, µ, ν) is given by (3) for some integer coefficients u 1 , . . . , u d such that 0 ≤ u s ≤ λ ks -λ ks+1 for all s. Denote by ω t the partition of t with t parts equal to one. Recall that we denoted by T s the standard tableau obtained by exchanging the entries k s and k s + 1 in T . It is straightforward to check that (ω ks , a Ts (ω ks ), b Ts (ω ks )) -(ω ks , a T (ω ks ), b T (ω ks )) = g ks+1 -g ks .

This allows to rewrite (3) as

(λ, µ, ν) = (θ, a T (θ), b T (θ)) + d s=1 u s (ω ks , a Ts (ω ks ), b Ts (ω ks )),
where θ = λ -d s=1 u s ω ks . Since u s ≤ λ ks -λ ks+1 for all s, this θ is again a partition. Since T and the T s are all additive, we know that g(θ, a T (θ), b T (θ)) = 1 and g(ω ks , a Ts (ω ks ), b Ts (ω ks )) = 1 for all s. In particular all these Kronecker coefficients are non zero, and from the semigroup property we deduce that g(λ, µ, ν) is positive. 2 Corollary 2 A strongly simplicial face is the non negative integral span of the additive faces it contains. Moreover any additive face is properly contained in some strongly simplicial face.

Proof. The first statement means that any stable triple in F can be obtained as a linear combination with positive integer coefficients, of some stable triples in the additive faces contained in F . This is what we established in the proof of the Theorem.

For the second statement, simply observe that at least one face of the Kronecker polyhedron, that contains f T and has dimension one more, must be simplicial. Indeed, these faces correspond to the minimal generators of the conormal cone, and they are simplicial exactly for those generators that are neither horizontal nor vertical. But the generators cannot be all horizontal or vertical, since otherwise inside T , the integer k + 1 would always be South-East of k, which is absurd. 2

Remarks. One can wonder if there can be non trivial arithmetic conditions on the strongly simplicial faces, for the Kronecker coefficients to be non zero? This would mean that L F is really a proper sublattice of the intersection of F with the weight lattice. This seems a priori possible but we have no example of such a phenomenon.

One can also wonder if stable Kronecker coeficients, when one considers strongly simplicial faces, count points in some polytopes, as they do on additive faces [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF]. In the proof above we indeed bounded the stretched Kronecker coefficients by numbers of points in some polytopes, but it is not clear that this bound coincides with the stable Kronecker coefficient. In the additive case this follows from an ampleness argument, which does not apply in this more general situation.

Strongly simplicial facets

In [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF] we gave a combinatorial description of the facets of the Kronecker polytope containing a given additive face f T . These facets are in bijection with the maximal relaxations compatible with T , where a maximal relaxation R is given by an additive (non standard) tableau defined by sequences x 1 ≤ • • • ≤ x a and y 1 ≤ • • • ≤ y b such that the sums R(i, j) = x i +y j are not necessarily distinct. What we require is that the set of vectors e i + f j -e k -f l , for R(i, j) = R(k, l), has maximal rank r = a + b -3. Such a family of vectors being given, the sequences x 1 ≤ • • • ≤ x a and y 1 ≤ • • • ≤ y b are uniquely defined up to translation, and multiplication by the same positive number. It is convenient to define the tableau R uniquely by letting x 1 = y 1 = 0, and asking the two sequences to be made of integers, with no common divisor. The compatibility condition with a standard tableau T is that R(i, j) < R(k, l) implies T (i, j) < T (k, l). Otherwise said, R defines a partial order on the boxes in the rectangle a × b, which is refined by the total order defined by T . The equation of the facet F R is then given by a i=1

x i µ i + b j=1 y j ν j = a i=1 b j=1 (x i + y j )λ T (i,j) ,
where T is any standard tableau compatible with R.

Can such a maximal relaxation R define a strongly simplicial facet? This would mean that R is defined by strictly increasing sequences, and that there exists exactly r = a + b -3 values of R appearing twice, the corresponding difference vectors being independent. In terms of the hyperplanes of equations x i + y j -x k -y l = 0, and the arrangement they define in the open cone defined by 0 = x 1 < • • • < x a and 0 = y 1 < • • • < y b , such an R corresponds to a point where exactly r hyperplanes meet transversaly. Recall that this transversality property implies that any of the 2 r standard tableaux T compatible with R is additive.

Another unexpected fact is that in general, there exist surprisingly many strongly simplicial facets! Proposition 4 P Kron(a, b, ab) contains at least a+b-4 b-2 strongly simplicial facets.

Proof. One can construct tableaux defining strongly simplicial facets by a simple induction: suppose that a tableau S defines a simplicial facet for the format a × (b -1). Then we get one for the format a × b by adding a column defined by y b = x a + y b-1 . Of course this also works for rows. So starting from the tableau defining the unique simplicial face in format 2 × 2, we can construct a+b-4 b-2 strongly simplicial facets in format a × b by choosing to apply the previous process on rows or columns successively, in all possible orders. 2

Examples

Let us examine in more detail the low dimension cases.

Example 1. For a = b = 2 there is exactly one additive face (up to symmetry). This additive face is the intersection of two facets, one of which is strongly simplicial. On the additive face we get g((λ 1 , λ 2 , λ 3 , λ 4 ), (λ

1 + λ 2 , λ 3 + λ 4 ), (λ 1 + λ 3 , λ 2 + λ 4 )) = 1,
and for the strongly simplicial facet we get the more general statement that

g((λ 1 , λ 2 , λ 3 , λ 4 ), (µ 1 , µ 2 ), (ν 1 , ν 2 )) = 1 when µ 1 -ν 2 = λ 1 -λ 4 and λ 1 + λ 3 ≤ µ 1 ≤ λ 1 + λ 2 .
Moreover all these triples are stable.

Example 2. For a = b = 3 there exist 42 standard tableaux fitting in a square of size three, among which 36 are additive. The number of maximal relaxations is 17. They are encoded in the following tableaux:

F + 1 0 0 0 0 0 0 1 1 1 F + 2 0 0 0 1 1 1 1 1 1 F + 3 0 0 1 1 1 2 2 2 3 F + 4 0 1 1 1 2 2 2 3 3 F + 5 0 1 2 2 3 4 3 4 5 F 6 0 0 1 0 0 1 1 1 2 F + 7 0 0 1 1 1 2 1 1 2 F 8 0 1 1 1 2 2 1 2 2 F 9 0 1 2 1 2 3 2 3 4 F + 10 0 1 2 1 2 3 3 4 5
and for each tableau F + i there is another one denoted F - i obtained by diagonal symmetry. Recall that additive faces have dimension four. A detailed analysis yields the following result: Proposition 5 For a = b = 3, the maximal strongly simplicial faces are, up to diagonal symmetry:

1. in codimension one, F + 5 and F + 10 , 2. in codimension two,

F + 3 ∩ F + 4 , F + 3 ∩ F 9 , F + 4 ∩ F 9 , 3. in codimension three, F 6 ∩ F + 7 ∩ F 9 and F + 7 ∩ F 8 ∩ F 9 .
Let us describe the sets of triples (λ, µ, ν) on these strongly simplicial faces. We will use the notations λ ij = λ i + λ j and λ ijk = λ i + λ j + λ k . There are no arithmetic constraints on these stongly simplicial faces, so the Kronecker coefficients are always equal to one and all these triples are stable.

F +
Note also that the additive face defined by the standard tableau 1 2 3 4 5 7 6 8 9 is contained in both F + 5 and F + 3 ∩ F + 4 , showing that an additive face can be contained in two maximal strongly simplicial faces of different dimensions! This indicates that the structure of the set of additive triples must be quite intricate in general.

Example 3. For a = b = 4 there are 6660 additive tableaux and 457 maximal relaxations, according to [START_REF] Klyachko | Quantum Marginal problem and representations of the symmetric group[END_REF][START_REF] Vergne | Moment cones of representations[END_REF]. Among these, we know 43 strongly simplicial ones, among which:

Symmetries

There exist two natural involutions on the set of additive tableaux. Recall that an additive tableau can be defined by increasing sequences x 1 < • • • < x a and y 1 < • • • < y b such that the sums x i + y j are distinct. Then we can replace each of this sequence by the opposite one, reordered increasingly. Since this preserves the set of hyperplanes of equations x i + y j = x k + y l , this defines two commuting involutions on the set of additive tableaux, and then also on the set of maximal relaxations, and on the subset of simplicial relaxations.

Non simplicial faces 3.1 The degree of a face

Recall that a stretched Kronecker coefficient g(kλ, kµ, kν) is quasipolynomial: there exists a collection of polynomials P 0 , . . . , P p-1 , such that g(kλ, kµ, kν) = P i (k) for k = i (mod p).

By the monotonicity property, P i+j (k + ℓ) ≥ P i (k) as soon as P j (ℓ) = 0. This implies that among the polynomials P 0 , . . . , P p-1 , those that are not identically zero have the same degree d, and the same leading term as well. We call d = d(λ, µ, ν) the degree of the triple (λ, µ, ν). For example weakly stable triples have degree zero, and a triple of degree zero is one that has a weakly stable multiple.

Another straightforward consequence of the monotonicity property, and of the convexity of the faces, is the following statement.

Proposition 6 Let F be a face of the Kronecker polyhedron. The degree is constant on the interior of F , and can only decrease, or remain the same, on its boundary faces.

Definition. Let F be a face of the Kronecker polyhedron. We define its degree as the degree of its interior points. For example, any additive face, more generally any strongly simplicial face, has degree zero.

The defect of simpliciality

For a non simplicial face, containing an additive face, we will show that the degree can be read off directly on the normal bundle.

Definition. A face F of the Kronecker polyhedron will be called δ-simplicial if there exists an additive face f T in F , such that the face f of the conormal cone corresponding to F is δsimplicial. By this we mean that f has dimension d, but contains d + δ weights of the conormal bundle, counted with their multiplicity.

Strongly simplicial is therefore the same as 0-simplicial. Note also that starting from a face F of the Kronecker polyhedron, the integer δ will not depend on the minimal face f T contained in F . This is a consequence of the following statement: Theorem 3 A δ-simplicial face F of the Kronecker polyhedron has degree δ.

Proof. We consider F with the additive face f T , and we identify F with the corresponding face of the conormal cone. We consider stretched Kronecker coefficients g(kλ, k(a T (λ)+σ), k(b T (λ)+θ)), where the weight (σ, θ) belongs to the linear span of the face. Denote this Kronecker coefficient by g k . It may a priori happen that the lattice generated by the weights of the conormal bundle belonging to the face does not contain (σ, θ). In general there exists a minimal integer e, depending on (σ, θ), such that e(σ, θ) belongs to this lattice. If k is not divisible by e, then k(σ, θ) does not belong to the lattice and g k = 0. If we restrict to those k that are divisible by e, then the number of ways to express k(σ, θ) as a non negative integer linear combination of weights of the conormal bundle certainly grows like k δ . By the injectivity of (2), this implies that the growth of g k is at most in k δ .

To get to the required conclusion, we must control the surjectivity of (2). The key point is the following general statement.

Lemma 1 Let L be an ample line bundle, M a globally generated line bundle on a smooth complex projective variety X. Let ι : Y ֒→ X be the embedding of a smooth subvariety, and denote by N the normal bundle. Then there exists integers m and n, not depending on M , such that the natural map If we apply this statement to ι T , we deduce that there exists integers m T , n T such that (2) is surjective up to degree d as soon as λ i -λ i+1 ≥ m T d + n T for each i. Replacing λ by kλ we get the surjectivity up to degree (k -n T )/m T . This yields a lower bound for g k of order (k -n T /m T ) δ , and the claim follows. 2

Proof of the Lemma. To get the surjectivity it is enough to prove that

H 1 (X, I d+1 Y ⊗ L a ⊗ M ) = 0.
Let π : Z → X be the blow-up of Y , and E the exceptional divisor. Since π * O Z (-iE) = I i Y and there are no higher direct images, we are reduced to proving that

H 1 (Z, O Z (-(d + 1)E) ⊗ π * (L a ⊗ M )) = 0.
The canonical line bundle of Z is K Z = π * K X ⊗ O Z ((c -1)E), if c denotes the codimension of Y in X. So we can rewrite the previous condition as

H 1 (Z, K Z ⊗ O Z (-(d + c)E) ⊗ π * (L a ⊗ M ⊗ K -1
X )) = 0. We can find an a 0 such that L a 0 ⊗ K -1 X is ample. Moreover the exists b 0 such that I Y ⊗ L b 0 is generated by global sections, hence also

O Z (-E) ⊗ π * L b 0 . Then O Z (-(d + c)E) ⊗ π * (L a ⊗ M ⊗ K -1
X ) is nef and big as soon as a ≥ a 0 + b 0 (d + c), and the required vanishing follows from the Kawamata-Viehweg vanishing theorem. 2

Proposition 2

 2 The set SKron a,b,c of weakly stable triples in Kron a,b,c is the intersection of Kron a,b,c with a union of faces of P Kron a,b,c .

F 6 ∩ F + 7 ∩

 7 F 9 is defined by the three equalities µ 1 + ν 1 = λ 125 + λ 136 , µ 2 = λ 348 , ν 2 = λ 247 and the inequalities λ 126 ≤ µ 1 ≤ λ 125 . F + 7 ∩ F 8 ∩ F 9 is defined by the three equalities µ 1 + ν 1 = λ 124 + λ 135 , µ 2 = λ 356 , ν 2 = λ 267 and the inequalities λ 125 ≤ µ 1 ≤ λ 124 .

(

  The symbols A, B and so on stand for 10, 11 and so on.) It would be interesting to have the complete list.

H

  0 (X, I d Y ⊗ L a ⊗ M ) -→ H 0 (Y, S d N * ⊗ ι * (L a ⊗ M ))is surjective when a ≥ md + n.

  [START_REF] Murnaghan | On the analysis of the Kronecker product of irreducible representations of S n[END_REF] is defined by the equation 2µ 2 + 3µ 3 + ν 2 + 2ν 3 = λ 2 + 2λ 3 + 2λ 4 + 3λ 5 + 3λ 6 + 4λ 7 + 4λ 8 + 5λ 9 and the inequalitiesλ 124 ≤ µ 1 ≤ λ 123 , λ 123 + λ 146 ≤ µ 1 + ν 1 ≤ λ 123 + λ 145 , λ 12 -λ 79 ≤ µ 1 -ν 3 ≤ λ 12 -λ 89 . F + 10 is defined by the equation µ 2 + 3µ 3 + ν 2 + 2ν 3 = λ 2 + λ 3 + 2λ 4 + 2λ 5 + 3λ 6 + 3λ 7 + 4λ 8 + 5λ 9 and the inequalities λ 13 -λ 89 ≤ ν 1 -µ 3 ≤ λ 12 -λ 89 , λ 569 + λ 789 ≤ µ 3 + ν 3 ≤ λ 469 + λ 789 , λ 789 ≤ µ 3 ≤ λ 689 . F + 3 ∩ F + 4 isdefined by the two equations ν 2 = λ 258 and µ 2 + 2µ 3 + ν 3 = λ 369 + λ 456 + 2λ 789 and the inequalities λ 124 ≤ µ 1 ≤ λ 123 , λ 789 ≤ µ 3 ≤ λ 689 . F + 3 ∩ F 5 is defined by the two equations ν 1 = λ 136 and µ 1 -µ 3 + ν 2 = λ 124 + λ 258 -λ 689 and the inequalities λ 125 ≤ µ 1 ≤ λ 124 , λ 689 ≤ µ 3 ≤ λ 679 . F + 4 ∩ F 5 is defined by the two equations ν 3 = λ 479 and µ 1 -µ 3 -ν 2 = λ 124 -λ 258 -λ 689 and the inequalities λ 134 ≤ µ 1 ≤ λ 124 , λ 689 ≤ µ 3 ≤ λ 589 .

Acknowledgements. This paper was 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. We also thank Mateusz Michalek for his help regarding the combinatorics of simplicial facets. In particular Proposition 6 is due to him.


Fabrice Legeai

Thomas Derrien

Valentin Wucher

David Audrey

Gaël Le Trionnaire

Denis Tagu

Long non--coding RNA in the pea aphid; identification and comparative expression in sexual and asexual embryos

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Long non coding RNA (lncRNA) have been shown to be involved in multiple mechanisms in particular in epigenetic regulation such as control of DNA methylation patterns or chromatin remodeling.

Aphids have the ability to produce extremely divergent phenotypes in response to environmental stimuli. Reproductive modes are condition-dependent in aphids since they are able to switch from asexual to sexual reproduction in response to autumnal photoperiod shortening. This means that aphids produce embryos developing into either sexual adults or asexual parthenogenetic females. Our group already deciphered the transcriptomic and post-transcriptomic regulations associated with this process and as a general understanding of epigenetic regulatory mechanisms potentially mediating phenotypic plasticity, we investigated the role of lncRNA in sexual and asexual embryos of the pea aphid.

We established a new bioinformatics pipeline for the detection of lncRNAs from RNA-Seq data, and produced the first pan-genomics catalogue of aphids lncRNAs.

Among 2,172 identified lncRNAs, 24 are differentially expressed between sexual and asexual embryos at different stages. Furthermore, we asserted for each lncRNA a classification of putative cis-interactions based on its genomic distance to neighboring mRNAs. Besides the identification of microRNAs and their putative targets, these results allow the constitution of a broad gene regulation network of the aphid phenotypic plasticity at the embryo level. This workflow is available in Galaxy on the BioInformatics Platform for Arthopods of Agroecosystems (www.inra.fr/bipaa) and can be applied to any organism for which an annotated genome sequence and RNA-Seq data are provided.