Maxime Pelletier
email: pelletier@math.univ-lyon1.fr

Production of faces of the Kronecker cone containing only stable triples

One way to study the Kronecker coefficients is to focus on the Kronecker cone, which is generated by the triples of partitions corresponding to non-zero Kronecker coefficients. In this article we are interested in producing particular faces of this cone, formed of stable triples (a notion defined by J. Stembridge in 2014), using many geometric notions -principally those of dominant and well-covering pairsand results of N. Ressayre. This extends a result obtained independently by L. Manivel and E. Vallejo in 2014 or 2015, expressed in terms of additive matrix. To illustrate the fact that it allows to produce quite a few new faces of the Kronecker cone, we give at the end of the article details about what our results yield for "small dimensions".

Introduction

The Kronecker coefficients are defined as the multiplicities arising in the decomposition of the tensor product of irreducible representations of symmetric groups. More precisely, if k is a positive integer, denote by S k the symmetric group of permutations of 1, k . Then the irreducible complex representations of S k are indexed by the partitions of the integer k (i.e. the non-increasing finite sequences of positive integers -that we will call parts -whose sum is k) and, for any such partition α, we denote by M α the corresponding irreducible complex S k -module. Then, if α and β are partitions of k,

M α ⊗ M β = γ⊢k M ⊕g α,β,γ γ
, and the coefficients g α,β,γ are the Kronecker coefficients. For simplicity of notations we can extend the definition of these coefficients to triples of partitions of different integers by setting g α,β,γ = 0 in that case. One axis of research concerning these Kronecker coefficients is to be interested in studying whether their value is 0 or not. As a consequence we fix two positive integers n 1 and n 2 and, denoting for any partition α its length -i.e. the number of its parts -by ℓ(α), we are interested in the following set:

The Kronecker coefficients are moreover known to possess an interesting stability property, discovered by F. Murnaghan in the 1930's: given three partitions λ, µ, ν, the sequence g λ+(d),µ+(d),ν+(d) d∈N is known to be constant when d ≫ 0. In order to generalise this property, J. Stembridge defined in [START_REF] Stembridge | Generalized Stability of Kronecker Coefficients[END_REF] the notion of a stable triple of partitions:

Definition 1.4. A triple (α, β, γ) of partitions is said to be stable if g α,β,γ = 0 and, for any triple (λ, µ, ν) of partitions, the sequence of general term g λ+dα,µ+dβ,ν+dγ is constant when d ≫ 0.

With this terminology, Murnaghan's stability just states that the triple (1), (1), (1) is stable. An interesting characterisation of this stability notion has been proven by the work of Stembridge (in [START_REF] Stembridge | Generalized Stability of Kronecker Coefficients[END_REF]) and S. Sam and A. Snowden (in [START_REF] Sam | Proof of Stembridge's conjecture on stability of Kronecker coefficients[END_REF])1 : Proposition 1.5. A triple (α, β, γ) of partitions is stable if and only if, for any positive integer d, g dα,dβ,dγ = 1.

This in particular leads us to define another notion close to this one: Definition 1.6. A triple (α, β, γ) ∈ PKron n 1 ,n 2 is said to be almost stable if, for any positive integer d, g dα,dβ,dγ ≤ 1.

Stable triples are in particular of interest because of the following well-known result, which can for instance be found in [Man15b, Proposition 2]:

Proposition 1.7. The set of stable triples in Kron n 1 ,n 2 is the intersection of Kron n 1 ,n 2 with a union of faces of the Kronecker cone PKron n 1 ,n 2 .

As a consequence we want to find ways to produce such faces of PKron n 1 ,n 2 , which contain only stable triples, or possibly almost stable ones. There already exists one result in this direction, proven independently by L. Manivel (in [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF]) and E. Vallejo (in [START_REF] Vallejo | Stability of Kronecker coefficients via discrete tomography[END_REF]), expressed in terms of additive matrices.

Definition 1.8. A matrix A = (a i,j ) i,j ∈ M n 1 ,n 2 (Z ≥0 ) is said to be additive if there exist integers x 1 > • • • > x n 1 and y 1 > • • • > y n 2 such that, for all (i, j), (k, l) ∈ 1, n 1 × 1, n 2 , a i,j > a k,l =⇒ x i + y j > x k + y l .

The result of Manivel and Vallejo is then that any such additive matrix gives an explicit face of PKron n 1 ,n 2 which contains only stable triples. This face is moreover regular, which means that it contains some triple (α, β, γ) of regular partitions (i.e. α, β, and γ have respectively n 1 , n 2 , and n 1 n 2 pairwise distinct parts, with the last one being possibly 0), and it has the minimal dimension possible for a regular face: n 1 n 2 .

In this article we obtain results producing, from an additive matrix, more faces of this kind. Actually, rather than looking precisely at an additive matrix, we look instead at what we call the order matrix, which sort of "encodes the type" of the additive matrix: considering an additive matrix A = (a i,j ) i,j whose coefficients are pairwise distinct2 , instead of the coefficients of A we write their rank in the decreasingly-ordered sequence of the a i,j 's. Then our first result (see Section 3.4) is: Theorem 1.9. Any configuration of the following type in the order matrix:

k k + 1 row i j j + 1 columns
gives an explicit regular face of the Kronecker cone PKron n 1 ,n 2 , of dimension n 1 n 2 , containing only stable triples. The same result is true for each configuration of the type

k k + 1 row i row i + 1 column j
Then we also obtain another result, concerning other types of configurations in the order matrix (called Configurations A

○ to E ○, and involving now three or four coefficients of the order matrix, see Section 3.5):

Theorem 1.10. Each one of the Configurations A

○ to E ○ in the order matrix gives a face -not necessarily regular and possibly reduced to zero -of the Kronecker cone PKron n 1 ,n 2 which contains only almost stable triples.

Obtaining these results is based on the notions of dominant and well-covering pairs, coming from the work of N. Ressayre, that we present in Section 2. At the end of this article (in Section 4), we apply our results as well as Manivel and Vallejo's to all possible order matrices of small size (namely 2×2, 3×2, and 3×3) in order to have a look at the number of new interesting faces of PKron n 1 ,n 2 that we can produce.

Acknowledgements: I would like to thank Nicolas Ressayre for his advice and for invaluable discussions during the preparation of this article. I also acknowledge support from the French ANR (ANR project ANR-15-CE40-0012).

2 Definitions and a few existing general results

Definitions in the general context

For now G is a connected complex reductive group acting on a smooth projective variety X. Let us consider a maximal torus T in G, τ a one-parameter subgroup of T (denoted by τ ∈ X * (T )), and C an irreducible component of X τ , the set of points in X fixed by τ . We denote by G τ the centraliser of τ (i.e. of Im τ ) in G and set

P (τ ) = {g ∈ G s.t. lim t→0 τ (t)gτ (t -1 ) exists}.
Notice that P (τ ) is a parabolic subgroup of G and that G τ is the Levi subgroup of P (τ ) containing T . Consider then

C + = {x ∈ X s.t. lim t→0 τ (t).x ∈ C},
which is a P (τ )-stable. For any x ∈ C, we define the following subspaces of T x X, the Zariski tangent space of X at x:

T x X >0 = {ξ ∈ T x X s.t. lim t→0 τ (t).ξ = 0}, T x X <0 = {ξ ∈ T x X s.t. lim t→0 τ (t -1 ).ξ = 0}, T x X 0 = (T x X) τ , T x X ≥0 = T x X >0 ⊕ T x X 0 , T x X ≤0 = T x X <0 ⊕ T x X 0 .
Theorem 2.1 (Białynicki-Birula).

(i) C is smooth and, for any x ∈ C, T x C = T x X 0 .

(ii) C + is smooth and irreducible and, for any x ∈ C, T x C + = T x X ≥0 .

We can now consider η :

G × P (τ ) C + -→ X [g : x] -→ g.x .
The following definition comes from [START_REF] Ressayre | Geometric Invariant Theory and the Generalized Eigenvalue Problem[END_REF].

Definition 2.2. The pair (C, τ ) is said to be dominant if η is, and covering if η is birational. It is said to be well-covering when η induces an isomorphism onto an open subset of X intersecting C.

In the context of Kronecker coefficients

We consider from now on V 1 and V 2 two complex vector spaces of dimension respectively n 1 and n 2 . We then set

G 1 = GL(V 1 ), G 2 = GL(V 2 ), G = G 1 × G 2 , Ĝ = GL(V 1 ⊗ V 2 ).
We also choose T 1 , T 2 , T = T 1 × T 2 , and T ⊃ T respective maximal tori, and

B 1 , B 2 , B = B 1 × B 2 , B
respective Borel subgroups containing the corresponding tori. We use moreover the following notation: if V ′ is a complex vector space of dimension n ′ and B ′ a Borel subgroup of GL(V ′ ), then any n ′ -tuple of integers λ gives a character of B ′ , and we denote by C λ (B ′ ) the one-dimensional complex representation of B ′ given by this character. Recall then a classical interpretation of the Kronecker coefficients (see for instance [Pel18, Section 2.1]):

Proposition 2.3. If α, β, and γ are partitions of lengths at most n 1 , n 2 and n 1 n 2 respectively, set:

L α = G 1 × B 1 C -α (B 1 ), L β = G 2 × B 2 C -β (B 2 ), L γ = Ĝ × B C -γ ( B),
which are line bundles on G 1 /B 1 , G 2 /B 2 , and Ĝ/ B respectively. Then one has a G-

linearised line bundle L α,β,γ = L α ⊗ L β ⊗ L * γ on G/B × Ĝ/ B such that: g α,β,γ = dim H 0 (G/B × Ĝ/ B, L α,β,γ ) G .
We consider in addition parabolic subgroups P of G and P of Ĝ containing the Borel subgroups. All corresponding Lie algebras will be denoted with lower case gothic letters. We consider X = G/P × Ĝ/ P , on which G acts diagonally. We finally denote by W and Ŵ the Weyl groups associated to G and Ĝ respectively, and by W P (resp. Ŵ P ) the Weyl group of the Levi subgroup of P (resp. P ) containing T (resp. T ). It is canonically a subgroup of W (resp. Ŵ ).

We also give notations concerning the root systems: let us denote by Φ (resp. Φ) the set of roots of G (resp. Ĝ), with Φ + and Φ -(resp. Φ+ and Φ-) the subsets of positive and negative ones with respect to the choice of B (resp. B). Finally, the set of roots of p is denoted by Φp .

Let us consider τ ∈ X * (T ). It is known that the irreducible components of X τ are the G τ v -1 P/P × Ĝτ v-1 P / P , for v ∈ W P \W/W P (τ ) and v ∈ Ŵ P \ Ŵ / Ŵ P (τ ) . We then fix two such v and v, and denote by C the corresponding irreducible component of X τ . Therefore, if (α, β, γ) is a triple of partitions such that L α,β,γ descends to a line bundle on X -that we will also denote by L α,β,γ -3 then, for any x ∈ C, C * acts via τ on the fibre (L α,β,γ ) x over x. This action is given by an integer n which, since C is an irreducible component, does not depend on x ∈ C. We then set µ L α,β,γ (C, τ ) = -n.

Lemma 2.4. For any dominant pair (C, τ ) we consider the set of all triples (α, β, γ) ∈ PKron n 1 ,n 2 such that µ L α,β,γ (C, τ ) = 0. Then it is a face of PKron n 1 ,n 2 (possibly reduced to zero). Moreover it can also be described as:

{(α, β, γ) s.t. X ss (L α,β,γ ) ∩ C = ∅}.
We denote this face by F(C).

Note that this result is actually valid in the general context of Section 2.1, where X is any smooth projective variety and G is a connected complex reductive group acting on X.

The equivalent of PKron

n 1 ,n 2 is then the cone L s.t. ∃N ∈ N * , H 0 (X, L ⊗N ) G = {0} . Proof. It comes directly from [Res10, Lemma 3].
Lemma 2.5. If P = B or P = B, and if (C 1 , τ 1 ) and (C 2 , τ 2 ) are two well-covering pairs such that F(C 1 ) = F(C 2 ), then there exists g ∈ G such that g.C 2 = C 1 .

Proof. This comes from [Res11, Lemma 6.5].

Ressayre also proved, in [START_REF] Ressayre | Geometric Invariant Theory and the Generalized Eigenvalue Problem[END_REF], that any regular face of PKron n 1 ,n 2 is given by a well-covering pair4 . Finally, another consequence of [START_REF] Ressayre | Geometric Invariant Theory and Generalized Eigenvalue Problem II[END_REF] is that, if C is a singleton and (C, τ ) is well-covering, then the face F(C) is a regular face of minimal dimension of PKron n 1 ,n 2 (i.e. n 1 n 2 ).

3 Application to obtain stable triples

Link between well-covering pairs and stability

Theorem 3.1. Assume that (C, τ ) is well-covering. Then, for all G-linearised line bundles L on X such that µ L (C, τ ) = 0,

H 0 (X, L) G ≃ H 0 (C, L| C ) G τ . Proof. It is Theorem 4 of [Res10].
Remark 3.2. If we only make the hypothesis that (C, τ ) is dominant (and µ L (C, τ ) = 0), we still have that

H 0 (X, L) G ֒→ H 0 (C, L| C ) G τ .
Corollary 3.3. Assume that (C, τ ) is well-covering, and that G τ has a dense orbit in C. Then the face F(C) contains only almost stable triples.

Proof. It is an immediate consequence of the previous theorem: let (α, β, γ) ∈ F(C).

Then µ L α,β,γ (C, τ ) = 0 and therefore

∀d ∈ Z >0 , g dα,dβ,dγ = dim H 0 (X, L ⊗d α,β,γ ) G = dim H 0 (C, L| ⊗d C ) G τ ≤ 1
since G τ has a dense orbit in C.

Remark 3.4. If we only make the hypothesis that (C, τ ) is dominant, Remark 3.2 tells us that we still have almost stable triples. But note that F(C) can be reduced to zero.

Remark 3.5. There is an important particular case when τ is dominant, regular (i.e. for all α ∈ Φ, α, τ = 0), and Ĝ-regular (i.e. for all α ∈ Φ, α, τ = 0). Then G τ = T and Ĝτ = T . As a consequence, C is a singleton -say {x 0 } -, and the condition "G τ has a dense orbit in C" is automatic. Moreover one then has:

dim H 0 (C, L| C ) G τ = 1 ⇐⇒ T acts trivially on L x 0 ⇐⇒ ∀σ ∈ X * (T ), µ L (C, σ) = 0.
All the previous results and remarks lead directly to the following main result:

Theorem 3.6. Assume that (C, τ ) is well-covering and that τ is dominant, regular, and Ĝ-regular (then C is a singleton). Then F(C) is a regular face of minimal dimension (i.e. n 1 n 2 ) of the Kronecker cone PKron n 1 ,n 2 and contains only stable triples.

Remark 3.7. When X has this form, there is a characterisation of the dominant and covering pairs with a Schubert condition. It can be found in [START_REF] Ressayre | Geometric Invariant Theory and the Generalized Eigenvalue Problem[END_REF]. Let us explain it quickly here: we use the cohomology ring, H * (G/P (τ ), Z), of G/P (τ ) and we denote, for any closed subvariety Y of G/P (τ ), by [Y ] ∈ H * (G/P (τ ), Z) its cycle class in cohomology. We also use, with the same notations, H * ( Ĝ/ P (τ ), Z).

Then, since P (τ ) = G ∩ P (τ ), G/P (τ ) identifies with the G-orbit of P (τ )/ P (τ ) in Ĝ/ P (τ ), which gives a closed immersion ι : G/P (τ ) ֒→ Ĝ/ P (τ ). It induces a map ι * in cohomology:

ι * : H * ( Ĝ/ P (τ ), Z) -→ H * (G/P (τ ), Z).
Then the result from [START_REF] Ressayre | Geometric Invariant Theory and the Generalized Eigenvalue Problem[END_REF] (Lemma 14) is:

Lemma 3.8. (i) The pair (C, τ ) is dominant if and only if [P vP (τ )/P (τ )] • ι * ([ P v P (τ )/ P (τ )]) = 0.
(ii) It is covering if and only if

[P vP (τ )/P (τ )] • ι * ([ P v P (τ )/ P (τ )]) = [pt],
i.e. if and only if the intersection between two generic translates in Ĝ/ P (τ ) of P vP (τ )/P (τ ) and P v P (τ )/ P (τ ) contains exactly one point.

A sufficient condition to get dominant pairs

We now want to see that we can indeed obtain dominant or well-covering pairs (C, τ ).

For this we consider respective bases of the vector spaces V 1 and V 2 : (e 1 , . . . , e n 1 ) and

(f 1 , . . . , f n 2 ). They give also a basis of V 1 ⊗ V 2 : (e i ⊗ f j ) i,j ordered lexicographically (i.e. (e 1 ⊗ f 1 , e 1 ⊗ f 2 , . . . , e n 1 ⊗ f n 2 )), sometimes denoted (ê 1 , . . . , ên 1 n 2 ) 5 .
Thanks to these bases we will often identify G 1 , G 2 , and Ĝ respectively with GL n 1 (C), GL n 2 (C), and GL n 1 n 2 (C). We finally take B and B the respective Borel subgroups of G and Ĝ formed by the upper-triangular matrices, and set from now on

X = G/B × Ĝ/ B.
Start now from a one-parameter subgroup τ of T which is supposed to be dominant, regular, and even Ĝ-regular. In particular, τ has the form

τ : C * -→ T t -→ (    t x 1 . . . t xn 1    ,    t y 1 . . . t yn 2   ) , with non-negative integers x 1 > • • • > x n 1 , y 1 > • • • > y n 2 .
We then create the matrix M = (x i + y j ) i,j ∈ M n 1 ,n 2 (R). Since τ was taken Ĝ-regular, it has the property of having its coefficients which are pairwise distinct. From this we define what we will call the "order matrix" of τ : it is a matrix having the same size as M but whose coefficient at position (i, j) is the ranking of the coefficient x i + y j when one orders the coefficients of M decreasingly (we will usually circle that ranking when we write the order matrix in order to highlight the difference with the coefficient x i + y j ).

Giving such an order matrix is equivalent to giving a flag ŵ. B/ B fixed by T (and then a well-defined ŵ ∈ Ŵ ): to each matrix position (i, j) ∈ 1, n 1 × 1, n 2 we associate the element e i ⊗ f j of the basis of V 1 ⊗ V 2 . Then we create a T -stable complete flag in V 1 ⊗ V 2 by ordering the elements e i ⊗ f j according to the numbers in the order matrix.

Example: For the one-parameter subgroup

τ : C * -→ T t -→ (   t 4 t 2 1   , t 3 1 ) , the matrix M is   7 4 5 2 3 0
  , and so the order matrix of τ is

  ➀ ➂ ➁ ➄ ➃ ➅   .
Then the flag ŵ. B/ B happens to be

Ce 1 ⊗ f 1 ⊂ Ce 1 ⊗ f 1 ⊕ Ce 2 ⊗ f 1 ⊂ Ce 1 ⊗ f 1 ⊕ Ce 2 ⊗ f 1 ⊕ Ce 1 ⊗ f 2 ⊂ Ce 1 ⊗ f 1 ⊕ Ce 2 ⊗ f 1 ⊕ Ce 1 ⊗ f 2 ⊕ Ce 3 ⊗ f 1 ⊂ Ce 1 ⊗ f 1 ⊕ Ce 2 ⊗ f 1 ⊕ Ce 1 ⊗ f 2 ⊕ Ce 3 ⊗ f 1 ⊕ Ce 2 ⊗ f 2 ⊂ Ce 1 ⊗ f 1 ⊕ Ce 2 ⊗ f 1 ⊕ Ce 1 ⊗ f 2 ⊕ Ce 3 ⊗ f 1 ⊕ Ce 2 ⊗ f 2 ⊕ Ce 3 ⊗ f 2 ,
which we denote by

fl(e 1 ⊗ f 1 , e 2 ⊗ f 1 , e 1 ⊗ f 2 , e 3 ⊗ f 1 , e 2 ⊗ f 2 , e 3 ⊗ f 2 ) ∈ Fℓ(V 1 ⊗ V 2 ).
This corresponds to ŵ = 1 2 3 4 5 6 1 3 2 5 4 6 = 2 3 4 5 notation as a product of transpositions

∈ S 6 ≃ Ŵ .
As usual with such a one-parameter subgroup, we get two parabolic subgroups P (τ ) and P (τ ) of G and Ĝ respectively. According to the hypotheses made on τ , P (τ ) = B in that case, and P (τ ) is a Borel subgroup denoted instead B(τ ). The set of positive (resp. negative) roots of Ĝ for this choice of Borel subgroup is denoted by Φ+ (τ ) (resp. Φ-(τ )). The unipotent radicals of B, B, and B(τ ) are respectively denoted U , Û , and Û (τ ), while those of the respective opposite Borel subgroups will be U -, Û -, Û -(τ ).

Consider now two elements v ∈ W and v ∈ Ŵ . They give

C = {x 0 } = {(v -1 B/B, v-1 B/ B)},
an irreducible component of X T . As usual we then have

C + = Bv -1 B/B × B(τ )v -1 B/ B and η : G × B C + -→ X [g : x] -→ g.x .
We need two more notations: let us denote by ρ the morphism of restriction of roots of Ĝ (which are morphisms from T to C) to morphisms from T to C and, for all

u ∈ W (resp. û ∈ Ŵ ), set Φ(u) = Φ -∩ uΦ + (resp. Φ(û) = Φ-∩ û Φ+ ). Proposition 3.9. If ρ ŵ Φ ((v ŵ) ∨ ) -1 ⊂ Φ(v -1 ) and ρ : ŵ Φ ((v ŵ) ∨ ) -1 -→ Φ(v -1 )
is a bijection, then the pair (C, τ ) is dominant.

Proof. The map η will be dominant if its tangent map at [e : x 0 ], T [e:x 0 ] η, is an isomorphism. Moreover, on root spaces for non-negative weights (i.e. on T

x 0 C + = T x 0 X ≥0 ), T [e:
x 0 ] η is just the identity (cf Theorem 2.1). As a consequence,

T [e:x 0 ] η is an isomorphism ⇐⇒ T [e:x 0 ] η| u -: u --→ u -∩ v -1 u -v ⊕ û-(τ ) ∩ v-1 û-v is an isomorphism. Then, if we define orb : U --→ X u -→ u.x 0 , we have T [e:x 0 ] η| u -= T e orb.
Moreover T e orb is an isomorphism if and only if it is injective, i.e. if and only if the isotropy subgroup U - x 0 of x 0 in U -is finite. As a consequence, T e orb is an isomorphism if and only if the Lie algebra of U -

x 0 is {0}. Therefore,

T [e:x 0 ] η is an isomorphism ⇐⇒ u -≃ u -∩ v -1 u -v ⊕ û-(τ ) ∩ v-1 û-v as T -modules ⇐⇒ β∈Φ -∩v -1 Φ + g β ≃ β∈ Φ-(τ )∩v -1 Φ- g β as T -modules.
Notice now that one has Φ-(τ ) = ŵ Φ-, where ŵ ∈ Ŵ is the element coming from the order matrix, as explained above. Then

Φ-(τ ) ∩ v-1 Φ-= ŵ Φ-∩ ((v ŵ) ∨ ) -1 Φ+ , denoted 6 by ŵ Φ (v ŵ) ∨ . Thus: T [e:x 0 ] η is an isomorphism ⇐⇒ ρ Φ-(τ ) ∩ v-1 Φ-⊂ Φ -∩ v -1 Φ + and ρ : Φ-(τ ) ∩ v-1 Φ--→ Φ -∩ v -1 Φ + is a bijection ⇐⇒ ρ ŵ Φ ((v ŵ) ∨ ) -1 ⊂ Φ(v -1 ) and ρ : ŵ Φ ((v ŵ) ∨ ) -1 -→ Φ(v -1 ) is a bijection, Remark 3.10. It is a classical result that, for u ∈ W (resp. û ∈ Ŵ ), the cardinal of the set Φ(u) (resp. Φ(û)) corresponds to the length of the element u (resp. û) of the Coxeter group W (resp. Ŵ ), denoted ℓ(u) (resp. ℓ(û)). As a consequence ♯Φ(v -1 ) = ℓ(v -1 ) = ℓ(v). In this context (C = {(v -1 B/B, v-1 B/ B)})
, there is a characterisation of wellcovering pairs given by Ressayre in [START_REF] Ressayre | Geometric Invariant Theory and the Generalized Eigenvalue Problem[END_REF], Proposition 11: Lemma 3.11. The pair (C, τ ) is well-covering if and only if it is covering and

v -1 .   α∈Φ -∩vΦ - α   + ρ   v-1 . α∈ Φ-∩v Φ-(τ ) α  = α∈Φ - α.
Lemma 3.12. If v and v are chosen as in Proposition 3.9, then:

(C, τ ) is covering =⇒ (C, τ ) is well-covering.
Proof. Assume that v and v are chosen as in Proposition 3.9 and that (C, τ ) is covering. Then:

v -1 .   α∈Φ -∩vΦ - α   + ρ   v-1 . α∈ Φ-∩v Φ-(τ ) α  - α∈Φ - α = α∈Φ -∩v -1 Φ - α - α∈Φ - α + ρ   α∈ Φ-(τ )∩v -1 Φ- α  = - α∈Φ -∩v -1 Φ + α + α∈Φ -∩v -1 Φ + α = 0
and, by the previous lemma, (C, τ ) is well-covering.

First case to apply Proposition 3.9: an existing result

The first and simplest case to satisfy the condition of Proposition 3.9 is the case when ♯Φ(v -1 ) = 0, i.e. ℓ(v) = 0. This means that v = 1 W , the unit in W . But then we also

need that ♯ Φ ((v ŵ) ∨ ) -1 = 0, i.e. ŵ-1 v-1 ŵ0 = ((v ŵ) ∨ ) -1 = 1 Ŵ . Thus v-1 = ŵ ŵ0 gives a dominant pair (C, τ ).
Moreover the Schubert condition given in Lemma 3.8 is not difficult to check here: according to the form of C = {(B/B, ŵ ŵ0 B/ B)}, the first Schubert variety to consider is just B/B, which is a single point, whereas the second one is B ŵ0 ŵ-1 B(τ )/ B(τ ) = B ŵ0 B/ B, which is the whole variety Ĝ/ B. Hence the product of the two Schubert classes is in fact the class of a point, and then (C, τ ) is well-covering by Lemma 3.8 and Lemma 3.12. We obtain the following: Theorem 3.13. Each order matrix corresponding to a dominant, regular, Ĝ-regular oneparameter subgroup τ of T gives a well-covering pair (C, τ ) with:

C = (B/B, ŵ ŵ0 B/ B) .
As a consequence the corresponding face F(C) of the Kronecker cone PKron n 1 ,n 2 contains only stable triples.

This theorem is actually an already existing result, evoked in the introduction and due independently to L. Manivel (see [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF]) and E. Vallejo (see [START_REF] Vallejo | Stability of Kronecker coefficients via discrete tomography[END_REF]). Let us now explain their result and why it corresponds to the previous one.

We consider a matrix A = (a i,j ) i,j ∈ M n 1 ,n 2 (Z ≥0 ). We call λ and µ its 1-marginals, i.e. the finite sequence of integers (λ 1 , . . . , λ n 1 ) and (µ 1 , . . . , µ n 2 ) given by

λ i = n 2 j=1 a i,j and µ j = n 1 i=1 a i,j ,
and we suppose that λ and µ are partitions (i.e. are non-increasing). Moreover, we denote by ν the π-sequence of A, i.e. the (finite) non-increasing sequence (ν 1 , . . . , ν n 1 n 2 ) formed by the entries of A.

Definition 3.14. The matrix A is said to be additive if there exist integers

x 1 > • • • > x n 1 and y 1 > • • • > y n 2 such that, for all (i, j), (k, l) ∈ 1, n 1 × 1, n 2 , a i,j > a k,l =⇒ x i + y j > x k + y l .
Theorem 3.15 (Manivel, Vallejo). Assume that the matrix A is additive. Then the triple (λ, µ, ν) of partitions is a stable triple.

Manivel and Vallejo gave different proofs of this result. What we want to highlight here is that it corresponds to Theorem 3.13.

Proof. The parabolic subgroup P of Ĝ we consider this time is the one corresponding to the "shape" of -ŵ0 .ν, i.e. the one such that L * ν is the pull-back of an ample line bundle on Ĝ/ P . Furthermore, we take P = B, and so

Y = G/B × Ĝ/ P .
The matrix A gives a flag in Ĝ/ P , similarly to what we explained about the order matrix of a one-parameter subgroup of T : the ordering of the coefficients a i,j in non-decreasing order (it is different from before) gives a partial (since some of these coefficients can be equal) ordering of the elements e i ⊗ f j of the basis of V 1 ⊗ V 2 . Then this ordering corresponds to a T -stable partial flag in V 1 ⊗ V 2 that is precisely an element of Ĝ/ P .

Example: The additive matrix 3 2 3 1 gives the flag

(Ce 2 ⊗ f 2 ⊂ Ce 2 ⊗ f 2 ⊕ Ce 1 ⊗ f 2 ⊂ Ce 2 ⊗ f 2 ⊕ Ce 1 ⊗ f 2 ⊕ Ce 1 ⊗ f 1 ⊕ Ce 2 ⊗ f 1 = V 1 ⊗ V 2 ) ∈ Fℓ(1, 2; V 1 ⊗ V 2 ), which we will denote by fl e 2 ⊗ f 2 , e 1 ⊗ f 2 , {e 1 ⊗ f 1 , e 2 ⊗ f 1 } .
The obtained flag is thus of the form û P / P , with û ∈ Ŵ ≃ S n 1 n 2 . In the previous example, the flag can for instance be written with û = (1 4 3).

Remark: This element û is in general not uniquely defined: what is unique is its class in Ŵ / Ŵ P , but it is sufficient to pick one representative û ∈ Ŵ of this one.

We then set x 0 = (B/B, û P / P ) ∈ Y.

The point x 0 is fixed by T , and we can check (this is an easy computation) that, for any one-parameter subgroup τ of T , µ L λ,µ,ν (x 0 , τ ) = 0.

Since A is additive, there exist integers

x 1 > • • • > x m and y 1 > • • • > y n such that, for all (i, j), (k, l) ∈ 1, m × 1, n , a i,j > a k,l =⇒ x i + y j > x k + y l .
This means that, if we consider the following one-parameter subgroup τ of T :

τ : C * -→ T t -→ (    t x 1 . . . t xm    ,    t y 1 . . . t yn    )
, it is dominant, regular, and verifies that, for all α ∈ Φ,

v-1 .α ∈ Φ \ Φp =⇒ α, τ > 0. (1)
Moreover, we can always assume that τ is Ĝ-regular7 . As a consequence, C = {x 0 } is an irreducible component of Y τ (cf Remark 3.5).

This pair (C, τ ) corresponds actually to the same pair as in Theorem 3.13: consider the order matrix of the one-parameter subgroup τ of T and ŵ ∈ Ŵ the well-defined (since τ is dominant, regular, Ĝ-regular) Weyl group element we associated to such an order matrix. First case: Assume that the coefficients of the matrix A are pairwise distinct. Then the relation between ŵ and û is simply û = ŵ ŵ0 . Then the pair (C, τ ) is exactly the one of Theorem 3.13 and thus the face F(C) of PKron n 1 ,n 2 contains only stable triples. Finally, considering what we have written before, (λ, µ, ν) ∈ F(C). Second case: Assume some of the coefficients of A are equal. Then the relation between ŵ and û is rather that û and ŵ ŵ0 are the same modulo multiplication by Ŵ P on the right. But this means that the two still define the same partial flag in Ĝ/ P and, for the same reasons as in the first case, the triple (λ, µ, ν) is on the face of the Kronecker cone PKron n 1 ,n 2 given by Theorem 3.13. As a consequence it is stable. Moreover, the face of PKron n 1 ,n 2 ∩ {(α, β, γ) s.t. L * γ is a line bundle on Ĝ/ P } given by (C, τ ) with C = {(B/B, û P / P )} ⊂ Y is simply the section of this former face by the subspace {(α, β, γ) s.t. L * γ is a line bundle on Ĝ/ P }.

The fact that an additive matrix gives in fact a face of minimal dimension of the cone PKron n 1 ,n 2 was already explained by Manivel in [START_REF] Manivel | On the asymptotics of Kronecker coefficients[END_REF].

Second case: length 1

At first we need some more notations: for i ∈ 1, n 1 , j ∈ 1, n 2 , and k ∈ 1, n 1 n 2 (which corresponds to a pair (i ′ , j ′ ) ∈ 1, n 1 × 1, n 2 , see Footnote 5), ε i , η j , and εk = ε(i ′ ,j ′ ) are the characters of T 1 (the set of diagonal matrices in G 1 ), T 2 (same in G 2 ), and T respectively, defined by

ε i :    t 1 . . . t n 1    -→ t i , η j :    t 1 . . . t n 2    -→ t j ,
and εk :

   t 1 . . . t n 1 n 2    -→ t k .
Assume that ♯Φ(v -1 ) = 1, i.e. ℓ(v) = 1. This means that v = v -1 = s α , with α a simple root of G (and then Φ(s α ) = {-α}). There are two kinds of such α's:

• the simple roots of G 1 = GL(V 1 ), which are the ε i -ε i+1 , for i ∈ 1, n 1 -1 ,
• the simple roots of G 2 = GL(V 2 ), which are the η iη i+1 , for i ∈ 1, n 2 -1 .

In addition, since we also want ♯ Φ ((v ŵ) ∨ ) -1 = 1, it is necessary that ((v ŵ) ∨ ) -1 = s α, for α a simple root of Ĝ, i.e. a εk -εk+1 for k ∈ 1, n 1 n 2 -1 . Then we have

ŵ. Φ(s α) = {ε ŵ(k+1) -ε ŵ(k) }.
As a consequence we see that α and α will be suitable if and only if

• ŵ(k) = (i, j) ∈ 1, n 1 × 1, n 2 ≃ 1, n 1 n 2 and ŵ(k + 1) = (i, j + 1),
• or ŵ(k) = (i, j) and ŵ(k + 1) = (i + 1, j).

In these cases it is easy to then express v -1 and v-1 and we will get the following result: Theorem 3.16.

• As soon as we have k ∈ 1, n 1 n 2 -1 such that ŵ(k) = (i, j) ∈ 1, n 1 × 1, n 2 ≃ 1, n 1 n 2 and ŵ(k + 1) = (i, j + 1), we have a well-covering pair (C, τ ) (and hence a regular face F(C) of the Kronecker cone PKron n 1 ,n 2 containing only stable triples), where

C = 1, (j j + 1) .B/B, ŵ(k k + 1) ŵ0 . B/ B . • Likewise, if k ∈ 1, n 1 n 2 -1 is such that ŵ(k) = (i, j
) and ŵ(k + 1) = (i + 1, j), the pair (C, τ ), where

C = (i i + 1), 1 .B/B, ŵ(k k + 1) ŵ0 . B/ B , is well-covering.
Proof. We have already seen why these two kinds of properties for ŵ give dominant pairs (thanks to Proposition 3.9). Then all that remains to be seen is whether these pairs are in fact well-covering, which will be done by looking at the Schubert condition (see Lemma 3.8): recall that we have an injective map

ι : G/B ֒-→ Ĝ/ B(τ ) gB -→ g B(τ ) (with B(τ )/ B(τ ) = ŵ B/ B).
We can be a little more precise while describing what ι does on flags:

ι : Fℓ(V 1 ) × Fℓ(V 2 ) ֒-→ Fℓ(V 1 ⊗ V 2 ) (E 1 ⊂ • • • ⊂ E n 1 -1 ), (F 1 ⊂ • • • ⊂ F n 2 -1 ) -→ (H 1 ⊂ • • • ⊂ H n 1 n 2 -1 )
, with:

∀k ∈ 1, n 1 n 2 -1 , H k = H k-1 +E i ⊗F j , where (i, j) = ŵ(k) and H 0 = {0}, E n 1 = V 1 , F n 2 = V 2 .
Then we want to look at the intersection between two generic translates of

Xv = Bv B(τ )/ B(τ ) and ι(X v ) = ι(BvB/B) (with the usual notation C = (v -1 B/B, v-1 B/ B) ).
Here, in both cases, there exists

k 0 ∈ 1, n 1 n 2 -1 such that Xv is of the form B ŵ0 s αk 0 ŵ-1 B(τ )/ B(τ ) = B ŵ0 s αk 0 B/ B, i.e.

is a Schubert variety of codimension 1, and hence a divisor of

Ĝ/ B = Fℓ(V 1 ⊗ V 2 ).
As a consequence it can be rewritten as

Xv = (H 1 ⊂ • • • ⊂ H n 1 n 2 -1 ) ∈ Fℓ(V 1 ⊗ V 2 ) s.t. dim H k 0 ∩ Vect(ê 1 , . . . , ên 1 n 2 -k 0 ≥ 1 .
Then all the translates of Xv correspond to

(H 1 ⊂ • • • ⊂ H n 1 n 2 -1 ) ∈ Fℓ(V 1 ⊗ V 2 ) s.t. dim(H k 0 ∩ S) ≥ 1 , for all vector subspaces S of V 1 ⊗ V 2 of dimension n 1 n 2 -k 0 .
First case: v = 1, (j 0 j 0 + 1) = s β j 0 (i.e. ŵ(k 0 ) = (i 0 , j 0 ) for some i 0 and ŵ(k 0 + 1) = (i 0 , j 0 + 1)). Then

X s β j 0 = Bs β j 0 B/B = Vect(e 1 ) ⊂ • • • ⊂ Vect(e 1 , . . . , e n 1 -1 ) , Vect(f 1 ) ⊂ • • • ⊂ Vect(f 1 , . . . , f j 0 -1 ) ⊂ F j 0 ⊂ Vect(f 1 , . . . , f j 0 +1 ) ⊂ • • • ⊂ Vect(f 1 , . . . , f n 2 -1 ) ∈ Fℓ(V 1 ) × Fℓ(V 2 ) ; F j 0 of dim j 0 .
As a consequence, ι(X s β j 0 ) is the set of all flags

(H 1 ⊂ • • • ⊂ H n 1 n 2 -1
) such that there exists a subspace F j 0 of dimension j 0 of V 2 verifying Vect(f 1 , . . . , f j 0 -1 ) ⊂ F j ⊂ Vect(f 1 , . . . , f j 0 +1 ) and, for all k ∈ 1, n 1 n 2 corresponding to some (i, j),

H k = H k-1 + Vect(e 1 , . . . , e i ) ⊗ F j 0 when j = j 0 H k = H k-1 + Vect(e 1 , . . . , e i ) ⊗ Vect(f 1 , . . . , f j ) otherwise .
Then a generic translate of Xv is

(H 1 ⊂ • • • ⊂ H n 1 n 2 -1 ) ∈ Fℓ(V 1 ⊗ V 2 ) s.t. dim(H k ∩ S) ≥ 1 for S of dimension n 1 n 2 -k 0 which does not intersect Vect(ê ŵ(1) , . . . , ê ŵ(k 0 ) ). Thus ι(X s β j 0 ) ∩ Y = (H 1 ⊂ • • • ⊂ H n 1 n 2 -1 ) ∈ ι(X s β j 0 ) s.t. H k ∩ S = {0} = (H 1 ⊂ • • • ⊂ H n 1 n 2 -1 ) ∈ ι(X s β j 0 ) s.t. F j 0 = Vect(f 1 , . . . , f j 0 -1 , f j 0 +1 ) (since ŵ(k 0 + 1) is (i 0 , j 0 + 1))
. This is a singleton, and the Schubert condition is then verified. As a consequence, the pair (C, τ ) is covering, and hence well-covering by Lemma 3.12.

Second case: v = (i 0 i 0 + 1), 1 = s α i 0 . It is sufficient to exchange the roles of V 1 and V 2 .

These kinds of properties of ŵ are really easy to "read" on the order matrix, which allows us to reformulate the previous theorem in the following equivalent way: Theorem 3.17. For any dominant, regular, Ĝ-regular one-parameter subgroup τ of T , each configuration of the following type in the order matrix of τ :

k k + 1 row i j j + 1 columns gives a well-covering pair (C, τ ), with C = 1, (j j + 1) .B/B, ŵ(k k + 1) ŵ0 . B/ B .
Hence we obtain a regular face of the Kronecker cone PKron n 1 ,n 2 , of dimension n 1 n 2 and containing only stable triples:

F(C) = (α, β, γ) ∈ PKron n 1 ,n 2 s.t.
T acts trivially on the fibre of L α,β,γ over C .

Likewise, each configuration of the type

k k + 1 row i row i + 1 column j
gives a well-covering pair (C, τ ) with

C = (i i + 1), 1 .B/B, ŵ(k k + 1) ŵ0 . B/ B .
Example: The order matrix ➀ ➂ ➁ ➃ which comes for instance from the one-parameter subgroup

τ : t -→ ( t 1 , t 2 1 )
and corresponds to ŵ = (2 3) ∈ S 4 ≃ Ŵ , gives two different well-covering pairs according to the previous theorem:

• one with C 1 = (1 2), 1 .B/B, ŵ(1 2) ŵ0 =(1 4 3) . B/ B , • the other with C 2 = (1 2), 1 .B/B, ŵ(3 4) ŵ0 =(1 2 4) . B/ B .
It is not difficult to see that these cannot be obtained by Theorem 3.13: with Lemma 2.5 in mind, we can "normalise" these well-covering pairs by action of G so that C is of the form {(B/B, û B/ B)}.

• (1 2), 1 .C 1 = B/B, (1 2 4). B/ B ,

• (1 2), 1 .C 2 = B/B, (1 4 3). B/ B .
Then we see that these û cannot be a ŵ ŵ0 , for a ŵ coming from an additive matrix, which is what one would get using Theorem 3.13. Hence, thanks to Lemma 2.5, these two examples give two new faces -F(C 1 ) and F(C 2 ) -of the Kronecker cone PKron 2,2 which contain only stable triples. The equations of the subspaces spanned by these faces in (α, β, γ)

s.t. |α| = |β| = |γ|, ℓ(α) ≤ 2, ℓ(β) ≤ 2, ℓ(γ)
≤ 4 are easy to write:

• α 1 = γ 1 + γ 4 β 1 = γ 1 + γ 2 for F(C 1 ), • α 1 = γ 2 + γ 3 β 1 = γ 1 + γ 2 for F(C 2 ).

Third case: length 2

We follow exactly the same reasoning as in the second case, and keep the same notations.

We assume here that ♯Φ(v -1 ) = 2, i.e. v -1 = s α s β , for α and β distinct simple roots of G.

If s α and s β commute, one then has

Φ(v -1 ) = {-α, -β}. If not, Φ(v -1 ) = {-α, -α-β}.
According to the different possibilities there are for α and β, we then have seven different types for Φ(v -1 ):

1. Φ(v -1 ) = {ε i+1 -ε i , η j+1 -η j }, with i ∈ 1, n 1 -1 , j ∈ 1, n 2 -1 , 2. Φ(v -1 ) = {ε i+1 -ε i , ε j+1 -ε j }, with i, j ∈ 1, n 1 -1 and |i -j| ≥ 2, 3. Φ(v -1 ) = {η i+1 -η i , η j+1 -η j }, with i, j ∈ 1, n 1 -1 and |i -j| ≥ 2, 4. Φ(v -1 ) = {ε i+1 -ε i , ε i+2 -ε i }, with i ∈ 1, n 1 -2 , 5. Φ(v -1 ) = {ε i+2 -ε i+1 , ε i+2 -ε i }, with i ∈ 1, n 1 -2 , 6. Φ(v -1 ) = {η i+1 -η i , η i+2 -η i }, with i ∈ 1, n 2 -2 , 7. Φ(v -1 ) = {η i+2 -η i+1 , η i+2 -η i }, with i ∈ 1, n 2 -2 .
Then we must also have ((v ŵ) ∨ ) -1 = s αs β, for α, β simple roots of Ĝ. As before, this

yields three kinds of Φ ((v ŵ) ∨ ) -1 : (a) Φ ((v ŵ) ∨ ) -1 = {ε k+1 -εk , εk ′ +1 -εk ′ }, with k, k ′ ∈ 1, n 1 n 2 -1 and |k -k ′ | ≥ 2, (b) Φ ((v ŵ) ∨ ) -1 = {ε k+1 -εk , εk+2 -εk }, with k ∈ 1, n 1 n 2 -2 , (c) Φ ((v ŵ) ∨ ) -1 = {ε k+2 -εk+1 , εk+2 -εk }, with k ∈ 1, n 1 n 2 -2 .
And finally some of the cases 1 to 7 are compatible with some of the cases (a) to (c), and will give -as in Paragraph 3.4 -configurations (concerning ŵ) providing v -1 and v-1 which verify the condition from Proposition 3.9. After removing those which are not possible for a ŵ coming from a dominant, regular, Ĝ-regular one-parameter subgroup (i.e. coming from an additive matrix), we obtain the following:

Configuration A ○ (corresponding to cases 1 and (a)):

There exist k, k ′ ∈ 1, n 1 n 2 -1 such that |k -k ′ | ≥ 2 and        ŵ(k) = (i, j) ŵ(k + 1) = (i + 1, j) ŵ(k ′ ) = (i ′ , j ′ ) ŵ(k ′ + 1) = (i ′ , j ′ + 1)
.

It corresponds to the following situation in the order matrix:

k ′ k ′ + 1 row i ′ j ′ j ′ + 1 k k + 1 row i row i + 1 j (k, k + 1, k ′ , k ′ + 1 pairwise distinct) columns
This gives:

v -1 = (i i + 1), (j ′ j ′ + 1) and v-1 = ŵ(k k + 1)(k ′ k ′ + 1) ŵ0 (with |k -k ′ | ≥ 2).

Configuration B ○ (cases 2 and (a)):

There exist k, k ′ ∈ 1, n 1 n 2 -1 such that |k -k ′ | ≥ 2 and        ŵ(k) = (i, j) ŵ(k + 1) = (i + 1, j) ŵ(k ′ ) = (i ′ , j ′ ) ŵ(k ′ + 1) = (i ′ + 1, j ′ ) , with |i -i ′ | ≥ 2.
It corresponds to the following situation in the order matrix:

k k + 1 row i row i + 1 column j k ′ k ′ + 1 row i ′ row i ′ + 1 column j ′ (k, k + 1, k ′ , k ′ + 1 pairwise distinct) (pairwise distinct) This gives v -1 = (i i + 1)(i ′ i ′ + 1), 1 and v-1 = ŵ(k k + 1)(k ′ k ′ + 1) ŵ0 (with |i -i ′ |, |k -k ′ | ≥ 2).

Configuration C ○ (cases 1 and (b)):

There exists k ∈ 1, n 1 n 2 -2 such that ŵ(k) = (i, j) ŵ({k + 1, k + 2}) = {(i + 1, j), (i, j + 1)} .
It corresponds to two types of situation in the order matrix:

k k + 1 k + 2 row i row i + 1 j j + 1 columns k k + 2 k + 1 row i row i + 1 j j + 1 columns And this gives v -1 = (i i + 1), (j j + 1) and v-1 = ŵ(k k + 1)(k + 1 k + 2) =(k k+1 k+2) ŵ0 .

Configuration D ○ (cases 1 and (c)):

There exists k ∈ 1, n 1 n 2 -2 such that ŵ({k, k + 1}) = {(i, j + 1), (i + 1, j)} ŵ(k + 2) = (i + 1, j + 1)
.

It corresponds to two types of situation in the order matrix:

k + 2 k k + 1 row i row i + 1 j j + 1 columns k + 2 k + 1 k row i row i + 1 j j + 1 columns And this gives v -1 = (i i + 1), (j j + 1) and v-1 = ŵ(k + 1 k + 2)(k k + 1) =(k k+2 k+1) ŵ0 .

Configuration E

○ (cases 4 and (b) on the one hand, and 5 and (c) on the other):

There exists k ∈ 1, n 1 n 2 -2 such that    ŵ(k) = (i, j) ŵ(k + 1) = (i + 1, j) ŵ(k + 2) = (i + 2, j) .
It corresponds to the following situation in the order matrix:

k k + 1 k + 2 row i row i + 1 row i + 2 column j
And this configuration gives two different pairs (v -1 , v-1 ):

v -1 = (i i + 1 i + 2), 1 and v-1 = ŵ(k k + 1 k + 2) ŵ0 , and v -1 = (i i + 2 i + 1), 1 and v-1 = ŵ(k k + 2 k + 1) ŵ0 .
Furthermore the Configurations B ○ and E ○ each have a "transposed configuration" in which the roles of the rows and columns are exchanged. Those two transposed configurations will be denoted respectively by b

○ and e ○. They also give a pair (v -1 , v-1 ) (or two, in the case of Configuration e ○) in which the roles of V 1 and V 2 are exchanged. In other words, v-1 does not change and -for instancev -1 = (i i + 1)(i ′ i ′ + 1), 1 (for configuration B ○) becomes v -1 = 1, (j j + 1)(j ′ j ′ + 1) (for configuration b ○).

Theorem 3.18. Let τ be a dominant, regular, Ĝ-regular one-parameter subgroup of T .

Let C = (v -1 B/B, v-1 B/ B) ,
with v -1 and v-1 coming from one of the Configurations A ○ to E ○ or one of their transposed configurations. Then the pair (C, τ ) is dominant and, as a consequence, gives a face -not necessarily regular and possibly reduced to zero -F(C) of the Kronecker cone PKron n 1 ,n 2 which contains only almost stable triples.

Remark 3.19. As we wrote, the two configurations . B/ B .

k k + 1 k + 2 row i row i + 1 row i + 2 column j k k + 1 k + 2 row i j j + 1 j + 2
Once again we can normalise these C's:

• (1 2), (1 2) .C 3 = B/B, (2 4). B/ B ,

• (1 2), (1 2) .C 4 = B/B, (1 3). B/ B .

The equations defining the subspaces spanned by the corresponding faces -possibly reduced to zero -F(C 3 ) and F(C 4 ) of the Kronecker cone PKron 2,2 are respectively:

α 1 = γ 1 + γ 4 β 1 = γ 2 + γ 4 and α 1 = γ 2 + γ 3 β 1 = γ 2 + γ 4 .
One can then check that F(C 3 ) and F(C 4 ) are indeed not reduced to zero:

(5, 5), (5, 5), (3, 3, 2, 2) ∈ F(C 3 ) ∩ F(C 4 ) (it is really a non-stable almost stable triple). But in fact, F(C 3 ) and F(C 4 ) are equal: the equations of subspace that they span can be rewritten as

       γ 1 = γ 2 γ 3 = γ 4 α 1 = γ 1 + γ 3 β 1 = γ 1 + γ 3 .
So we have actually found only one face of PKron 2,2 , which is not regular and contains only almost stable triples.

4 Application to all cases of size 2x2, 3x2, and 3x3

All order matrices of size 2x2

For a dominant, regular, Ĝ-regular one-parameter subgroup τ of T , there are only two possible order matrices (i.e. types of additive matrices) in this case:

➀ ➁ ➂ ➃ and ➀ ➂ ➁ ➃ ,
corresponding -for instance -respectively to the one-parameter subgroups

τ 1 : t → ( t 2 1 , t 1 ) and τ 2 : t → ( t 1 , t 2 1
), which we will from now on denote by τ 1 = (2, 0|1, 0) and τ 2 = (1, 0|2, 0). Each one of these order matrices gives one face of PKron 2,2 , that we will call "additive", coming from the result of Manivel and Vallejo (i.e. Theorem 3.13). They are respectively given by Then Theorems 3.17 and 3.18 enable us to find other faces (all those coming from the first order matrix will be denoted with an exponent (1) and all others with an exponent (2)). While giving them, we will at the same time "normalise" the singletons C's as we did in the previous examples: they will all have the form (B/B, û B/ B) with û ∈ Ŵ . (It allows to apply Lemma 2.5 when the pairs are well-covering).

C (1) 0 = B/B, ( 1
Theorem 3.17 applied to the first possible order matrix gives two well-covering pairs, with C

(1) Let us do the same for the second possible order matrix (it is actually what we did in the examples of the previous section). With Theorem 3.17:

1 = B/B, (1 4 2 3) B/ B , C (
C (2) 1 = B/B, (1 2 4) B/ B , C (2) 2 = B/B, (1 4 3) B/ B .
And with Theorem 3.18:

C (2) 3 = B/B, (2 4) B/ B , C (2) 4 = B/B, (1 3) B/ B .
These examples being small, it is then not difficult to look in details at every possibly non-regular face (i.e. those coming from Theorem 3.18). What we find is that these four dominant pairs actually define all the same non-regular and non-zero face of PKron 2,2 . As before, the subspace of (α, β, γ) s.t. |α| = |β| = |γ|, ℓ(α) ≤ 2, ℓ(β) ≤ 2, ℓ(γ) ≤ 4 spanned by this face has the following equations:

       γ 1 = γ 2 γ 3 = γ 4 α 1 = γ 1 + γ 3 β 1 = γ 1 + γ 3 .
In total, we have then found 4 new distinct (by Lemma 2.5) regular faces of PKron 2,2 which contain only stable triples, whereas 2 others were already known. And we have also found 1 non-regular face containing only almost stable triples.

All order matrices of size 3x2

Let us do exactly as in the previous case. Five order matrices are possible here:

  ➀ ➁ ➂ ➃ ➄ ➅   ,   ➀ ➃ ➁ ➄ ➂ ➅   ,   ➀ ➁ ➂ ➄ ➃ ➅   ,   ➀ ➂ ➁ ➃ ➄ ➅   , and
  ➀ ➂ ➁ ➄ ➃ ➅   .
We number them in that order from 1 to 5 and will denote accordingly some possible corresponding one-parameter subgroups:

τ 1 = (4, 2, 0|1, 0), τ 2 = (2, 1, 0|3, 0), τ 3 = (4, 1, 0|2, 0), τ 4 = (4, 3, 0|2, 0), τ 5 = (4, 2, 0|3, 0).

We then have 5 additive faces with: Theorem 3.17 furthermore gives 15 well-covering pairs: since we normalise them by writing C as (B/B, û B/ B) , we give in the following table the list of elements û obtained, together with the name of the singleton C that they give. name of C element û giving C C

C (1) 0 = B/B, ( 1
(1) 1

(1 5 2 6) C

(1) 2

(1 5)(2 6)(3 4) C

(1) 3

(1 6 2 5) C

(2) 1

(1 6)(3 4 5) C

(2) 2

(1 6 3 2 4 5) C

(2) 3

(1 4 5 3 2 6) C

(2) 4

(1 6)(2 4 3) C

(3) 1

(1 5 2 3 6) C

(3) 2

(1 4 3 5 2 6) C

(3) 3

(1 6)(2 4 5) C (4) 1

(1 6)(2 5 3) C (4) 2

(1 6 3 4 2 5) C (4) 3

(1 6 2 5 4) C

(5) 1

(1 6)(3 5) C

(5) 2

(1 6)(2 4) (The numbers written in exponent between parentheses indicate which order matrix the considered well-covering pairs come from.) Theorem 3.18 gives also 20 dominant pairs, written in the same kind of table: name of C element û giving C C

(1) 4

(1 5 2 6 3) C

(1) 5

(1 3 4 5)(2 6) C

(1) 6

(1 5)(2 6 4 3), C

(1) 7

(1 4 6 2 5) C

(2) 5

(1 4 5 3 6) C

(2) 6

(1 2 6)(3 4 5) C

(2) 7

(1 6 5)(2 4 3) C

(2) 8

(1 6 3 2 4) C

(3) 4

(1 5)(2 4 6) C

(3) 5

(1 5 2)(3 6) C

(3) 6

(1 4 6 2 3 5) C

(3) 7

(1 3 6)(2 5) C

(3) 8

(1 5 2 3 4 6) C

(4) 4

(1 6 2 5 4 3) C (4) 5

(1 6 4)(2 5) C (4) 6

(1 5 3)(2 6) C (4) 7

(1 5)(2 6 4) C (4) 8

(1 4)(2 5 6) C

(5) 3

(1 5 6 2 4) C

(5) 4

(1 5 3 6 2)

If we check here one by one whether these dominant pairs are actually well-covering, we find that eight of them indeed are: those given by C

α 1 = γ 1 + γ 6 α 2 = γ 2 + γ 4 β 1 = γ 1 + γ 2 + γ 3 .
We can for instance notice that (4, 3, 2), (8, 1), (4, 3, 1, 1) belongs to F(C

(2) 5 ), whereas (4, 3, 1), (7, 1), (4, 2, 1, 1) is in F(C

(2) 6 ). Then, since these faces contain each some triple (α, β, γ) of partitions which is such that either α and β are regular (meaning that the parts of α are pairwise distinct, as are those of β), or γ is regular (likewise), [START_REF] Ressayre | Geometric Invariant Theory and the Generalized Eigenvalue Problem[END_REF]Theorem 12] assures that these two dominant pairs are in fact well-covering. Moreover, Lemma 2.5 proves that the two (thus regular) faces are distinct. This is interesting because they come from the same Configuration E ○, appearing in the first column of the order matrix number 2. Hence this kind of configuration can indeed give two different regular faces of PKron n 1 ,n 2 (cf Remark 3.19).

Lemma 2.5 furthermore assures that all the regular faces corresponding to the 28 well-covering pairs that we presented are pairwise distinct. Looking in more details at the 12 other dominant pairs, which are not well-covering, we can see that they in fact give only two distinct non-regular faces of PKron 3,2 containing almost stable triples: the equations of the subspaces that they respectively span are

           γ 1 = γ 2 γ 3 = γ 4 γ 5 = γ 6 α 1 = α 2 = γ 1 + γ 3 β 1 = γ 1 + γ 3 + γ 5 and                γ 1 = γ 2 γ 3 = γ 4 γ 5 = γ 6 α 1 = 2γ 1 α 2 = γ 3 + γ 5 β 1 = γ 1 + γ 3 + γ 5 .
In total we then obtained 23 new regular faces of PKron 3,2 which contain only stable triples, whereas 5 others were already known. We also got 2 other non-regular faces, containing only almost stable triples. ○ yielded only dominant pairs which were not well-covering (our examples were too small to observe a Configuration B ○). One could wonder whether this is always the case.

All order matrices of size 3x3

For this case the numbers begin to become much larger: there are 36 possible order matrices (i.e. 36 types of additive matrices) of size 3×3. As a consequence we do not write all of them here8 . First Manivel and Vallejo's theorem yields 36 additive faces of PKron 3,3 with this 36 additive matrices. But then if we look in details at these matrices, we find that Theorem 3.17 gives 144 well-covering pairs, i.e. 144 regular faces containing only stable triples. Moreover, Theorem 3.18 adds 232 dominant pairs to this, i.e. 232 faces of PKron 3,3 -possibly non-regular and not necessarily pairwise distinct -containing only almost stable triples. Considering what happened in the two previous cases, we can hope that some of those dominant pairs are in fact well-covering. It would be of course possible to check whether this is true, but it is far too tedious to do it here.

Let us nevertheless give one detailed example of a new face of PKron 3,3 that we can obtain with our results: look at the order matrix

  ➀ ➁ ➆ ➂ ➄ ➇ ➃ ➅ ➈  

=

  former and e ○ for the latter) each give two -a priori different -dominant pairs . We will see later (cf Paragraph 4.2) that such configurations can indeed give two different faces of PKron n 1 ,n 2 .Example: When one applies Theorem 3.18 to the same order matrix ➀ ➂ ➁ ➃ as in Paragraph 3.4, one gets two new dominant pairs: a Configuration C ○ and a Configuration D ○ can be observed, which give respectivelyC 3 =(1 2), (1 2) .B/B, ŵ(1 2 3) ŵ0

  1) 2 = B/B, (1 3 2 4) B/ B . Theorem 3.18 gives two dominant ones, with C /B, (1 2 3) B/ B .

  /B, (1 6)(2 4)(3 5) B/ B .

  us give two examples, for C equations defining the subspaces of (α, β, γ)s.t. |α| = |β| = |γ|, ℓ(α) ≤ 3, ℓ(β) ≤ 2, ℓ(γ) ≤ 6 spanned respectively by F(C γ 1 + γ 5 α 2 = γ 2 + γ 6 β 1 = γ 1 + γ 2 + γ 3 and   

Remark 4. 1 .

 1 While doing the previous computations, we noticed that Configurations A ○ and E ○ ultimately gave a well-covering pair each time, whereas Configurations C ○ and D

Other proofs of this by different methods exist in[START_REF] Paradan | Stability property of multiplicities of group representations[END_REF] and[START_REF] Pelletier | A Geometric Approach to the stabilisation of certain sequences of Kronecker coefficients[END_REF].

This assumption is in fact not a restriction on the faces that we produce at the end.

This corresponds to a simple condition concerning the forms of the partitions, which must correspond to the types of the partial flag varieties G/P and Ĝ/ P .

He even proved it in a much more general setting than PKronn 1 ,n 2 .

The ordering of the basis of V1 ⊗ V2 gives in particular an explicit bijection between 1, n1 × 1, n2 and 1, n1n2 , which we will regularly use to identify the two in what follows.

for any û ∈ Ŵ , û∨ is defined as ŵ0 û, where ŵ0 is the longest element of the Weyl group Ŵ

The set of one-parameter subgroups of T verifying condition (1) is an open convex polyhedral cone and, among those subgroups, the not Ĝ-regular ones are elements of some hyperplanes. Thus the set of dominant Ĝ-regular one-parameter subgroups of T verifying condition (1) is not empty.

For the interested reader, they can be found online at http://math.univ-lyon1.fr/~pelletier/recherche/matrices_addi along with the number of well-covering and dominant pairs that each provides.

coming for instance from the dominant, regular, Ĝ-regular one-parameter subgroup τ = (4, 1, 0|7, 5, 0) of T . Theorem 3.18 tells us that the pair with C = B/B, (1 9 4 2 6)(5 8) B/ B is dominant (it comes from a Configuration E ○ in the third column of the matrix). And we can compute the equations of the subspace spanned by

Then one can notice that (6, 5, 4), (7, 6, 2), (3, 2 6 ) ∈ F(C). As a consequence [START_REF] Ressayre | Geometric Invariant Theory and the Generalized Eigenvalue Problem[END_REF]Theorem 12] assures that the pair (C, τ ) is well-covering, and F(C) is indeed a regular face of PKron 3,3 containing only stable triples.


The Reduced Basis Method Applied to Aerothermal Simulations

Main Idea

Weak formulation of the model :

a(u(µ), v ; µ) = f (v ; µ)
FEM Approximation:

X N = span{φ 1 , . . . , φ N } FEM approximation space -→ u N (µ) = N i=1 u N i φ i N ≈ 10 6 -→ A N (µ)u N (µ) = F N (µ) N × N system expensive to solve RB Approximation: u N (µ) ≈ u N (µ) : linear combination of FEM solution W N = span{u N (µ 1 ), . . . , u N (µ N )} RB approximation space → u N (µ) = N i=1 u N i (µ)u N (µ i ) 10 N 100 → A N (µ)u N (µ) = F N (µ) N × N system cheaper to solve u N (µ 1 ) u N (µ 3 ) u N (µ 4 ) u N (µ 5 ) u N (µ 2 ) u N (µ) Ingredients • Set of parameters : D µ • FEM 'truth" approximation X N : finite element approximation space of dimension N >> 1 u N (µ) ∈ X N is solution of a(u N (µ), v ; µ) = f (v ; µ) ∀v ∈ X N • RB approximation Sample : S N = {µ 1 ∈ D µ , ..., µ N ∈ D µ } Approximation space : W N = span{u N (µ 1 ), ..., u N (µ N )} with N << N Galerkin projection on W N to determine RB coefficients Efficient offline-online strategy u N (µ) = N i=1 u N i (µ)u N (µ i ) N × N system to solve : N i=1 a(u N (µ i ), v k ; µ)u N i (µ) = f (v k ; µ), 1 k N If the parameter dependance can be
expressed as an affine decomposition :

a(u, v ; µ) = Q a q θ a q (µ) a q (u, v ) and f (v ; µ) = Q f q θ f q (µ) f q (v ) ⇒ N i=1    Q a q=1 θ q a (µ) a q (u N (µ i ), u N (µ j )) precomputed    u N i (µ) = Q f q=1 θ q f (µ) f q (u N (µ j )) precomputed Airbus Use-Case
Propose in the context of the ANR Project CHORUS Objective : Apply the Certified Reduced Basis Methods on an aerothermal simulation in an avionic bay

Model :

• Steady Navier-Stokes/Heat transfer

• Incompressible Newtonian Fluid • Boussinesq Approximation • Turbulent Flow Mathematical Model Governing Equations          ρu • ∇u + ∇p -µ∆u = ρβ(T -T 0 )g , in Ω × [0, T f ], ∇ • u = 0, in Ω × [0, T f ], u • ∇T -κ∆T = 0, in Ω × [0, T f ], +Boundary Conditions.
(1) with • ρ : fluid density (kg .m -3 ),

• κ ≈ 10 -5 m 2 .s -1 : thermal diffusivity (m 2 .s -1 ),

• µ ≈ 10 -5 m 2 .s -1 : dynamic viscosity (kg .m -1 .s -1 ),

• β : thermal expansion coefficient (K -1 ),

• Length Scale L ≈ 1 m,
• ⇒ Re ≈ 10 5 , Pr ≈ 1, Pe ≈ 10 5 . The main objective of this project is to construct an e-infrastructure that provides, in a user-driven, integrative way, tailored access to the necessary services, resources and even tools for the fast prototyping, providing the service producers with the mathematical frameworks as well.

Solving Strategy

•Perspectives•

  Finite Element Discretization • Newton Algorithm with transient continuation • Parallel implementation using Feel++ library: http://www.feelpp.org/ Challenge and Difficulties • Multi-physic coupled model: simultaneous construction of the different reduced spaces • High Reynolds flow: use of stabilization methods (SUPG/GLS) in the FEM and the RB model • Non-Linearity: Newton algorithm with an affine decomposition of the Jacobian/Residual • Non-affine terms: use of Empirical Interpolation Method (EIM) for discrete operators • Non-Linearity: Use of Simultaneous EIM and RB (SER) algorithm to generate an affine approximation of the non-linear terms (stabilization terms) • Complex Formulation: Due to geometric parameters, use of EIM to automatically recover the affine decomposition. Development of Efficient Error Estimators for the Reduced Model • Reduction of the Coupled Turbulence Model Numerical Results: Cooling of an Printed Circuit Board, Reduced Model of the neighboring of an electronic component submitted to a cooling air flow. Physical Model • Air thermal diffusivity: κ a = 2.7 • 10 -5 • Air kinematic viscosity: µ a = 1.9 • 10 -5 Parameters • Q 1 and Q 2 : Heat sources from the two Integrated Circuits: [0, 10 6 ] • κ 1 and κ 2 :Thermal conductivity of the two Integrated Circuits: [0.2, 150] • D: The inflow rate: [5 • 10 -4 , 10 -2 ] to the size of the basis. Maximum of the error, evaluated on 100 RB approximations compared with the FEM approximations Sponsor MSO4SC: Mathematical Modeling, Simulation and Optimization for Societal Challenges with Scientific Computing

  JB. Wahl, C. Prud'homme, Y. Hoarau

	The Reduced Basis Method
	Motivations
	• Modeling : multi-physics non-linear models, complex geometries, genericity
	• Uncertainty management / Risk analysis
	• Optimization in early design, certification or operating phases
	Objective 1: Fast	Objective 2: Reliable
	• Complex geometries	• Field quality
	-→ Large number of dofs	• Design optimization
	• Uncertainty quantification	-→ Certified bounds
	-→ Large number of runs	-→ Reach material limits


Ivar Farup. Interpolation of the MacAdam ellipses Emmanuel Chevallier
email: emmanuelchevallier1@gmail.com

Ivar Farup

Interpolation of the MacAdam ellipses

Keywords: chromaticity space, MacAdam ellipses, Riemannian geometry, interpolation

come

Introduction

This paper addresses the study of the geometry of the space of chromaticities, the geometry of interest being the one arising from the human perception. The way the brain compares chromaticities is modeled by a mathematical distance. It is well known that attempts to choose this distance with a Euclidean structure lead to significant distortions of the perception, see [START_REF] Kuehni | Color ordered: a survey of color systems from antiquity to the present[END_REF]. Although there are indications that the space of human colour perception is not even Riemannian [START_REF] Alleysson | Neurogeometry of color vision[END_REF], it is often assumed to be the case, see [START_REF] Schrödinger | Grundlinien einer Theorie der Farbenmetrik im Tagessehen[END_REF][START_REF] Wyszecki | Color science (ch. 8)[END_REF][START_REF] Pant | Riemannian formulation and comparison of color difference formulas[END_REF] among others. There exist various approaches to determine its Riemannian structure. The first consists in deriving the metric from a set of axioms on color properties [START_REF] Provenzi | A differential geometry model for the perceived colors space[END_REF][START_REF] Resnikoff | Differential geometry and color perception[END_REF]. Although this yields a consistent mathematical system, assumptions have to be made on the axioms that does not necessarily represent human color vision in limiting cases, such as very low or high intensity light and close to spectral stimuli. Another approach is to construct a more heuristic metric function based on perceptual observations [START_REF] Wyszecki | Color science (ch. 8)[END_REF]. In this case, it is more difficult to get the perceptual attributes to correspond with the resulting global geometry [START_REF] Schrödinger | Grundlinien einer Theorie der Farbenmetrik im Tagessehen[END_REF]. Another alternative is to construct a global geometry based on observations from color order experiments [START_REF] Judd | Ideal color space: Curvature of color space and its implications for industrial color tolerances[END_REF]. This can give good results for the overall perceptual color attributes, but does not always fit well with local color metric experiments. At last, it is possible model the unit balls of the Riemannian metric by just noticeable difference ellipses, which was suggested by MacAdam and Silberstein, see [START_REF] Macadam | Specification of small chromaticity differences[END_REF][START_REF] Silberstein | Investigations on the intrinsic properties of the color domain[END_REF]. Due to the complexity of the experimental procedure, only 25 ellipses have been measured. As depicts Fig. 1, the ellipses do not cover the whole space. Thus it is important to be able to interpolate them, see for instance [START_REF] Kohei | On curvature of color spaces and its implications[END_REF]. This paper addresses the problem of the choice of the interpolation strategy. Although these ellipses have been used for many decades, only a very few works paid attention to the interpolation [START_REF] Macadam | Analytical approximations for color metric coefficients[END_REF][START_REF] Macadam | Analytical approximations for color metric coefficients. II. Friele approximation[END_REF][START_REF] Macadam | Analytical Approximations for Color Metric Coefficients. III. Optimization of Parameters in Frieles Formulas[END_REF][START_REF] Gravesen | The metric of colour space[END_REF]. A rigorous study of the choice of the interpolation is more complicated than it seems. When considered as unit balls, the MacAdam ellipses can be expressed in any parametrization of the chromaticity plane. The interpolation can thus be performed in different coordinate systems, which lead to different interpolations. The difficulty is then to compare the quality of the different interpolations in a way which depends as little as possible on a coordinate system. The main goal of this paper is to study and propose ways to compare interpolations in a framework as rigorous as possible.

Sections 2 and 3 provide notions on color spaces and Riemannian geometry. Section 4 addresses the problem of the comparison of the interpolations. The comparison tool defined in section 4 is used to compare various interpolation strategies of the MacAdam ellipses in section 5.

Basics on colors spaces and chromaticities

Let s(λ) be the spectral radiance of a light wave. In the CIE XYZ color space [START_REF]CIE Standard Colorimetric Observers[END_REF], the coordinates X,Y and Z of the color corresponding to s are defined as The function y models the perception of the luminance: if s(λ) is a Dirac at the frequency λ, the perceived luminance is given by y(λ).

     X = s(λ)x(λ)dλ Y = s(λ)y(λ)dλ Z = s(λ)z(λ)
Consider now the projective plane of the vector space defined by the three coordinates X,Y and Z. The chromaticites are defined as the part of the projective plane corresponding to positive distributions s(λ). Chromaticities are usually parametrized by the xy coordinates defined by the CIE [START_REF]CIE Standard Colorimetric Observers[END_REF] as

x = X X+Y +Z y = Y X+Y +Z .
A main draw back of the coordinates X,Y ,Z,x and y is that they lead to parametrizations whose Euclidean structures do not reflect the visual perception. The Lab and Luv coordinates are non linear transformations of X,Y and Z, which have been defined in such a way that their Euclidean structures is closer to the the visual perception, Lab:

     L = 116f ( Y Yn ) -16 a = 500(f ( X Xn ) -f ( Y Yn )) b = 200(f ( Y Yn ) -f ( Z Zn ))
,

Luv:      L = 116f ( Y Yn ) -16 u = 13L( 4X X+15Y +3Z -u n ) v = 13L( 9Y X+15Y +3Z -v n ) , where (X n , Y n , Z n ) are coordinates of a white reference in CIE XYZ, u n = 4Xn Xn+15Yn+3Zn , v n = 9Yn Xn+15Yn+3Zn and f (t) = t 3 if t > ( 6 29 ) 3 1 3 ( 29 6 ) 2 t + 4 29 otherwise .
At fixed L, it can be checked that there is a homeomorphism between xy, ab and uv. Thus ab and uv can be used to parametrize chromaticities. A MacAdam ellipse is an approximation of the set of colors which are indistinguishable to the human eye from a reference color, at constant luminance. These ellipses were introduced by MacAdam in 1942, see [START_REF] Macadam | Visual sensitivities to color differences in daylight[END_REF]. The set of colors of luminance 48cd/m 2 can be identified with the space of chromaticities and the MacAdam ellipses are then interpreted as sets of chromaticities. When the distance on chromaticities is modeled by a Riemannian distance, it is natural to use the ellipses as unit balls of the local scalar products. Note that since the distances perceived by the brain are not invariant under scaling, there is unfortunately no canonical distance on chromaticities. The metric induced by the MacAdam ellipses is thus associated with the luminance 48cd/m 2 .

The issue tackled in this paper is the interpolation of the ellipses, seen as the local scalar products of a Riemannian distance. In this interpretation the ellipses are naturally represented by 2 by 2 symmetric positive definite matrices. The next two sections provide general considerations on the interpolation of the scalar products of a Riemannian distance. These considerations are then put in practice in section 5, where the underlying space is the space of chromaticities and the local scalar products are the MacAdam ellipses.

Basics of Riemannian geometry

Let M be a topological space, homeomorphic to an open subset of R n . An homeomorphism is a continuous map whose inverse is also continuous. Let φ be an homeomorphism from M to an open subset U φ ⊂ R n . The pair (M, φ) is called a chart. A C k -Riemannian metric is a C k field of scalar products on U φ . In other words, a Riemannian metric (M, φ, G) associates a matrix G x ∈ Sym + (n) to each point x ∈ U φ , where Sym + (n) is the set of n by n real symmetric positive definite matrices.

We shall denote

u x = u, u x = G x (u, u). A continuously dif- ferentiable path is a map γ : [a, b] → M such that φ • γ is continuously differentiable.
The length of a continuously differentiable path γ is defined by the integral

L(γ) = b a (φ • γ) (t) φ•γ(t) dt.
The infimum of the lengths of the paths joining two points defines a distance between these two points. Geodesics are constant speed paths which are locally shortest paths. Geodesic paths can be seen as straight segments or lines on M.

Let V be an open subset of R n and ψ an homeomorphism from M to V such that θ = φ • ψ -1 is a diffeomorphism. Recall that a diffeomorphism is an differentiable homeomorphism whose inverse is also differentiable. Let now G U denotes the field of scalar products defined on U . The Riemannian metric G U defined on U can be transported on V by the following transformation,

∀x ∈ U, G V θ -1 (x) = dθ x G U x (dθ x ) T
where dθ x is the matrix of the differential of θ at x. It can be checked that the metrics G U and G V define the same distance and area on M.

Let us define a distance between scalar products on R n , independently of a basis. Let δ be the distance on Sym + (n) induced by the Riemanian metric

G Σ (X, Y ) = tr(Σ -1 XΣ -1 Y ). Given A, B ∈ Sym + (n), the expression of the distance is δ(A, B) = ||log(A -1/2 BA -1/2 )||, ( 1
)
where log is the matrix logarithm and ||.|| is the Frobenius norm on matrices. Let Gl n (R) denote the n by n real and invertible matrices. δ is an affineinvariant distance, ∀P ∈ Gl n (R), δ(A, B) = δ(P AP T , P BP T ).

Indeed, it is easy to check that G Σ is affine-invariant, see [START_REF] Pennec | A Riemannian framework for tensor computing[END_REF] for more details. Thus, δ is invariant by basis change. Let G U x and GU x be two scalar products at x ∈ U . Then,

δ(G U x , GU x ) = δ(G V θ -1 (x) , GV θ -1 (x) ).
However, it is important to note that the comparison between the scalar products at two different points generally depends on the parametrization,

δ(G U x , G U y ) = δ(G V θ -1 (x) , G V θ -1 (y) )
since dθ x and dθ y are not related. A property of (M, ψ) is called intrinsic if it invariant under a diffeomorphic change of chart.

Interpolation of a sampled Riemannian metric

Let M be a topological space, homeomorphic to an open subset of R n , and let φ be such a homeomorphism from M to an open subset U φ ⊂ R n . Let G be a Riemannian metric on (M, φ). Let p 1 , ..., p k ∈ M k . Assume that the Riemannian metric is only known at {φ(p 1 ), ..., φ(p k )} = (x 1 , ..., x k ) ∈ (U φ ) k . We are interested in the question of the interpolation of the Riemannian metric for arbitrary points p ∈ M. An interpolation rule is a function

R φ : ∪ k∈N (U φ × Sym + ) k × U φ → Sym + ((x 1 , G x 1 ), ..., (x k , G x k ), x) → Ĝx
To simplify the notations, we drop the subscript φ. U φ will be noted U and R φ ((x 1 , G x 1 ), ..., (x k , G x k ), x) will be noted R(x) or Ĝx .

Classical evaluation of an interpolation

Usually, the quality of an interpolation of a function is judged by its regularity. In our discussion, the notion of regularity includes local and non-local rates of variation. The local regularity notion is represented by the order of differentiability. The non-local notion can be evaluated by quantities such as the Lipschitz constant. Recall that a map

f is α-Lipschitz if ∀(x, y), d 2 (f (x), f (y)) < α.d 1 (x, y) (2)
where d 1 is a distance on the domain of f and d 2 a distance on the image. Generally, the evaluation of the non-local regularity requires the existence of distances d 1 and d 2 . Recall that a Riemannian metric is a function which associates to each x ∈ U a scalar product G U x . In the case of the Riemannian metric interpolation, there is no intrinsic way of evaluating the non-local regularity. Firstly, there is no intrinsic distance on M other than the unknown metric G. In Eq. 2 d 1 is unknown, since it is the interpolated quantity. Secondly, according to the previous discussion there is no intrinsic way of comparing G U

x and G U y when x = y. In Eq. 2, there is no intrinsic d 2 . Among the classic ways of evaluating the interpolation of a function, only the differentiability order can be used in the case of Riemannian metric interpolation. However it is clear that when the metric is known at a finite number of points, there exist many smooth interpolations of the known data. Furthermore, these interpolations can be significantly different. This is formalized in the rest of the paragraph.

Let G be a reference metric on (M,φ), such as the perceptual metric in the case of the manifold of colors. Given two Riemannian metrics G 1 and

G 2 , let D(G 1 , G 2 ) = sup x∈U δ(G 1 x , G 2 x )
be a distance between G 1 and G 2 . Note that this distance is intrinsic.

Theorem 4.1. Given a set of points {x 1 , ..., x k } ∈ U k and a set of scalar products G x 1 , ..., G x k , there are infinitly many C ∞ Riemannian metrics Ĝ

such that Ĝx i = G x i for all 1 ≤ i ≤ n. Furthermore, for all A ∈ R + , there exist C ∞ metrics Ĝ and Ĝ such that Ĝx i = Ĝ x i = G x i and A < D( Ĝ, Ĝ ). Proof. Let u i = {x i } ∪ (U \ {x 1 , ..., x k }) and (α i ) i∈{1,...,k} be smooth func- tions such that supp(α i ) ⊂ u i , ∀x ∈ U, i α i (x) = 1. Let Ĝx = i α i (x)G x i .
Since Sym + is convex, Ĝx ∈ Sym + . Ĝ defines thus a smooth field of Sym + matrices such that Ĝx i = G x i . By construction, it is possible to impose any arbitrary value Ĝy ∈ Sym + at a point y / ∈ {x i }. Hence for all A > 0, it is possible to construct two smooth metrics Ĝ and Ĝ which interpolate the scalar products on the {x i } and such that D( Ĝ, Ĝ ) > A.

Two observations can be made. Using the standard evaluations of an interpolation, that is to say the order of differentiability, there are infinitely many interpolations of a Riemannian metric of maximal quality. Furthermore, among these possible interpolations some of them are significantly different. It is thus interesting to try to define additional intrinsic criteria.

Adapted intrinsic evaluation

Intrinsic criteria

A possible intrinsic way to evaluate the quality of the interpolation of a set of scalar products is to analyzing the curvature of the interpolated metric. In the considered situation where M is homeomorphic to R n , interpolations whose curvature have low absolute values are preferred.

Another approach consists in evaluating the interpolation rule itself and not directly the interpolation. Using a "leave one out" cross-validation approach, it is possible to evaluate the generalization capacity of an interpolation rule R. Following the discussion of section 3, it is not possible to intrinsically compare the scalar products defined at different points p = q ∈ M. However, it is possible to compare two scalar products defined at the same point independently from a chart. Let S be the set of pairs (x i , G x i ), 1 ≤ i ≤ k. Let S i be the set of pairs obtained by removing the pair (x i , G x i ). It is possible to compare R(S i , x i ) and G x i since they are both defined at the same point. Let d be a distance on Sym + (n). An interpolation rule R can be evaluated by the set of distances d(R(S i , x i ), G x i ) for all i. In the case of the MacAdam ellipses, d(R(S i , x i ), G x i ) is the distance between the ellipse at p i and the interpolated ellipse at p i from the 24 other ellipses. These terms can be seen as playing the role of the evaluation of the non-local regularity of the interpolation. Given a set of pairs S, let

Q(R, S) = i d(R(S i , x i ), G x i ) (3)
be a measure of the quality of an interpolation R. When d is independent of a chart, this evaluation of the interpolation rule R is intrinsic. It evaluates how well the rule R can predict the measured ellipses.

Limits of the intrinsic evaluation

Despite the fact that both previous criteria bring useful information, both suffer from the limitations described in theorem 4.1. This is formalized by theorem 4.2 and 4.3. Although the results are only proved in the two dimensional case, we are convinced that they hold in any dimension.

4.2. Let U ⊂ R 2 and S be a set of k pairs (x i , G i ) ∈ U ×Sym + (2).
There always exist infinitely many C ∞ interpolations Ĝ such that Ĝx i = G x i for all 1 ≤ i ≤ k and such that G is a flat metric. Furthermore, for all A ∈ R, there exist C ∞ flat metrics Ĝ and Ĝ such that Ĝx i = Ĝ

x i = G x i and A < D( Ĝ, Ĝ ).
Proof. The idea of the proof is to built a smooth diffeomorphism f :

U → V whose differential at each x i is G 1/2 x i . It can be checked that in the chart (M, f • φ), G V x i = I.
The Euclidean metric on V is then a flat metric that interpolates the G V

x i . The pushforward of this Euclidean metric by f -1 provides a flat interpolation of the scalar products G x 1 , ..., G x k . The construction of such diffeomorphism is described in the appendix for the dimension 2. Despite the technical difficulties of their constructions, we are convinced that such diffeomorphisms exist in any dimension. Using a similar argument as in theorem 4.1, it is possible to construct interpolations that differ as greatly as possible.

Theorem 4.3 is a simple adaptation of theorem 4.2.

Theorem 4.3. Let U ⊂ R 2 and S be a set of k pairs (x i , G i ) ∈ U ×Sym + (2).
There always exist infinitely many interpolation rules R such that Q(R, S) = 0 and such that Ĝ is C ∞ and flat. Furthermore, for all A ∈ R, there exist two such rules R and R with A < D( Ĝ, Ĝ ).

Proof. Let f be a diffeomorphism as in the proof of theorem 4.2. Let R be the rule such that R(., x) = df T x df x . Let S and S i be as defined in section 4.2.1. Since R(S, x i ) = R(S i , x i ) = G x i , Q(R, S) = 0. Like in theorem 4.2, R(., x) is flat and C ∞ . Again, using a similar argument as in theorem 4.1, it is possible to construct interpolations that differ as greatly as possible.

As previously, these results tell us two things. Firstly, there are always infinitely many optimal interpolations. Secondly, among these possible interpolations some of them are significantly different. These results are expected to be similar for all the interesting intrinsic criteria.

In practice however, the interpolation should not be evaluated in a fully intrinsic way. In the case of the space of chromaticities under the perceptual Riemannian metric, it is clear that specific parametrizations have to be privileged. Indeed, it is natural to assume that the perceptual metric is related to the physical properties of the eye. On that regard, the interpolations arising in theorem 4.2 and 4.3 are not satisfying since they are based on a chart that have been artificially created and which are unlikely to be related to the human perception. The perceptual metric is expected to be related to the CIE XYZ color space. All the color spaces defined by the CIE, or other simple transformations of the CIE XYZ color space are preferred to arbitrary parametrization. The study is restricted to interpolations that have a regular behavior with respect to these charts. Then it is important to dispose of intrinsic criteria to compare them. In the next section, the interpolation are evaluated using the cross validation criteria defined in section 4.2.1.

Interpolation of the MacAdam ellipses

M is the space of chromaticities, the points p i ∈ M are the chromaticities of the centers of the MacAdam ellipses and the G x i are the scalar products represented by the MacAdam ellipses.

The MacAdam ellipses are assumed to be exact measurements of the local scalar products. Thus all the considered interpolation rules fit exactly the observations: R(S, x i ) = G x i . The considered interpolations and their evaluations depend on the followings elements,

• a parametrization of M • a choice of parameters of the scalar products

• a type of interpolation

• a distance d on the set of scalar products, see Eq. 3.

Color spaces

Three parametrizations of colors are considered: xyY , Lab and Luv. At fixed Y there is a unique correspondance between xy, ab and uv. In this paper Y is set at 48cd/m 2 since it is the luminance at which the ellipses are measured. The three couples are reasonable parametrizations of M in which the different interpolations are computed.

Parameters

Two types of representations of the scalar products are considered.

The first one is based on the matrix of the bilinear form in a chart. Three representations of this matrix are considered, the matrix itself (M ), its logarithm (M L) and its inverse (M I).

The second type of representations is based on the parameters of the ellipses. The parameters of an ellipse are the lengths of its axes a and b with a ≥ b and the direction of its main axis. Let θ ∈ [0, 2π[ be an angle between abscissa axis and the main axis as depicted in Fig. 3. A direction is a point on the projective line, which is isometric to a circle. The point representing the direction of the main axis is parametrized by the angle

α = 2.(θ[π]).

Interpolation type

Four types of interpolation are considered: nearest neighbor (N ), barycentric (B), kernel (K) and spline (S).

In order to define the barycentric and the kernel interpolation, it is necessary to define a notion of average between scalar products. The notion of average is dependent of the choice of parameters used to represent the scalar products. When the parameters have a matrix form (M -M L-M I), the average is chosen to be the standard Euclidean average. When the scalar products are represented by ellipses (P -P L-P I), the average on the first two parameters is also chosen to be Euclidean. The coordinate α represents a point on a circle, the most adapted average is thus the Karcher-Fréchet average ( [START_REF] Karcher | Riemannian center of mass and mollifier smoothing[END_REF]). Since the computation of the Fréchet average requires a gradient descent, we chose instead to compute the Euclidean average and to project the result on the circle.

Barycentric

The barycentric interpolation relies on a triangulation of the centers of the MacAdam ellipses. This triangulation is chosen to be the Delaunay triangulation in the given chart. When the point at which the data is interpolated lies inside a triangle, the interpolated ellipse is a weighted average of the ellipses at the vertices. When the point is outside of the triangulation it is orthogonaly projected on the triangulation. The interpolation is then computed at the projected point. The barycentric interpolation provides a continuous field of ellipses. Note however that as for the nearest neighbor interpolation, this field is not twice differentiable and thus cannot be used to estimate the curvature.

Kernel

Given a kernel k, the kernel interpolation at x is defined as an average of all the MacAdam ellipses weighted by k( x-x i ) j k( x-x j ) where x j are the centers of the MacAdam ellipses and . is the canonical Euclidean norm of the chart. Since the interpolated scalar product at x i should be the observed MacAdam ellipse, the kernel should tend to infinity in zero. The weights are then prolongated by continuity. The function used in the experiments is

k λ ( x -x i ) = e -x-x i λ x -x i .
λ enables to control the area of influence of each x i . In the experiments, λ is chosen by cross validation.

Spline

The last type of interpolation is based on bi-harmonic splines. For parameters of type M or P , the coefficients are interpolated independently. The set of symmetric positive definite matrices is a cone of the symmetric matrices.

In practice the bi-harmonic spline interpolation produces matrices outside of this cone, which is incompatible with the scalar product interpretation. Thus only the logarithm of the matrix is used for the spline interpolation.

Recall indeed that the logarithm of the cone of symmetric positive definite matrices is the vector space of symmetric matrices. Since it is a circular data, the parameter α encoding the direction of the main axis of the ellipse requires a specific treatment. Instead of using an intrinsic interpolation on the circle, α is represented by a vector (cos(α), sin(α)) ∈ R 2 whose coordinates are be interpolated using bi-harmonic splines. The interpolation is then projected on the circle to give the interpolated α.

Distance evaluation

After defining the interpolation rules, we need to define the distance d in the evaluation function Q of Eq. 3,

Q(R, S) = i d(R(S i , x i ), G x i ),
regardless the choice of d.

The affine-invariant distance δ defined in section 3 has the advantage of being independent of the parametrization in which it is computed. However, since the scalar product G x i is the ground truth, it is natural to compare the scalar products in an orthonormal basis of G x i . We define two measures of the distance to G x i . The first is

d B x i (G) = Gx i (v,v)=1 |G(v, v) -G x i (v, v)|dv x i
where dv x i is the measure induced by G x i on its unit ball. The second is

d M x i (G) = M Gx i G -I ,
where

M Gx i G
is the matrix of G in an orthonormal basis of G x i and . the Frobenius norm. We can see that d M

x i does not depend on the choice of the orthonormal basis. The upper index M stands for matrix and the upper index B stands for ball. d B

x i and d M x i are independent of a chart. The evaluation of the quality of an interpolation rule becomes

Q M (R, S) = i d M x i (R(S i , x i )) (4)
and

Q B (R, S) = i d B x i (R(S i , x i )) (5)
Note that Eq. 4 and Eq. 5 differ slightly from Eq. 3 since d depends on the ground truth scalar product.

Results and discussion

Tables 1 and2 summarize the notations defined in the previous section. The quality of the different interpolation rules are gathered in tables 3 and 4.

The two evaluation distances provide similar results. Recall that the spline interpolation in the cone of symmetric positive definite matrices might produce values outside of the cone. Thus splines are used only in the logarithmic representation.

Independently of the choice of parameters, tables 3 and 4 clearly indicate that interpolations of type P are better than the ones of type P L and P I, and that interpolations of type M I are better than the ones of type M and M L. These two results are consistent because the lengths of an ellipse axis are the square of the eigenvalues of the inverse of the matrix of the scalar product and not of eigenvalues of the matrix itself. In his 1943 paper, see [START_REF] Macadam | Specification of small chromaticity differences[END_REF], MacAdam did the interpolation of the coefficients of the matrix of the scalar product in the xy parametrization. It turns out to be the worst setting among the interpolations studied here.

A set of observations can be made about the influence of the parametrization.

• Except the lines BP -BP L -BP I of table 3, uv parameters provide better results than the ab parameters. The plots of errors by ellipses presented in Fig. 5 and 6 and in appendix B show that the ellipse on top of the gamut is always poorly predicted in the ab parametrization. The same ellipse is always properly predicted in the uv parameters.

• Except lines BP -BP L-SP of table 3 and BM I -BP -BP L-SP of table 4, uv parameters provide better results than the xy parameters.

• Except lines BM I -BP -BP L -SP of table 3 and 4, ab parameters provide better results than the xy parameters.

Thus, even if the uv parameters seem to give better results in general, it is not possible to state a clear rule.

(BP, xy) and (KM I, uv) are the two best interpolations according to Q B and Q M . It is sensible to privilege KM I in the uv parameters over BP in the xy parameters due to its higher regularity. Unlike the barycentric interpolation, the differentiability of the metric obtained with the kernel interpolation enables to define the Riemannian curvature and the geodesic equations.

Fig. 5 and 6 depicts the individual d B x i scores obtained by each ellipse for interpolations of types BM -BM I and KM -KM I. The center of each ellipse is represented by a black circle. For visualization purposes the d B x i scores are interpolated between the ellipses using a barycentric interpolation. Note that again this interpolation is dependent on the parametrization, but it does not deserve a careful study. The colorbar is depicted in Fig. 4. Fig. 7 depicts the interpolation of the ellipses according to different interpolation strategies: (KM I, uv), (BP, xy) and (BM, xy). (KM I, uv) and (BP, xy) have the two best scores according to Q B and Q M respectively, while (BM, xy) has the worst scores according to both criteria. It appears that the ellipses of the two best score interpolations are reasonably similar. The barycentric frameworks tend to produce poor extrapolations outside of the convex hull of the MacAdam ellipses. Other choices of triangulation of the ellipse centers and their impacts on the quality of the extrapolations should be investigated in future works.

Several geometric aspects of the different interpolations, such as the Riemannian curvature and the geodesics remain to be studied. Given an origin and an initial tangent vector, the kernel and spline interpolations enable establish the geodesic equation. Fig. 8 depicts preliminary results on geodesics. The shape of triangles is characteristic of negatively curved space, which confirms the results obtained in [START_REF] Farup | Hyperbolic geometry for colour metrics[END_REF]. Recall that since the curvature of the space is not expected to be uniformly negative, the geodesics are not expected to be unique. Even though the geodesics joining two fixed points tend to follow similar trends, a more in depth analysis of the different region of the gamut should be performed in a future work.

Conclusion

There exist several sensible parametrizations of the chromaticity plane. In this paper we focused on parameters xy and their transformation in the ab and uv planes at 48cd/m 2 . A set of simple interpolation strategies of the MacAdam ellipses described in section 5 are considered in each parametrization. In order to determine in which of these coordinates the interpolation should be achieved, it is necessary to dispose of an intrinsic criteria, that is to say a criteria independent of the choice of a parametrization. Curvature and cross-validation are examples of such criteria. Theorems 4.2 and 4.3 show that when the set of scalar products is known on a finite set, there are always infinitely many interpolations which are optimal for both criteria. This result indicates that a purely intrinsic evaluation would lead to irrelevant results and that it is thus important to study only interpolations which are regular with respect to the natural parametrizations.

Cross-validation is used to compare the different interpolations described in section 5, the results are summarized in tables 3 and 4. The main information that can be pulled out of these tables is that the interpolation of the ellipses should be made on the parameters of the ellipses or on the inverse of the matrix of the corresponding scalar product and not directly on the matrix of the scalar product as MacAdam suggested in [START_REF] Macadam | Specification of small chromaticity differences[END_REF]. Although this is not the case for all interpolations, the uv parameters tend to present better results.

The experiments presented in this paper are based on pointwise comparisons of interpolated ellipses. On a local level, the two best interpolations provide consistent results: the differences between the ellipses are small. Our future work will mainly focus on the study of the curvature and on the geodesics of the interpolated metrics. Since all the constructions can be generalized to three dimensions, we also intend to analyze of the ellipsoids datasets [START_REF] Melgosa | Suprathreshold color-difference ellipsoids for surface colors[END_REF] describing the geometry of the full color space. At last we plan to carry on our work on statistics on the manifold of colors, see [START_REF] Chevallier | Histograms of images valued in the manifold of colours endowed with perceptual metrics[END_REF].

A Construction of the diffeomorphism of theorems 4.2 and 4.3

Given x 1 , .., x k in R n and A 1 , .., A k a set of n × n symmetric positive definite matrices, we search for a smooth diffeomorphism f of R n such that df x i = A i .

Let R be such that the closed balls B(x i , R) do not intersect. We construct a diffeomorphism f which is equal to the identity outside of the B(x i , R).

The main step is to find a diffeomorphism g which is the identity outside of the ball B(0, R) such that dg = A for an arbitrary symmetric positive definite matrix A. Then, using the translations t i : x → x + x i , we obtain

k diffeomorphisms g i = t i • g A i • t -1 i
that we glue together to obtain the diffeomorphism f = g 1 • ... • g k . Since the balls B(x i , R) are disjoint, we have that f is equal to g i in each ball B(x i , R), hence df x i = A i .

Construction of g A . By a linear change of coordinate we can suppose that the matrix A is diagonal with positive coefficients. We start by the one dimensional case.

Let a be a positive number. Set φ : R → R be a C ∞ even function such that φ

(0) = 1 ≥ φ(x) ≥ -for all x in R, R 0 φ(t)dt = 0, R 0 |φ(t)|dt ≤ R/10 and sup φ ⊂ [-R, R]
. The constant will be chosen later. Set

g a (x) = ϕ(a, x) = x 0 (1 + (a -1)φ(t))dt. • i. g is C ∞ and g a (x) = 1 + (a -1)φ(x), hence g a (0) = a. • ii. Since R 0 φ(t)dt = 0, we have g a (x) = x + (a -1) x 0 φ(t)dt = x when |x| ≥ R.
• iii. Let us now choose the constant such that g a (x) > 0 for all x.

When a ≥ 1, we have 1 + (a -1)φ(x) ≥ 1 2 iff φ(x) ≥ -1 2(a-1) , it is then enough to take =

1 2(a-1) ( = ∞ if a = 1). When a ≤ 1, 1 + (a -1)φ(x) ≥ a iff (1 -a)(1 -φ(x)) ≥ 0 which is true because φ ≤ 1.
Therefore g a is a diffeomorphism such that g a (0) = a and g a (x) = x when |x| ≥ R.

In dimension 2, the idea is to take F (x 1 , x 2 ) = (ϕ(a 1 , x 1 ), ϕ(a 2 , x 2 )). This almost works but F is not the identity outside of [-R, R] × [-R, R]. We have F (x 1 , x 2 ) = (x 1 , x 2 ) when |x 1 | and |x 2 | ≥ R while we would like the same result when |x 1 | or |x 2 | ≥ R. We modify the function F by varying a 1 and a 2 . We consider even C ∞ functions α i , i = 1, 2, such that Case 1 |x 1 | ≥ R. Then we also have |y 1 | ≥ r and thus G(x 1 , x 2 ) = (x 1 , ϕ(α 2 (x 1 ), x 2 )) = G(y 1 , y 2 ) = (y 1 , ...) thus y 1 = x 1 . We deduce that ϕ(α 2 (y 1 ), y 2 ) = ϕ(α 2 (x 1 ), y 2 ) = ϕ(α 2 (x 1 ), x 2 ) and thus that y 2 = y 1 because the functions x → ϕ(a, x) are injective.

α i (x) =      a i for x ∈ [0, R]
Case 2 |x 1 | < R. According to the remark |y 1 | < R. We need to discuss now the second variable.

Sub-case 1. |x 2 | ≥ R comes down to case 1 by changing variables 1 and 2.

Sub-case 2. |x 2 | < R. Using the remark we also have |y 2 | < R. Therefore G(x 1 , x 2 ) = (ϕ(a 1 , x 1 ), ϕ(a 2 , x 2 )) = (ϕ(a 1 , y 1 ), ϕ(a 2 , y 2 )) and thus (x 1 , x 2 ) = (y 1 , y 2 ). Jacobian ∂ x 1 G(x 1 , x 2 ) = (1 + (α 1 (x 2 ) -1)ψ(x 1 ), α 2 (x 1 ) -α 1 (x 2 )

x 1 0 ψ(t)dt α 2 (x 1 )

x 2 0 ψ(t)dt.

Using inequalities of item iii, and then the inequality

B Error maps

These plots depict the d B error per ellipses for several interpolations. The error is interpolated between the ellipses as described in paragraph 5.5. Notations are defined in tables 1 and 2. Fig. 9 provides the color code.

Figure 1 :

 1 Figure 1: MacAdam ellipses in the xy chromaticity diagram. To ease their visualization, the ellipses are enlarged 10 times.

  dλ , where x, y and z are real positive functions determined by the CIE, see Fig 2.

Figure 2 :

 2 Figure 2: The CIE standard observer color matching functions.

Theorem

Figure 3 :

 3 Figure 3: θ is an angle between the abscissa axis and the main axis of the ellipse. α = 2.(θ[π]) parametrizes the direction of the main axis.

Figure 4 Figure 5 :Figure 6 :Figure 7 :

 4567 Figure 4: Colorbar

Figure 8 :

 8 Figure 8: Geodesics joining fixed points are computed using the geodesic equation and by adjusting the initial tangent vector. Although only one geodesic per pairs of points and interpolations is plotted, recall that the uniqueness of the geodesics is not guaranteed. (a): geodesics joining centers of the central and the extremal MacAdam ellipses using the (KM I, uv) interpolation. (b): geodesics joining centers of extremal ellipses according to different interpolations. The (KM, xy) upper geodesics is missing due to some numerical instabilities. (b,c,d): although the (KM I, uv) and (KP, uv) geodesics are usually very close one from another, it is not the case in the upper right corner of the uv gamut.

  monotonous on [R, 2R] with derivative ≤ 2a i /r in absolute value 1 for x ≥ 2R. The map we are looking for is defined byG(x 1 , x 2 ) = (ϕ(α 1 (x 2 ), x 1 ), ϕ(α 2 (x 1 ), x 2 )) Since G coincides with F on [-R, R] × [-R, R] the differential in 0 is the diagonale (a 1 , a 2 ). By construction G is the identity outside of the square [-2R, 2R] × [-2R, 2R]. It remains to check that G is a diffeomorphism.It is enough to prove that G is injective and that its Jacobian is always positive. Indeed, the surjectivity is implied by the injectevity and the fact that G is the identity outside of the square [-2R, 2R] × [-2R, 2R]. Let us start with the injectivity. Assume that G(x 1 , x 2 ) = G(y 1 , y 2 ).Remark:|x 1 | ≥ R is equivalent to |y 1 | ≥ R. Indeed |x 1 | ≥ Rand |y 1 | < R is not possible because we would have |ϕ(α 1 (x 2 ), x 1 )| ≥ R and |ϕ(α 1 (y 2 ), y 1 )| < R.

∂ x 2 G(x 1 , x 2 ) = (α 1 (x 2 ) x 1 0ψ

 21211 (t)dt, 1 + (α 2 (x 1 ) -1)ψ(x 2 )) thus Jac G(x 1 , x 2 ) = (1 + (α 1 (x 2 ) -1)ψ(x 1 ))(1 + (α 2 (x 1 ) -1)ψ(x 2 ))

Figure 9

 9 Figure 9: Colorbar

Figure 14

 14 Figure 14: Barycentric

Figure 16

 16 Figure 16: MI (without splines)

Table 1 :

 1

		Interpolation type	Table 2: Interpolation parameters
	N nearest neighbor	M	matrix
	B	barycentric	P ellipse parameters
	K	kernel	L	logarithm
	S	spline	I	inverse

Table 3 :

 3 Evaluation of the different interpolation rules using Q B .

		xyY	Lab	Luv
	N	0.292 0.256 0.216
	BM	0.337 0.228 0.227
	BM L 0.178 0.178 0.171
	BM I 0.168 0.185 0.165
	BP	0.16 0.186 0.198
	BP L 0.183 0.193 0.198
	BP I 0.225 0.205 0.208
	KM	0.293 0.242 0.199
	KM L 0.247 0.207 0.174
	KM I 0.205 0.195 0.163
	KP	0.229 0.204 0.18
	KP L 0.246 0.211 0.185
	KP I 0.265 0.219 0.192
	SM L 0.265 0.208 0.178
	SP	0.175 0.207 0.184
	SP L 0.254 0.215 0.184
	SP I	0.263	0.2	0.188

Table 4 :

 4 Evaluation of the different interpolation rules using Q M .

		xyY	Lab	Luv
	N	1.461 1.232 0.918
	BM	1.814 1.024 0.971
	BM L 0.792 0.769	0.7
	BM I 0.624 0.738 0.644
	BP	0.62 0.799 0.79
	BP L	0.74 0.854 0.823
	BP I 0.956 0.931 0.87
	KM	1.514 1.121 0.809
	KM L 1.128 0.941 0.705
	KM I 0.776 0.795 0.602
	KP	1.004 0.911 0.721
	KP L 1.118 0.963 0.761
	KP I 1.242 1.006 0.799
	SM L 1.440 0.894 0.706
	SP	0.665 0.863 0.694
	SP L 1.232 0.931 0.715
	SP I	1.308 0.826	0.7