Antoine Médoc

A REFINEMENT OF HORN'S CONJECTURE by

Keywords: Horn inequalities, Horn conjecture, Littlewood-Richardson coefficients, Kirwan cone, Schubert calculus

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

1. Introduction 1.1. Horn's conjecture. -Let A and B be two square matrices of the same order. A natural question (also coming out in physics for example) is to know the relations between the eigenvalues of A, B and A + B. If A and B are diagonalizable and commute, then they are simultaneously diagonalizable and the spectrum of their sum is well known. In this text we will study the more delicate case of Hermitian matrices with complex coefficients and the related Horn conjecture. This problem and Horn's conjecture (proven in 1999) are exposed in some famous papers [START_REF]Eigenvalues, invariant factors, highest weights, and Schubert calculus[END_REF][START_REF] Brion | Restriction de représentations et projections d'orbites coadjointes [d'après Belkale, Kumar et Ressayre[END_REF][START_REF] Kumar | A survey of the additive eigenvalue problem (with appendix by M. Kapovich)[END_REF]. A pedagogical introduction can be found in [START_REF] Bhatia | Algebraic geometry solves an old matrix problem[END_REF].

Since Hermitian matrices have real eigenvalues, we will see the spectrum of these matrices as tuples with real entries ranked in decreasing order. The previous question can now be reformulated : what are the families (Λ 1 , Λ 2 , Λ 3 ) of real tuples such that Λ 1 (resp. Λ 2 ) is the spectrum of an Hermitian matrix A (resp. B) and that Λ 3 is the spectrum of -(A + B) ? In 1962, Alfred Horn conjectured about the fact that a set of finite inequalities defined by induction are sufficient to describe all possible spectrums for Hermitian matrices and their sum [START_REF] Horn | Eigenvalues of sums of Hermitian matrices[END_REF].

For all i ∈ N * we denote [i] the set of integers j ∈ N such that 1 ⩽ j ⩽ i. Let r ∈ N * . We denote by R r ⩾ the set of all λ := (λ(i)) i∈[r] ∈ R r such that λ(1) ⩾ • • • ⩾ λ(r). For all λ ∈ R r ⩾ we denote O λ the set of all hermitian matrices of order r and of real spectrum with multiplicities λ (this notation comes from the fact that this set is an orbit for the conjugation by the unitary matrices subgroup). We will consider an arbitrary number of matrices, not necessarily three : let s ∈ N * . We define the Kirwan cone as the set of all Λ ∈ (R r ⩾ ) s such that there exists s hermitian matrices with a sum equal to 0 and spectrums corresponding to the s real sequences Λ 1 , . . . , Λ s :

LR(r, s) := Λ ∈ (R r ⩾ ) s | 0 ∈ s l=1 O Λ l .
We will consider linear equations for the elements of the Kirwan cone LR(r, s). Let λ ∈ R r , d ∈ [r] and J ⊂ [r] a subset of cardinality d. We denote |λ| := r j=1 λ(j) and |λ| J := j∈J λ(j). We identify J with the unique strictly growing map [d] → [r] of image J and we denote µ(J) := (J(d) -d, . . . , J(1) -1) ∈ R d ⩾ . Let 1 d := (1) k∈[d] be the constant sequence equal to 1. Theorem 1.1 (Horn's conjecture). -Let Λ ∈ (R r ⩾ ) s . The tuple Λ is in LR(r, s) if and only if the two following conditions hold :

1. s l=1 |Λ l | = 0 ; 2. for all d ∈ [r -1] and all s-tuple (J l ) l∈[s] ∈ [r] s of subsets of cardinality d such that (µ(J 1 ), . . . , µ(J s-1 ), µ(J s ) -(s -1)(n -r)1 r ) ∈ LR(d, s), This theorem confirms Horn's conjecture and gives less than r s +1 equations to describe the cone LR(r, s). The inductive description of these inequalities is simple enough to allow us to compute them for small dimensions (see section 5). The first proof of Horn's conjecture is found in two main works by Anton A. Klyachko in 1998 [START_REF] Klyachko | Stable bundles, representation theory and Hermitian operators[END_REF] (using the geometric invariants theory) and by Allen Knutson and Terence Tao in 1999 (about the saturation property using combinatorics) [START_REF] Knutson | The honeycomb model of GL n (C) tensor product I: Proof of the staturation conjecture[END_REF]. A stronger theorem is presented in [START_REF] Belkale | Local systems on P 1 -S for S a finite set[END_REF] and [START_REF] Knutson | The honeycomb model of GL n (C) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone[END_REF] (we will see an application in section 5).

Saturation property.

-There exists an interesting relation between the Kirwan cone LR(r, s) and representation theory [START_REF] Knutson | The honeycomb model of GL n (C) tensor product I: Proof of the staturation conjecture[END_REF]. Let U(r) be the set of all unitary matrices of order r and Z r ⩾ be the semi-group R r ⩾ ∩ Z r . For all λ ∈ Z r ⩾ we denote V (λ) the irreducible representation of U(r) with the highest weight λ. For any representations V of U(r) we denote V U(r) the linear subspace of U(r)-invariant vertors of V . Theorem 1.2 (Knutson-Tao, Saturation property)

For all Λ ∈ (Z r ⩾ ) s , Λ ∈ LR(r, s) if and only if ( s l=1 V (Λ l )) U(r) ̸ = {0}.
The original proof of this theorem can be found in [START_REF] Knutson | The honeycomb model of GL n (C) tensor product I: Proof of the staturation conjecture[END_REF] and another can be found in [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF]. A generalization to quivers was presented in [START_REF] Derksen | Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients[END_REF].

Remark 1.3. -We denote LR Z (r, s) := LR(r, s) ∩ (Z r ) s . The set LR Z (r, 3)
is the semigroup made of all highest weights triplets (λ, µ, ν) such that the Littlewood-Richardson coefficient dim (V (λ) ⊗ V (µ) ⊗ V (ν)) U(r) is strictly positive.

1.

3. A refinement of Horn's conjecture. -Remark that we are handling many s-sequences of elements in the same set. For any set X, we consider the natural left action of the symmetric group S s on the Cartesian product X s : for all σ ∈ S s and x :

= (x l ) l∈[s] ∈ X s we denote σ • x := (x σ -1 (l) ) l∈[s] . Let σ ∈ S n . Definition 1.4. -Let X be a set. An element x := (x l ) l∈[s] ∈ X s is σ-stable if x = σ • x. For all subset A ⊂ X s , the set of all σ-stable elements in A is denoted A σ .
Theorem 1.1 gives a set of inequalities to describe the cone LR(r, s) ; checking these inequalities on the σ-stable s-tuples J allows us to describe LR(r, s) σ . But theorem 1.5 below assures that a smaller number of these inequalities is enough to describe the cone LR(r, s) σ . Remark that the second condition in theorem 1.1 is not adapted to σ-stability ; for all s-stuples J := (J l ) l∈[s] ∈ [r] s of subsets of cardinality d we denote

M(J ) := µ(J l ) -(r -d) s -1 s 1 d l∈[s] ∈ (R d ⩾ ) s .
This new notation is interesting because J is σ-stable if and only if M(J ) is σ-stable. Theorem 1.5 (Refined Horn's conjecture). -Let Λ ∈ (R r ⩾ ) s be a σstable tuple. The tuple Λ is in LR(r, s) σ if and only if the following conditions hold :

1.

s l=1 |Λ l | = 0 ; 2. for all d ∈ [r -1] and all s-tuple (J l ) l∈[s] ∈ [r] s of subsets of cardinality d such that M(J ) ∈ LR(d, s) σ , s l=1 |Λ l | J l ⩽ 0. Proof. -See subsection 4.2.
This is the main result of this paper. Some computations are presented in section 5 in the case s = 3 and σ = (1 2 3) (i.e. with triplets of three equal spectrums). For example, theorem 1.1 gives 539 inequalities to describe LR(6, 3) while theorem 1.5 gives only 10 of them to describe the triplets

(λ, λ, λ) of LR(6, 3). Example 1.6. -Let λ ∈ R 6 . The representation V (λ)⊗V (λ)⊗V (λ) admits a nonzero U (6)-invariant vector if and only if λ(1) ⩾ • • • ⩾ λ(6) and        λ(1) + λ(2) + λ(3) + λ(4) + λ(5) + λ(6) = 0, λ(1) + λ(5) + λ(6) ⩽ 0, λ(2) + λ(4) + λ(6) ⩽ 0, ( * ) λ(3) + λ(4) + λ(5) ⩽ 0.
Remark that inequalitie ( * ) is in fact a consequence of the others : we can remove ( * ) from the example. This shows that the number of inequalities given by theorem 1.5 is not minimal. We will see how to reduce this number again in Belkale's theorem 2.7 and in the main theorem 3.1.

Acknowledgements. -I would like to thank my doctoral advisor Paul-Émile Paradan for his ideas and constant help during the writing of this paper.

Notations and settings. -Most of the notations we use come from [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF].

-We fix s ∈ N * (the size of the tuples we study) and σ ∈ S s (the permutation preserving the tuples we want to describe). -For all n, r ∈ N * such that n ⩾ r, Subsets(r, n) denote the set of all subsets of [n] made of r elements and Subsets(d, r, s) denote the Cartesian product Subsets(d, r) s . -In all this paper, m, d, r, n will be positive integers satisfying the three inequalities m ⩽ d ⩽ r ⩽ n, I (resp. J ) (resp. K) will be an element of Subsets(r, n, s) (resp. of Subsets(d, r, s)) (resp. of Subsets(m, d, s)), U will be a complex vector space of finite dimension n and V will be a r-dimensional linear subspace of U .

-We denote by Gr(r, U ) the Grassmannian of all r-dimensional linear subspaces of U and we denote by Flag(U ) the set of all complete flags E := (E(i))

Ω I (E) := {V ∈ Gr(r, U ) | ∀j ∈ [r], rk(V ∩ E(I(j))) ⩾ j} the corresponding Schubert subvariety. Its dimension is dim I := r j=1 (I(j) -j)
(see the beggining of section 4 in [START_REF]Eigenvalues, invariant factors, highest weights, and Schubert calculus[END_REF] or lemma 3.1.7 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF]) and its class in the integral cohomology ring H * (Gr(r, U )) is denoted by ω I ; the dimension dim I and the cohomology class ω I only depends on I and does not depend on the flag E.

The class of the point [pt] is in H 2r(n-r) (Gr(r, U )) and, more precisely,

H 2r(n-r) (Gr(r, U )) = Z[pt]. With a = r(n -r) -dim I, ω I ∈ H 2a (Gr(r, U )).
We will be interested in the product of such cohomology classes. Definition 2.1. -Let Intersecting(r, n, s) (resp. Intersecting 0 (r, n, s)) (resp. Intersecting 00 (r, n, s)) be the set of all I ∈ Subsets(r, n, s) such that s l=1 ω I l is not null (resp. is a multiple of the class of a point) (resp. is the class of a point) in H * (Gr(r, U )). The elements of Intersecting(r, n, s) are intersecting.

This is the point of view from [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF]. In [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF], I is said to be intersecting if, for all E := (E l ) l∈[s] ∈ Flag(U ) s , the intersection

Ω I (E) := s l=1 Ω I l (E l )
is nonempty : these two definitions are equivalent. The first one can be seen as the most natural one and the second one as the easier one.

edim I := r(n -r) - s l=1 (r(n -r) -dim I l ).
A geometrical interpretation of the expected dimension is discussed in subsection 1.1 of [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF] and in lemma 2.15 of [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF]. Let I ∈ Subsets(r, n, s), E ∈ Flag(U ) and E ∈ Flag(U ) s . Since Ω I (E) is of dimension dim I, this subvariety is locally described by dim Gr(r, U ) -dim I = r(n -r) -dim I equations. Assume that I is intersecting, i.e. Ω I (E) is nonempty. Let C be an irreducible component of Ω I (E). It is locally described by s l=1 (r(n -r) -dim I l ) equations and its dimension is at least edim I. In fact, there is an equality if the intersection is proper. As we see in definition-proposition 2.14, there is

a dense subset Good(U, s) of Flag(U ) s such that, if E ∈ Good(U, s) and I is intersecting, any irreducible component of Ω I (E) is of dimension edim I. Remark 2.5. -We have Intersecting 0 (r, n, s) = {I ∈ Intersecting(r, n, s) | edim I = 0} .
2.1.2. The theorem. -Belkale's theorem tells us that a tuple is intersecting (a geometric property) if and only if it satisfies Horn's inequalities (an arithmetic property defined by induction). We give a refinement of Belkale's theorem in subsection 3.1.

For all d ∈ [r] and J ∈ Subsets(d, r), we denote IJ the composition of maps I • J ∈ Subsets(d, n). Let I ∈ Subsets(r, n, s). For all d ∈ [r] and J ∈ Subsets(d, r, s) we denote IJ := (I l J l ) l∈[s] wich is an element of Subsets(d, n, s). Example 2.6. -If I = {2, 3, 4} and J = {2}, then IJ = {3}. Theorem 2.7 (Belkale). -Let I ∈ Subsets(r, n, s). The following assertions are equivalent.

1. The tuple I is intersecting.

2. We have edim I ⩾ 0 and, for all d ∈ [r -1] and J ∈ Intersecting(d, r, s), edim IJ ⩾ edim J .

3. We have edim I ⩾ 0 and, for all d ∈ [r-1] and J ∈ Intersecting 0 (d, r, s), edim IJ ⩾ 0.

4. We have edim I ⩾ 0 and, for all d ∈ [r-1] and J ∈ Intersecting 00 (d, r, s), edim IJ ⩾ 0.

Proof. -See subsection 2.3

In [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF], the fact that the expected dimension is non negative is the codimension condition (0.1). Using example 2.3, we could have included this first inequality in the set of the others by taking d = r (we will use this in subsection 2.3).

Thanks to remark 2.5, the third assertion of the theorem allows us to compute Intersecting(r, n, s) by induction with an easy computation. In [START_REF] Ressayre | A Cohomology-Free Description of Eigencones in Types A, B, and C[END_REF], the author gives "an inductive algorithm to decide if a given Littlewood-Richardson coefficient equals to one or not". In section 5 we will use [Buc] to do some computations.

Main objects and relations

. -Here we present a very brief resume of the objects used to prove Belkale's theorem 2.7. We are following the presentation made in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF]. Let n ∈ N * and U a complex vector space of finite dimension n.

2.2.1. Intersecting tuples. -Let r ∈ [n]. Lemma 2.8. -For all d ∈ [r], Intersecting(r, n, s) • Intersecting(d, r, s) ⊂ Intersecting(d, n, s).
This comes from proposition 1 in [START_REF]Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients[END_REF] or from lemma 2.16 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF]. Lemma 2.9. -For all I ∈ Intersecting(r, n, s), edim I ⩾ 0. This is lemma 4.2.6 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF] which is proven using dominance in algebraic geometry.

2.2.2.

Harder-Narasimhan lemma on slopes. -First, we want to prove that an intersecting tuple satisfies Horn's inequalities. Definition 2.10.

-Let E ∈ Flag(U ) s , r ∈ [n] and V ∈ Gr(r, U ). The Schubert position of V with respect to E is the tuple Pos(V, E) ∈ Subsets(r, n, s) such that, for all l ∈ [s] and j ∈ [r], Pos(V, E) l (j) = min j ′ ∈ [r] | dim E l (j ′ ) ∩ V = j .
For all I ∈ Subsets(r, n, s) we denote

Ω 0 I (E) := {V ∈ Gr(r, U ) | Pos(V, E) = I} . Definition 2.11. -Let r ∈ [n] and θ ∈ (R r ) s . The slope associated to θ is defined by, for all d ∈ [r] and all J ∈ Subsets(d, r, s), slope θ (J ) := 1 d s l=1 j∈J l θ l (j) ∈ R. For all V ∈ Gr(r, U ), F ∈ Flag(V ) s and W ∈ Gr(d, V ) we denote slope θ (V, F) := slope θ (Pos(V, F)).
The Harder-Narasimhan lemma 2.12 refers to a classic method used in algebraic geometry. Here, it allows us to compute expected dimensions in a convenient way. Remark that, with the notations µ, S, V, F, w and Ṽ from section 6 of [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF], µ(S, Ṽ ) = -slope -w (V, F). Lemma 2.12 (Harder-Narasimhan). -Let r ∈ [n], V ∈ Gr(r, U ) a linear subspace, F ∈ Flag(V ) s and θ ∈ (R r ) s such that, for all l ∈ [s], θ l is increasing. There exists a unique linear subspace W * of V such that

slope θ (W * , F) = min W ∈Gr(V ),W ̸ ={0} slope θ (W, F ) =: m * , 0 < dim W * = max W ∈Gr(V ),slope θ (W,F )=m * dim W =: d * .
This is a slightly stronger result than the one found in lemma 4.3.4 from [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF] but the proof is exactly the same. A version of the Harder-Narasimhan lemma is also used in subsection 6.1 of [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF].

For all r ∈ [n], I ∈ Subsets(r, n) and I ∈ Subsets(r, n, s) we denote

λ(J) := (n -r + j -I(j)) j∈[r] ∈ R r ⩾ and Λ(I) := (λ(I 1 ), . . . , λ(I s-1 ), (j -I s (j)) j∈[r] ) which is an element of (R r ⩾ ) s . A link with M is seen in remark 4.5. Lemma 2.13. -Let r ∈ [n] and d ∈ [r]. Let I ∈ Subsets(r, n, s) and J ∈ Subsets(d, r, s). Then we have edim IJ -edim J = d slope -Λ(I) (J ).
It corresponds to the equation following equation (6.1) in [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF]. This is also lemma 4.3.9 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF], proven by a direct computation. Definition-Proposition 2.14. -There is a dense subset Good(U, s) of Flag(U ) s which satisfies the following properties.

1. For all E ∈ Good(U, s) and r ∈ [n],

Intersecting(r, n, s) = {Pos(V, E); V ∈ Gr(r, U )} .

Let r ∈

[n] and I ∈ Intersecting(r, n, s). For all E, E ′ ∈ Good(U, s), Ω 0 I (E) and Ω 0 I (E ′ ) have the same number of irreducible components and each one of them is of dimension edim I.

3. For all I ∈ Intersecting(r, n, s) and E ∈ Good(U, s),

Ω 0 I (E) is dense in Ω I (E).
This comes from propositions 1.1 and 2.3 from [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF]. It is also proven in lemma 4.3.1 from [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF].

Remark that, because of the geometrical interpretation of the expected dimension we discussed after definition 2.4, this proposition gives us lemma 2.9 again. Lemma 2.15. -Let r ∈ [n] and I ∈ Subsets(r, n, s) such that, for all d ∈ [r] and J ∈ Intersecting 00 (d, r, s), edim IJ ⩾ 0. Then we have ∀d ∈ [r], ∀J ∈ Intersecting(d, r, s), edim IJ ⩾ edim J .

A proof can be found in section 6.1 from [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF]. This lemma is also proposition 4.3.10 from [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF] (with the remark following it). It is a consequence of the Harder-Narasimhan lemma 2.12, lemma 2.13 and definitionproposition 2.14 : we will adapt this proof to obtain the refined lemma 3.9. This is corollary 4.3.11 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF]. We will adapt this proof to obtain lemma 3.10. Remark that lemma 2.9 is a direct consequence of example 2.3 and lemma 2.16.

2.2.3.

Dimensions and positions of any tuple. -We have already defined the expected dimension of a tuple and we will see two other dimensions : the true dimension in definition 2.18 and the kernel dimension in definition 2.21. Both are integers. Finally, we introduce the kernel position of a given tuple in definition 2.23 : it is another tuple. Let r ∈ [n] and I ∈ Subsets(r, n, s).

Let d ∈ [r]. For all I ∈ Subsets(r, n) and J ∈ Subsets(d, r) we denote

I J := {(IJ)(k) -J(k) + k; k ∈ [d]} ∈ Subsets(d, n -r + d).
For all J ∈ Subsets(d, r, s) we denote

I J := (I J l l ) l∈[s] . Lemma 2.17. -Let m ∈ [d], J ∈ Subsets(d, r, s) and K ∈ Subsets(m, d, s). We have edim I J K -edim K = edim (IJ K) -edim(J K).
This is proven by a direct computation in relation 4.2.10 from [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF]. Let V ∈ Gr(r, U ) and Q ∈ Gr(n -r, U ) such that U = V ⊕ Q. We denote

B := Flag(V ) s × Flag(Q) s .
As in definition 2.2 from [START_REF] Belkale | Geometric proofs of Horn and saturation conjectures[END_REF], we denote L(V, Q) the C-linear maps from V to Q and, for all (F, G) ∈ B, the nonempty set

L I (F, G) := s l=1 {φ ∈ L(V, Q) | ∀j ∈ [r], φ(F l (j)) ⊂ G l (I l (j) -j)} .
We also denote This comes from the last relation of section 3 in [START_REF] Sherman | Geometric proof of a conjecture of King, Tollu, and Toumazet[END_REF]. This is also equation 5.3.3 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF], proven with topological arguments. Remark 2.27. -While the algebraic varieties B, P(I), P t (I) and P kt (I) depends on V, Q, U (and I), the integers edim I, tdim I, kdim I and the tuple kPos(I) only depends on I.

P(I) := {(F, G, φ) ∈ Flag(V ) s × Flag(Q) s × L(V, Q) | φ ∈ L I (F, G)} .
2.3. The proof. -Let (1.), (2.), (3.) and (4.) be the four assertions of Belkale's theorem 2.7.

What we have already proven. -Let r ∈ N * and n ⩾ r. By lemma 2.16, (1.) ⇒ (2.). By remarks 2.2 and 2.5, (2.) ⇒ (3.) ⇒ (4.). To prove the last implication (4.) ⇒ (1.) we follow the proof by induction on r ∈ N * given in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF].

Starting the induction. -For all r ∈ N * and n ⩾ r, we define Horn(r, n, s) as the set of all I ∈ Subsets(r, n, s) such that ∀d ∈ [r], ∀J ∈ Intersecting 00 (d, r, s), edim IJ ⩾ 0.

Remark that, because of example 2.3, for all I ∈ Subsets(r, n, s), ∀J ∈ Intersecting 00 (r, r, s), edim IJ ⩾ 0 ⇔ edim I ⩾ 0. By example 4.3.12 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF], H(1) is true. Let r ⩾ 2 such that H(r -1) is true. Let n ⩾ r.

Contraposition and conclusion. -Let I ∈ Subsets(r, n, s) not intersecting and d := kdim I. We want to prove that I does not satisfy one of the Horn inequalities. If edim I < 0, I is not in Horn(r, n, s) (using example 2.3). Assume that edim I ⩾ 0.

By lemma 2.22, d ∈ [r -1] and we can consider the kernel position J := kPos I. Using lemma 2.19 and Sherman's relation 2.26, since I is not intersecting, then I J is not intersecting either. By induction hypothesis H(r-1), since d ⩽ r -1, I J is not in Horn(d, n -r + d, s) : there exists m ∈ [d] and K ∈ Intersecting 00 (m, d, s) such that edim I J K < 0. Using the formula of lemma 2.17, edim I(J K) -edim J K < 0.

By lemma 2.24, J is intersecting. But K is also intersecting : by lemma 2.8 we deduce that J K is intersecting. Thus, by the contrapositive of lemma 2.15, there exists d ∈ [r] and J ∈ Intersecting 00 ( d, r, s) such that edim I J < 0 : hence I is not in Horn(r, n, s). From this we deduce H(r) and, by induction, Belkale's theorem 2.7. Remark that, for all ρ ∈ S s , the map I ∈ Subsets(r, n, s) σ → ρ • I ∈ Subsets(r, n, s) ρσρ -1 is a bijection. Let c 1 , . . . , c p be the disjoint cycles of σ ranked by increasing length

A refinement of

l 1 ⩽ • • • ⩽ l p . We have Subsets(r, n, s) σ = p m=1
Subsets(r, n, s) cm and the only thing interesting about σ is its type (l 1 , . . . , l p ) which is a partition of the integer s = l 1 + • • • + l p . From now on, we could replace σ by its conjugate (1

• • • l 1 ) . . . (s -l p • • • n).
The theorem below is a refinement of Belkale's theorem 2.7 which will allow us to prove theorem 1.5 using Belkale's method.

Theorem 3.1. -Let I ∈ Subsets(r, n, s) σ . The following assertions are equivalent.

1. The tuple I is intersecting.

2. We have edim I ⩾ 0 and, for all J ∈ r-1 d=1 Intersecting(d, r, s) σ , edim IJ ⩾ edim J .

3. We have edim I ⩾ 0 and, for all J ∈ r-1 d=1 Intersecting 0 (d, r, s) σ , edim IJ ⩾ 0.

4. We have edim I ⩾ 0 and, for all J ∈ r-1 d=1 Intersecting 00 (d, r, s) σ , edim IJ ⩾ 0. Proof. -The first assertion comes from the fact that the product is commutative in definition 2.1. The second one is a direct consequence of lemma 2.8 and remark 3.5.

In relation with slopes and the Harder

-Narasimhan lemma. - Remark 3.7. -Let θ ∈ (R r ) s . For all d ∈ [r] and J ∈ Subsets(d, r, s), slope θ (J ) = slope σ•θ (σ • J ) and slope -Λ(I) (J ) = µ -Λ(σ•I) (σ • J ).
Lemma 3.8. -Let θ, F, d * and W * as in the Harder-Narasimhan lemma 2.12. Assume that F ∈ Good(U, s) and that, for all J ∈ Subsets(d * , r, s), we have µ θ (J ) = µ θ (σ • J ). Then the position Pos(W * , F) is intersecting and σ-stable.

Proof. -Let J * := Pos(W * , F). Since F is good, J * is intersecting. In addition to this, σ • J * is also intersecting and there exists W ∈ Gr(d * , V ) such that σ • J * = Pos(W, F ). The linear subspace W is of dimension d * and, by hypothesis,

slope θ (W * , F) = slope θ (W, F)
By unicity in the Harder-Narasimhan lemma 2.12, W = W * hence J * = σ • J * .

Lemma 3.9. -Assume that I is σ-stable and that, for all d ∈ [r] and J ∈ Intersecting 00 (d, r, s) σ , edim IJ ⩾ 0. Then we have

∀d ∈ [r], ∀J ∈ Intersecting(d, r, s) σ , edim IJ ⩾ edim J . Proof.
-Using example 2.3, the conclusion of the lemma holds for d = r. Assume that there exists d ∈ [r -1] and J ∈ Intersecting(d, r, s) σ such that edim IJ < edim J i.e., using lemma 2.13, slope -Λ(I) (J ) < 0.

Let V ∈ Gr(r, U ) and F ∈ Good(V, s). Since J is intersecting, there exists a nonzero W ∈ Gr(V ) such that J = Pos(W, F). From this we deduce that slope -Λ(I) (W, F) < 0.

Using the Harder-Narasimhan lemma 2.12, there exists a unique nonzero W * of minimal slope m * with respect to -Λ(I) and maximal dimension d * . Remark that, because of the last equation, m * < 0. By remark 3.7 and lemma 3.8,

J * := Pos(W * , F) ∈ Intersecting(d * , r, s) σ . For all W ′ ∈ Ω 0 J * (F), dim W ′ = d * and slope -Λ(I) (W ′ , F ) = m * . Since W * is unique, Ω 0 J * (F) = {W * } .
From this and definition-proposition 2.14 we have Ω J * (F) = {W * }. Since F is generic, s l=1 ω (J * ) l is the class of a point, i.e. J * ∈ Intersecting 00 (d * , n, s). In particular, edim J * = 0 i.e. edim IJ * = edim IJ * -edim J * i.e., using lemma 2.13,

edim IJ * = d * m * < 0.
This is in contradiction with the hypothesis on I.

Lemma 3.10 (Horn inequalities). -Assume that I is intersecting and σstable. Let d ∈ [r] and J ∈ Intersecting(d, r, s) σ . We have edim IJ ⩾ edim J . Proof. -For all d ′ ∈ [r] and all J ′ ∈ Intersecting(d ′ , r, s) σ , using lemma 2.8 IJ ′ is intersecting and, using lemma 2.9, edim IJ ′ ⩾ 0. Particularly, I satisfies the hypothesis of lemma 3.9 and edim IJ ⩾ edim J .

3.2.3.

In relation with tuples dimensions and the kernel position. -We use the notations of subsubsection 2.2.3. Lemma 3.11. -1. For all E ∈ Flag(U ) s and (F, G) ∈ B, we have 4. Assume that tdim I ⩾ 1. We have kPos(σ•I) = σ•kPos(I). In particular, if I is σ-stable, then so is kPos I. Proof. -We prove the first point of the lemma by reindexing the intersections defining the sets

Ω 0 σ•I (σ • E) = Ω 0 I (E) and L σ•I (σ • F, σ • G) = L I (F, G). 2. Let (F, G) ∈ B and φ ∈ L(V, Q). The triplet (F, G, φ) is in P(I) (resp. P t (I)) (resp. P kt (I)) if and only if (σ • F, σ • G, φ) is in P(σ • I) (resp. P t (σ • I)) (resp. P kt (σ • I)).
Ω 0 σ•I (σ • E) and L σ•I (σ • F, σ • G).
In the same way, we prove that the expected dimension is invariant by reindexing the defining sum. In the rest of this proof we consider the now proven equation

( * ) ∀(F, G) ∈ B, L σ•I (σ • F, σ • G) = L I (F, G).
We have Let (F, G) ∈ B and φ ∈ L(V, Q). We have seen that kdim I = kdim(σ • I) and that (F, G, φ) ∈ P t (I) if and only if

B = (σ -1 • F, σ -1 • G); (F, G) ∈ B .
(σ • F, σ • G, φ) ∈ P t (σ • I). Hence (F, G, φ) ∈ P kt (I) if and only if (σ • F, σ • G, φ) ∈ P kt (σ • I).
From this we deduce that, for all l ∈ [s] and k ∈ [kdim I], the integers kPos(I) σ -1 (l) (k) and kPos(σ • I) l (k) are the minimum of the same set, hence are equal. Finally, kPos(σ • I) = σ • kPos(I).

Remark 3.12. -Using lemma 2.24 and the fourth point of lemma 3.11, if I is σ-stable and kdim I ⩾ 1, then kPos(I) ∈ Intersecting(kdim I, n, s) σ .

3.3. Adaptation of Belkale's proof. -We prove the main theorem 3.1 juste like we proved Belkale's theorem 2.7 in subsection 2.3, using the point of view of [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF].

What we have already proven. -Let r ∈ N * and n ⩾ r. By lemma 3.10, (1.) ⇒ (2.). By remarks 2.2 and 2.5, (2.) ⇒ (3.) ⇒ (4.). We prove the last implication (4.) ⇒ (1.) by induction on r ∈ N * .

Sarting the induction. -In subsection 3.2 we have seen the adaptation of the necessary tools to the σ-stable case : the main ones are lemma 3.9 (playing the role of lemma 2.15) and remark 3.12 (about the σ-stability of the kernel position). For all r ∈ N * and n ⩾ r, we define Horn σ (r, n, s) as the set of all I ∈ Subsets(r, n, s) σ such that

∀d ∈ [r], ∀J ∈ Intersecting 00 (d, r, s) σ , edim IJ ⩾ 0.
In fact, because of remark 3.4, for all I ∈ Subsets(r, n, s),

∀J ∈ Intersecting 00 (r, r, s) σ , edim IJ ⩾ 0 ⇔ edim I ⩾ 0.
For all r ∈ N * we denote by H(r) the assertion

∀d ∈ [r], ∀n ⩾ d, Horn σ (d, n, s) ⊂ Intersecting(d, n, s) σ . By Belkale's theorem 2.7 in the case r = 1, for all n ∈ N * , Horn(1, n, s) is a subset of Intersecting(1, n, s) hence, by intersecting with Subsets(1, n, s) σ , H(1) is true. Let r ⩾ 2 such that H(r -1) is true. Let n ⩾ r.
Contraposition and conclusion. -Let I ∈ Subsets(r, n, s) σ not intersecting and d := kdim I. We want to prove that I does not satisfy one of the Horn inequalities. Just as in subsection 2.3, we can suppose that d ∈ [r -1] and we know that, with J := kPos(I), I J is not intersecting. But, using remarks 3.12 and 3.5, I J is σ-stable and we can use the induction hypothesis H(r -1) : since d ⩽ r -1, I J is not in Horn σ (d, n -r + d, s). Hence there exists m ∈ [d] and K ∈ Intersecting 00 (m, d, s) σ such that edim I J K < 0. Using the formula of lemma 2.17, edim I(J K) -edim J K < 0. By remarks 3.12 and 3.5 again, J K is σ-stable. In addition to this, just like in subsection 2.3, J K is intersecting. Thus, by the contrapositive of lemma 3.9, there exists d ∈ [r] and J ∈ Intersecting 00 ( d, r, n) σ such that edim I J < 0 : hence I is not in Horn σ (r, n, s). From this we deduce H(r) and, by induction, the main theorem 3.1.

The Kirwan cone with repetitions

4.1. Back to Horn's conjecture. -The link between Belkale's theorem 2.7 and Horn's conjecture is presented in subsection 6.3 of [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF]. We will use it to prove corollary 4.7. Let n ∈ N * and r ∈ [n]. Theorem 4.1 (Klyachko, Horn inequalities). -For all Λ ∈ (R r ⩾ ) s , the following assertions are equivalent.

1. The tuple Λ is in the Kirwan cone LR(r, s). Proof. -We adapt the proof of corollary 6.3.3 from [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF]. Let Λ be a σ-stable element of (R r ⩾ ) s . Assume that Λ ∈ LR(r, s) : just as in [BVW18] (corollary 2.13), the trace s l=1 |Λ l | is null and, for all d ∈ [r -1], Λ satisfies the Horn inequalities for all tuples in Intersecting(d, r, s) hence for all tuples in Intersecting(d, r, s) σ . We deduce that (1.) ⇒ (2.). The second implication (2.) ⇒ (3.) is clear.

The tuple Λ satisfies the Horn inequalities :

To prove that (3.) ⇒ (1.), we use the main theorem 3.1. First we consider the case of integers. Assume that Λ ∈ (Z r ⩾ ) s , that s l=1 |Λ l | = 0 and that, for all d ∈ [r -1] and J ∈ Intersecting 00 (d, r, s

) σ , s l=1 |Λ l | J l ⩽ 0. Let a ∈ N * be greater than |Λ 1 (s)| , . . . , |Λ n-1 (s)| and |Λ n (1)| . Let Λ := (Λ 1 + a1 r , . . . , Λ s-1 + a1 r , Λ s -(s -1)a1 r ) ∈ (Z r ⩾ ) s .
By definition of a and Λ, Λs (1) ⩽ 0 and, for all l ∈ [s-1], Λl (r) ⩾ 0. In addition to this, with n := r + as, n -r ⩾ max{ Λ1 (1), . . . , Λs-1 (1), -Λs (r)}. Hence there exists (a unique) I ∈ Subsets(r, n, s) such that Λ = Λ(I). We now want to show that I is in fact a σ-stable intersecting tuple of expected dimension 0. First, remark that Λ = (λ(I l ) -a1 r ) l∈ [s] . Then, since Λ is σ-stable, I is σ-stable.

Secondly, remark that, for all d ∈ [r] and all J ∈ Subsets(d, r), |1 r | J = d. From this we know that s l=1 | Λl | = s l=1 |Λ l | hence, using lemma 2.13, edim I = 0. In the same way, for all d ∈ [r -1] and J ∈ Intersecting 00 (d, r, s) σ ,

s l=1 | Λl | J l = s l=1 |Λ l | J l hence edim IJ ⩾ edim J = 0.
Finally, I is of expected dimension 0 and, by the main theorem 3.1, I is intersecting. By corollary 4.6, Λ ∈ LR(r, s) and, by remark 4.4, Λ ∈ LR(r, s).

Let K σ (r, s) be the set of all σ-stable Λ ∈ (R r ⩾ ) s such that s l=1 |Λ l | = 0 and, for all d ∈ [r -1] and J ∈ Intersecting 00 (d, r, s) σ , s l=1 |Λ l | J l ⩽ 0. At the beginning of the proof we have seen that LR(r, s) σ ⊂ K σ (r, s). Then we have proven that K σ (r, s) ∩ (Z r ) s is a subset of LR(r, s) σ . But LR(r, s) is invariant under rescaling by any

x ∈ R + so K σ (r, s) ∩ (Q r ) s ⊂ LR(r, s) σ . Since K σ (r, s) is a polyhedral cone, its euclidean closure is K σ (r, s) ∩ (Q r ) s = K σ (r, s).
On the other hand, LR(r, s) σ is closed for the euclidean topology. Finally, we have the converse inclusion K σ (r, s) ⊂ LR(r, s) σ and LR(r, s) = K σ (r, s).

A. MÉDOC

Remark 4.8. -Using corollary 4.7, if Λ ∈ (R r ⩾ ) s is σ-stable and satisfies the Horn inequalities for a set of tuples containing r-1 d=1 Intersecting 00 (d, r, s) σ , then Λ is in LR(r, s). See example 4.9 below. Remark 4.9. -As in example 4.3, if s = 3 and σ = (1 2 3), corollary 4.7 and the main theorem 3.1 gives a convenient description of the set LR Z (r, 3) σ . Let λ ∈ Z r ⩾ . The triplet (λ, λ, λ) is in LR Z (r, 3) σ if and only if |λ| = 0 and, for all d ∈ [r -1] and J := (J, J, J) ∈ Intersecting 0 (d, r, 3) σ , |λ| J ⩽ 0. Finally, together with corollary 4.10, this gives us an inductive description of LR Z (r, 3) σ . Corollary 4.10. -For all I ∈ Subsets(r, n, s), I ∈ Intersecting 0 (r, n, s) σ if and only if M(I) ∈ LR(r, s) σ . Proof. -The tuple I is σ-stable if and only if M(I) is σ-stable. We conclude by corollary 4.6. Theorem 1.5 is a consequence of corollaries 4.7 and 4.10. This refinement of Horn's conjecture is a consequence of the refinement of Belkale's theorem 2.7.

5.

Examples for s = 3 and σ = (1 2 3) 5.1. Numbers of equations. -In this subsection we consider the following questions for small values of r : how to describe (with a list of inequalities) the triplets (Λ 1 , Λ 2 , Λ 3 ) of real spectra Λ l (1) ⩾ • • • ⩾ Λ l (r) such that there exists Hermitian matrices X 1 , X 2 , X 3 of order r, of spectra Λ 1 , Λ 2 , Λ 3 and of sum X 1 + X 2 + X 3 = 0 ? and if Λ 1 = Λ 2 = Λ 3 ? This correspond to Horn's conjecture with s = 3 and σ = (1 2 3). In the tabular below, we compute some of the following numbers using the main theorem 3.1 and [Buc].

-The integer l 0 (r, 3) is the number of inequalities given by theorem 1.1 to describe LR(r, 3) : the 3(r-1) inequalities Λ l (i+1) ⩾ Λ l (i) (for all l ∈ [3] and i ∈ [r -1]) ; the two inequalities 3 l=1 |Λ l | ⩽ 0 and 3 l=1 |Λ l | ⩾ 0 ; the inequalities 3 l=1 |Λ| J l ⩽ 0 for all J ∈ d∈[r-1] Intersecting 0 (d, r, 3). -The integer l min (r, 3) is the minimal number of inequalities taken from the previous ones to describe the cone LR(r, 3). Using the Belkale-Knutson-Tao-Woodward theorem 5.2, if r ⩾ 2 we remove the inequalities

3 l=1 |Λ| J l ⩽ 0 for all d ∈ [r -1] and J / ∈ Intersecting 00 (d, r, 3). If r = 2, we also remove the three inequalities Λ l (2) ⩾ Λ l (1) (for all l ∈ [3]).
-The integer l 0 σ (r, 3) is the number of inequalities given by theorem 1.5 to describe the cone LR(r, 3) σ , i.e. the element of the form (λ, λ, λ) in LR(r, 3) : the r -1 inequalities λ(i + 1) ⩾ λ(i) (for all i ∈ [r -1]) ; the two inequalities |λ| ⩽ 0 and |λ| ⩾ 0 ; the inequalities 3 l=1 |Λ| J l for all J ∈ d∈[r-1] Intersecting 0 (d, r, 3) σ . Clearly, l σ (r, 3) ⩽ l(r, 3) = l Id (r, 3).

-The integer l 00 σ (r, 3) is the smaller number of inequalities we found in corollary 4.7 to describe the cone LR(r, 3) σ . We remove from the previous inequalities the ones of the form 3 l=1 |Λ| J l ⩽ 0 for all d ∈ [r -1] and J / ∈ Intersecting 00 (d, r, 3). We do not know if they are irredudant. Let Λ ∈ (R r ) 3 . The tuple Λ is in LR(r, 3) if and only if the following conditions hold :

1. for all l ∈ [3] and i ∈ [r -1], Λ l (i + 1) ⩾ Λ l (i) ; 2. In addition to this, all of these inequalities are essential. Proof. -The fact that these inequalities are enough to describe LR(r, s) comes from Klyachko's theorem 4.1. The fact that these inequalities are essential is theorem 4 from [START_REF] Knutson | The honeycomb model of GL n (C) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone[END_REF]. 5.2. Equations. -For all r, d ∈ N such that 1 ⩽ d < r ⩽ 5 we give the list of the elements (J 1 , J 2 , J 3 ) ∈ Intersecting 00 (d, r, 3) (remember example 5.1) up to permutation ; the σ-stable ones are in bold. Other examples can be found in appendix A from [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF] and in subsection 1.3 from [START_REF] Klyachko | Stable bundles, representation theory and Hermitian operators[END_REF]

sl=1

  |Λ l | J l ⩽ 0. Proof. -See subsection 4.1.

  Remark 2.2. -Definition 2.1 gives us Intersecting 00 (r, n, s) ⊂ Intersecting 0 (r, n, s) ⊂ Intersecting(r, n, s). Example 2.3. -We have the simple case Subsets(n, n, s) = Intersecting 00 (n, n, s) = ([n]) l∈[s] . Definition 2.4. -The expected dimension of a tuple I ∈ Subsets(r, n, s) is

  Lemma 2.16 (The Horn inequalities). -Let r ∈ [n] and d ∈ [r]. Let I ∈ Intersecting(r, n, s) and J ∈ Intersecting(d, r, s). Then we have Horn's inequality edim IJ ⩾ edim J .

Definition 2 .

 2 18. -The true dimension of I is min (F ,G)∈B dim L I (F, G) and denoted tdim I. Lemma 2.19. -We have tdim I ⩾ edim I and the tuple I is intersecting if and only if this is an equality. This is the first point of lemma 2.4 and the equivalence (α) ⇔ (γ) of proposition 2.3 from [Bel06]. Example 2.20. -If tdim I = 0 and edim I ⩾ 0, then I is intersecting. By fixing the true dimension we now consider the nonempty (since tdim I is a reached minimum) set P t (I) := {(F, G, φ) ∈ P(I) | dim L I (F, G) = tdim I} . Definition 2.21. -The kernel dimension of I is min (F,G,φ)∈Pt(I) dim Ker φ and is denoted kdim I. Lemma 2.22. -If edim I ⩾ 0 and kdim I ∈ {0, r}, I is intersecting. This comes from lemma 5.3.4 and corollary 5.2.6 in [BVW18]. By fixing the kernel dimension we finally consider the nonempty (since kdim I is a reached minimum) set P kt (I) := {(F, G, φ) ∈ P t (I) | dim Ker φ = kdim I} . Definition 2.23. -Assume kdim I ⩾ 1. The kernel position of I is the unique tuple kPos(I) ∈ Subsets(kdim I, r, s) such that, for all l ∈ [s] and all k ∈ [kdim I], kPos(I) l (k) = min (F ,G,φ)∈P kt (I) Pos(Ker φ, F l )(k). Lemma 2.24. -If 1 ⩽ kdim I ⩽ r -1, kPos(I) is intersecting. This is corollary 5.2.10 in [BVW18]. Remark 2.25. -Using lemma 2.24 and the first point of definitionproposition 2.14, the kernel position of a given tuple is indeed the Schubert position of a linear subspace of U with respect to a sequence of flags on U . Lemma 2.26 (Sherman's relation). -If 1 ⩽ kdim I ⩽ r -1, 0 ⩽ tdim I -edim I ⩽ tdim(I kPos I ) -edim(I kPos I ).

For

  all r ∈ N * we denote by H(r) the assertion ∀d ∈ [r], ∀n ⩾ d, Horn(d, n, s) ⊂ Intersecting(d, n, s).

  Belkale's theorem 3.1. Main theorem. -We have introduced the action of the symmetric group S s in subsection 1.3. Let n ∈ N * and r ∈ [n].

  Proof. -See subsection 3.3.Example 3.2. -If σ = Id, theorem 3.1 is Belkale's theorem 2.7. Example 3.3. -The set Intersecting(5, 10, 3) is made of 718, 738 elements while Intersecting(5, 10, 3) (1 2 3) is made of 49 elements (and none of them is of expected dimension null). Remark 3.4. -As in example 2.3 we haveSubsets(n, n, s) = Intersecting 00 (n, n, s) σ = ([n]) l∈[s] .3.2. The action of the symmetric group. -In this new subsection, we explain how some of the objects and relations of[START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF] (introduced in subsection 2.2) behave accordingly to the action of S s . Let n ∈ N * and r ∈ [n]. Let I ∈ Subsets(r, n, s).3.2.1.In relation with operations on tuples. -We are interested in σstability. Let d ∈ [r] and a tuple J ∈ Subsets(d, r, s). Remark 3.5. -The two operations we have seen on tuples preserves the σ-stability : if I and J are σ-stable, IJ and I J are σ-stable. Lemma 3.6. -The tuple I is intersecting if and only if σ • I is intersecting and Intersecting(r, n, s) σ • Intersecting(d, r, s) σ ⊂ Intersecting(d, n, s) σ .

3.

  The three dimensions on tuples we have seen satisfy edim(σ•I) = edim I, tdim(σ • I) = tdim I and kdim(σ • I) = kdim I.

  From this and equation ( * ) we know that the true dimensions of σ • I and I are the minimum of the same set, hence are equal. Let (F, G) ∈ B and φ ∈ L(V, Q). Because of equation ( * ), (F, G, φ) ∈ P(I) if and only if (σ • F, σ • G, φ) ∈ P(σ • I). From this and equations ( * ) and tdim(σ • I) = tdim I we deduce that (F, G, φ) ∈ P t (I) if and only if (σ • F, σ • G, φ) ∈ P t (σ • I). Thus the kernel dimensions of σ • I and I are the minimum of the same set, hence are equal.

  s l=1 |Λ l | = 0 and, for all d ∈ [r -1] and J ∈ Intersecting(d, r, s), s l=1 |Λ l | J l ⩽ 0. 2. The tuple Λ satisfies the Horn inequalities : s l=1 |Λ l | = 0 and, for all d ∈ [r -1] and J ∈ Intersecting(d, r, s), s l=1 |Λ l | J l ⩽ 0. 3. The tuple Λ satisfies the reduced and symmetric Horn inequalities : s l=1 |Λ l | = 0 and, for all d ∈ [r -1] and J ∈ Intersecting 00 (d, r, s) σ , s l=1 |Λ l | J l ⩽ 0.

  1. -For all d, r ∈ N such that 1 ⩽ d ⩽ r ⩽ 5,Intersecting 0 (d, r, 3) = Intersecting 00 (d, r, 3).

  Theorem 5.2 (Belkale-Knutson-Tao-Woodward)

3

  l=1 |Λ l | = 0 (seen as two inequalities) ; 3. if r ⩾ 3, for all d ∈ [r -1] and all J ∈ Intersecting 00 (d, r, 3),3 l=1 |Λ l | J l ⩽ 0.

.

  7 and this tabular we know that (λ, λ, λ) is inLR(5, 3) if and only if λ(1) ⩾ • • • ⩾ λ(5) and  + λ(2) + λ(3) + λ(4) + λ(5) = 0 λ(2) + λ(5) ⩽ 0 λ(3) + λ(4) ⩽ 0 λ(1) + λ(4) + λ(5) ⩽ 0 λ(2) + λ(3) + λ(5) ⩽ 0Example 5.7. -In example 1.6, we can also see that equation ( * ) is the consequence of the others because ({2, 4, 6} , {2, 4, 6} , {2, 4, 6}) is in Intersecting 0 (3, 6, 2) σ but not in Intersecting 00 (3, 6, 2) σ

  i∈[n] on U . Intersecting tuples. -Let n ∈ N * , r ∈ [n] and I ∈ Subsets(r, n). Let U be a complex vector space of finite dimension n. For all E ∈ Flag(U ) we denote

	2. Belkale's point of view on theorems 1.1 and 1.2
	2.1. Belkale's theorem. -In 2005, Prakash Belkale answered a question
	from William Fulton [Ful98] and proposed a geometric proof of the Horn con-
	jecture [Bel06] using Schubert calculus. Belkale's geometric point of view
	is well adapted to prove the refinement presented in this paper. In 2018,
	Nicole Berline, Michèle Vergne and Michael Walter presented Belkale's proof
	[BVW18] in a different way and the present text is based on this new redac-
	tion.
	2.1.1.

  who shows how some historical equations are consequences of Klyachko's theorem 4.1.Example 5.3. -Description of Intersecting 00 (d, 2, 3) for d ∈ [1]. d J 1 J 2 J 3 1 {1} {2} {2} Example 5.4. -Description of Intersecting 00 (d, 3, 3) for d ∈ [2].Description of Intersecting 00 (d, 4, 3) for d ∈ [3]. {1, 2, 3} {2, 3, 4} {2, 3, 4} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 3, 4} {1, 3, 4} {1, 3, 4} Example 5.6. -Description of Intersecting 00 (d, 5, 3) for d ∈ [4]. {1, 2, 3, 4} {2, 3, 4, 5} {2, 3, 4, 5} {1, 2, 3, 5} {1, 3, 4, 5} {2, 3, 4, 5} {1, 2, 4, 5} {1, 2, 4, 5} {2, 3, 4, 5} {1, 2, 4, 5} {1, 3, 4, 5} {1, 3, 4, 5} Let λ ∈ R 5 . Using corollary 4.

	Example 5.5. -d	{1, 3, 4} {1, 3, 4} J 1 {1, 3, 5}	{2, 3, 5} {2, 4, 5} J 2 {1, 3, 5}	{3, 4, 5} {2, 4, 5} J 3 {3, 4, 5}
	1	{1} {1, 3, 5}	{4} {1, 4, 5}	{4} {2, 4, 5}
		{2} {1, 3, 5}	{3} {2, 3, 5}	{4} {2, 4, 5}
		{3} {1, 4, 5}	{3} {1, 4, 5}	{3} {1, 4, 5}
	2	{1, 2} {1, 4, 5}	{3, 4} {2, 3, 5}	{3, 4} {2, 3, 5}
		{1, 3} {2, 3, 4}	{2, 4} {2, 3, 4}	{3, 4} {3, 4, 5}
		{1, 4} {2, 3, 4}	{1, 4} {2, 3, 5}	{3, 4} {2, 4, 5}
		{1, 4} {2, 3, 5}	{2, 4} {2, 3, 5}	{2, 4} {2, 3, 5}
	4	{2, 3}	{2, 3}	{3, 4}
		{2, 3}	{2, 4}	{2, 4}
	3 d	J 1		J 2		J 3
	1	{1}		{5}		{5}
		{2}		{4}		{5}
		{3}		{3}		{5}
		{3}		{4}		{4}
	2	{1, 2}		{4, 5}		{4, 5}
		{1, 3}		{3, 5}		{4, 5}
		{1, 4}		{2, 5}		{4, 5}
		{1, 4}		{3, 5}		{3, 5}
		{1, 5}		{1, 5}		{4, 5}
		{1, 5}		{2, 5}		{3, 5}
		{2, 3}		{3, 4}		{4, 5}
		{2, 3}		{3, 5}		{3, 5}
		{2, 4}		{2, 4}		{4, 5}
		{2, 4}		{2, 5}		{3, 5}
		{2, 4}		{3, 4}		{3, 5}
		{2, 5}		{2, 5}		{2, 5}
		{2, 5}		{2, 5}		{3, 4}
		d {3, 4}	J 1	J 2 {3, 4}	J 3	{3, 4}
	1 {1} 3 {1, 2, 3}	{3} {3, 4, 5}	{3} {3, 4, 5}
		{2} {1, 2, 4}	{2} {2, 4, 5}	{3} {3, 4, 5}
		2 {1, 2} {2, 3} {2, 3} {1, 2, 5} {1, 4, 5} {3, 4, 5}
		{1, 3} {1, 3} {2, 3} {1, 2, 5} {2, 4, 5} {2, 4, 5}

A. MÉDOC 3. The tuple Λ satisfies the reduced Horn inequalities : s l=1 |Λ l | = 0 and, for all d ∈ [r -1] and J ∈ Intersecting 00 (d, r, s), s l=1 |Λ l | J l ⩽ 0. Proof. -A proof is presented in the first point of corollary 6.3.3 from [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF]. It is also a consequence of corollary 4.7.

For all λ ∈ R r we denote λ * := (-λ(r), . . . , -λ(1)). Remark 4.4. -Let Λ ∈ (R r ) s . Let I r be the identity matrix of order r and (a 1 , . . . , a s ) ∈ R s such that s l=1 a l = 0.

Remark 4.5. -Let I ∈ Subsets(r, n, s). Using previous remark 4.4, we know that Λ(I) ∈ LR(r, s) if and only if M(I) ∈ LR(r, s).

The proof of corollary 6.3.3 in [BVW18] also gives us the following result which allows an inductive description on r ∈ N * of LR(r, s). Corollary 4.6. -For all I ∈ Subsets(r, n, s), I ∈ Intersecting 0 (r, n, s) if and only if M(I) ∈ LR(r, s). Proof. -By remark 4.5 it is enough to prove that I ∈ Intersecting 0 (r, n, s) if and only if Λ(I) ∈ LR(r, s). Assume that I ∈ Intersecting 0 (r, n, s). Using formula 6.3.1 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF], the linear subspace ( n l=1 L(Λ(I) l )) U(s) is nonnull. Hence, by the saturation property seen in Knutson-Tao theorem 1.2, Λ(I) ∈ LR(r, s).

We now prove the converse. Assume that Λ(I) ∈ LR(r, s). By corollary 2.13 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF], it satisfies the Horn inequalities. Using the equations of lemma 4.3.9 in [START_REF] Berline | The Horn inequalities from a geometric point of view[END_REF], this means that I ∈ Horn(r, n, s). We conclude by Belkale's theorem 2.7. Theorem 1.1 is a consequence of Klyachko's theorem 4.1 and corollary 4.6. -For all σ-stable Λ ∈ (R r ⩾ ) s , the following assertions are equivalent.

A consequence of the refinement of

1. The tuple Λ is in the Kirwan cone LR(r, s).


Amira Bouabid
email: amira.bouabid@gmail.com

Sophie Lepreux

Christophe Kolski

Distributed tabletops: Study involving two RFID tabletops with generic tangible objects

Keywords: Tangible interaction, tabletop, distributed UI, remote collaboration, tangible object, tangiget, RFID

This paper describes a study on an innovative system designed to support remote collaborative. This system is a game running on tabletops with tangible interaction. Twelve test groups, each composed of three participants, tested a distributed application for learning and recognition of colors. We propose a set of generic tangible objects. They model a set of collaborative styles which are possible between users. Our goal is to obtain objects that provide remote collaboration among users of such interactive tabletops. This study is supported by observations, trace analysis and questionnaires. In this study, we analyze if the use of generic objects is easy and understandable by users in case of remote collaboration. The user satisfaction when using the distributed tangible tabletops is also studied.

Introduction

Collaborative work proves important within a team, or more generally within a group of users. In fact, team people often needs to exchange ideas [START_REF] Qin | uEmergency: A Collaborative System for Emergency Management on Very Large Tabletop[END_REF], to work on common tasks [START_REF] Robinson | Distributed Tabletops: Supporting Remote and Mixed-Presence Tabletop Collaboration[END_REF] or to be informed about the progress of a task [START_REF] Yamashita | Improving Visibility of Remote Gestures in Distributed Tabletop Collaboration[END_REF]. Much work has been done on the subject concerning the flow of information between users and platforms [START_REF] Brave | Tangible interfaces for remote collaboration and communication[END_REF]. We seek in this work to facilitate the collaboration between different people working together. Our objective is to propose a system that provides remote collaboration throw interactive tabletops with tangible interaction [START_REF] Bouabid | Context-sensitive and Collaborative Application for Distributed User Interfaces on Tabletops[END_REF]. Collaboration in this system is based on a set of generic tangible objects, called tangigets, initially defined by Lepreux et al. in [START_REF] Lepreux | From Centralized Interactive Tabletops to Distributed Surfaces: The Tangiget Concept[END_REF] and Caelen and Perrot in [START_REF] Caelen | Bibliothèque d'objets. Deliverable, IMAGIT ANR project. LIG[END_REF]. These tangigets will materialize a set of collaboration styles listed by Isenberg et al. in [START_REF] Isenberg | An exploratory study of co-located collaborative visual analytics around a tabletop display[END_REF].

In the paper we describe the design of the system and the interaction, as well as principles for remote collaboration on tangible tabletops. The study is then presented, and the first results are explained. Finally the paper concludes and proposes research perspectives.

System Design: DUI on two RFID tabletops

As Distributed User Interface (DUI), we use two TangiSense interactive tabletops allowing tangible interaction [START_REF] Kubicki | RFID-driven situation awareness on TangiSense, a table interacting with tangible objects[END_REF] (designed by the RFidees Company; see www.rfidees.fr). These tabletops use the RFID technology to recognize objects placed on the surface, as shown in Fig. 1.

The application used in our study is a distributed version of an application on RFID tabletop allowing the learning and recognition of colors; the initial version was presented in [START_REF] Kubicki | RFID interactive tabletop application with tangible objects: exploratory study to observe young children'behaviors[END_REF]). A set of business objects is used; they present various colorless pictures. For this application the final users are very young children learning colors; they have to arrange the business objects in color areas according to dominant colors. These objects are divided into 4 categories; each category contains 8 objects of varying difficulty levels. A set of generic tangible objects, called tangigets, is used to ensure remote collaboration between interconnected tabletops. These objects give users the ability to interact remotely using the features provided by the interactive tabletop.

Interaction Design: use of generic tangible objects

Remote collaboration between users on tabletops is carried out by the use of a set of generic objects (tangigets).The aim of these generic objects is to support an interuser dialogue using only features offered by interactive tabletops: Table 1 shows the correspondence between the action of each tangiget and one of the collaboration styles.

Table 1. Tangigets contributing to collaboration styles. [START_REF] Isenberg | An exploratory study of co-located collaborative visual analytics around a tabletop display[END_REF] Representative action on the application by the use of a generic object

Collaboration styles proposed in

Active discussion

Case study

In the application, we propose to use a set of tangigets useful for remote collaboration. Table 2 shows those objects and their main functionalities in the application.

A presentation of the system and its functioning as well as the functioning of each tangiget was given for each group of three participants. It was followed by a familiarization phase with the interactive tabletops during which the participants were encouraged to try the application and all items offered and to freely ask questions. After this phase of familiarization with the system, tests were started with different scenarios provided. We designed three conditions that varied aspects of the use of the Help request and the correction of the exercise. In these different conditions, instructions were provided to users of the child tabletop. We aimed to get a definite number of mistakes and requests for help. Fig. 3 provides an illustration of the participants of a test group set in relation to their different roles.

To simulate a remote collaboration, participants were in the same room; the two tabletops were separated by a folding screen to prevent users from each tabletop from seeing the contents of the other tabletop. Moreover, to prevent that the Parent participant from being disturbed by possible natural discussions coming from the child tabletop, he/she had a headset with music. After the study, each participant had to complete a questionnaire. The questionnaire firstly concerned information on the usability aspects and participant satisfaction with the system. Secondly, he or she had to fill more specific information on generic objects, ease of use and their significance in relation to their role set by the designer.

Fig. 3. A participant testing the application (adult tabletop)

Finally, the evaluator used a trace file in which all the games played by the group were recorded in order to understand how they had addressed the problem and t o get their reactions on the technologies and principles used. The analysis of the trace file is based mainly on a set of reactions of the user following the action by the other user of the remote tabletop. According to the response received, we classified them into three categories: expected answer, acceptable answer and incoherent answer.

As an example, we illustrate our analyses with two tangigets (1) Erase object used by the user of the adult tabletop and (2) Identification object used by the user of the child tabletop. The results of the questionnaires for Identification and Erase objects are summarized in Fig. 4; the score of each answer is shown for each question. We can find from the figure that participants give high marks on the global situation. They think these tangigets are easy to use and have a meaningful form. Also they have understood the goal of collaboration. To study if the use of tangigets by the participants confirms or not their subjective answers, we analyzed all the trace files in which we recorded the events concerning all games played. We extracted all uses of tangigets; after that we classified them as expected use, acceptable use or incoherent use. Analyses relative to Identification and erase objects are summarized in Fig. 5.

Answers Analyses for

Conclusion

In this paper, we introduce an innovative distributed application for the learning and recognition of colors. Generic tangible objects, called tangigets, are used to facilitate collaboration and exchange between distant participants about the exercise. This system takes advantage of large-scale tangible tabletops, 1) providing a simple user interface easy to manipulate; 2) enabling several users to collaborate remotely in each step of the exercise; 3) providing the possibility to cover a set of collaboration styles (in the sense of Isenberg et al. [START_REF] Isenberg | An exploratory study of co-located collaborative visual analytics around a tabletop display[END_REF]) by the use of tangigets. A study was conducted with twelve groups of three users. The results are promising and show the interest of the distributed approach. In future works we aim to test such tangigets with other more complex applications to verify if collaboration remains easy/possible.

Fig. 1 .

 1 Fig. 1. Two different user interfaces shown on the interactive tabletops with some of used (business and generic) tangible objects: on the left, user Interface displayed on the child tabletop during the correction of exercise; on the right, interface displayed on the adult tabletop during the supervision of the exercise realization



  Identification tangiget: Used to identify users who are currently using the collaborative application and want to enter in collaboration with other users.  Task assignment tangiget: Used to organize tasks between different users of the collaborative application.  Starting synchronization tangiget: Used to synchronize the start of the activity distributed on connected tabletops.  Display Mode tangiget: Used to change the display of the main interface according to the user needs.  Request help tangiget: Used to ask for help or ask a question about a step or a detail of the collaborative activity.  Provide help tangiget: Used to offer help about a step or detail following a request.  End task tangiget: Used to mark the end of a task and/or to switch to another task.  Criticism tangiget: Used to work on all of the activity (not on one task).

Fig. 2

 2 Fig. 2 shows two examples of Tangigets used in this application: they correspond respectively to Criticism tangiget (Magician object) and Provide help tangiget (Erase object).

Fig. 2 .

 2 Fig. 2. Some of used Tangigets in the application: on the left, Provide help tangiget (represented by Erase object); on the right, Criticism tangiget (represented by Magician object)

Fig. 4 .Fig. 5 .

 45 Fig. 4. Subjective answers of participants who used Identification and Erase tangigets

Table 2 .

 2 Instances of tangigets used for the application.

	Type of tangiget	Instantiation	Definition of its role
	Identification	Identification	Used to identify the person present remotely and ready to
			play
	Starting Synchronization	Start	Start the game on the two connected tabletops
	Task assignment	Category	Assign a category of object to the person identified
	Request help	Collaboration	Request remote assistance by placing in this area
		area	object(s) requiring help
	Provide help	Erase	Offer help by crossing color areas
	Display mode	Focus	Display the results of the exercise (textual representation)
	End task	End exercise	Indicate the end of the exercise for each user
	Criticism	Magician	Correct remotely the exercise

Acknowledgements

The authors thank warmly Sebastien Kubicki for his work on the first centralized version of the application [5]. They thank also the 36 participants and Steve Gabet for his contribution to the distributed version.


Lazhar Kassa-Baghdouche
email: kassabaghdouche_lazhar@yahoo.com

Eric Cassan

Mid-infrared refractive index sensing using optimized slotted photonic crystal waveguides

Keywords: Mid-infrared, Slotted photonic crystal waveguides, Resonators structures, Refractive index sensor, Photonic crystals *

Slotted photonic crystal waveguides (SPCW) were designed to act as refractive index sensing devices at mid-infrared (IR) wavelengths around λ = 3.6 µm.

In particular, effort was made to engineer the input and output slot waveguide interfaces in order to increase the effective sensitivity through resonant tapering.

A slotted PhC waveguide immersed in air and liquid cladding layers was considered. To determine the performance of the sensor, the sensitivity of the device was estimated by calculating the shift in the upper band edge of the output transmission spectrum. The results showed that the sensitivity of a conventionally designed SPCW followed by modifications in the structure parameter yielded a 510 nm shift in the wavelength position of the upper band edge, indicating a sensitivity of more than 1150 nm per refractive index unit (RIU) with an insertion loss level of -0.3 dB. This work demonstrates the viability of photonic crystal waveguide high sensitivity devices in the Mid-IR, following a transposition of the concepts inherited from the telecom band and an optimization of the design, in particular a minimization of photonic device insertion losses.

Introduction

There has been considerable focus on silicon photonics in the last several years, largely owing to its applications in near-IR data communications in the telecommunication wavelength band. However, silicon is also optically transparent in the wavelength range between 1.1 µm and 8 µm. Therefore, it offers a range of uses in environmental and biochemical sensing [START_REF] Soref | Mid-infrared photonics in silicon and germanium[END_REF]. Conversely, because of the high absorption loss of silicon dioxide (SiO 2 ) material above λ ∼ 3.6 µm, moving to the mid-infrared (mid-IR) wavelength range is difficult within the silicon-on-insulator (SOI) technology. Consequently, for longer wavelength ranges, alternative materials or free-standing membrane structures have to be used.

Research dedicated to refractive index (RI) -based photonic sensors has increased in recent years, with most of these researches aiming at engineering new structures suitable for detection of a wide range of physical parameters, such as pressure or temperature. In particular, PhC cavities and waveguides structures have gained considerable attention for applications in biosensing [START_REF] Skivesen | Photonic-crystal waveguide biosensor[END_REF][START_REF] Lee | Two-dimensional silicon photonic crystal based biosensing platform for protein detection[END_REF][START_REF] Kurt | Various photonic crystal bio-sensor configurations based on optical surface modes[END_REF]. In Ref. [START_REF] Skivesen | Photonic-crystal waveguide biosensor[END_REF], a biosensor device is designed for detection of protein-concentrations and concentrations around 10 µg/ml are measured with excellent signal to noise ratio. On the other hand, coating the internal surface of two-dimensional PhC microcavities with proteins of different sizes produces a different amount of resonance redshift and a molecule monolayer was detected with a total mass as small as 2.5 f g [START_REF] Lee | Two-dimensional silicon photonic crystal based biosensing platform for protein detection[END_REF]. A new biosensor concept incorporating photonic crystal surface modes to sense small refractive index changes was designed [START_REF] Kurt | Various photonic crystal bio-sensor configurations based on optical surface modes[END_REF]. In this work, a strongly confined mode interacted with the active medium while propagating along a PhC-air interface. The detection sensitivity for silicon PhC cavities was also improved by introducing multiple-hole defects (MHDs) in order to increase the surface area available for label-free detection without degrading the cavity quality factor [START_REF] Kang | Photonic crystal slab sensor with enhanced surface area[END_REF][START_REF] Kang | Photonic crystal with multiple-hole defect for sensor applications[END_REF].

In another direction, an increasing number of works was devoted to hollow core slotted waveguides in the telecom range for their specific mode confinement properties [START_REF] Y.-N. Zhang | Optimizing the slow light properties of slotted photonic crystal waveguide and its application in a high-sensitivity gas sensing system[END_REF][START_REF] Zou | Mid-infrared holey and slotted photonic crystal waveguides in silicon-on-sapphire for chemical warfare simulant detection[END_REF][START_REF] Di Falco | Photonic crystal slotted slab waveguides[END_REF][START_REF] Caer | Dispersion engineering of wide slot photonic crystal waveguides by bragg-like corrugation of the slot[END_REF][START_REF] Caer | Enhanced localization of light in slow wave slot photonic crystal waveguides[END_REF]. Several slotted photonic crystal geometries have been proposed to demonstrate cavities with high quality factors (Q) for uses in chemical and gas sensing in the near-IR telecom wavelengths near λ = 1.55 µm [START_REF] Di Falco | Chemical sensing in slotted photonic crystal heterostructure cavities[END_REF][START_REF] Yamamoto | Design of a high-q air-slot cavity based on a widthmodulated line-defect in a photonic crystal slab[END_REF][START_REF] Gao | Demonstration of an air-slot mode-gap confined photonic crystal slab nanoca-vity with ultrasmall mode volumes[END_REF][START_REF] Kwon | Optimization of photonic crystal cavity for chemical sensing[END_REF][START_REF] Jágerská | Refractive index sensing with an air-slot photonic crystal nanocavity[END_REF]. On the contrary, few works have been devoted for now to silicon slotted waveguides for sensing applications in the Mid-IR. Among recent works dealing with that objective, the transmission characteristics of slotted photonic crystal waveguides (SPCWs) suspended on a silicon-on-sapphire substrate was demonstrated experimentally at λ = 3.43 µm in Ref. [START_REF] Zou | Mid-infrared holey and slotted photonic crystal waveguides in silicon-on-sapphire for chemical warfare simulant detection[END_REF]. In order to minimize the optical absorption path length and achieve high sensitivity in onchip optical absorption spectroscopy, slotted photonic crystal structures, were also investigated in [START_REF] Lai | Photonic crystal slot waveguide absorption spectrometer for on-chip near-infrared spectroscopy of xylene in water[END_REF][START_REF] Lai | On-chip methane sensing by near-ir absorption signatures in a photonic crystal slot waveguide[END_REF]. In this work, the objective was to achieve the selective detection of volatile organic compounds [START_REF] Lai | Photonic crystal slot waveguide absorption spectrometer for on-chip near-infrared spectroscopy of xylene in water[END_REF] and greenhouse gases [START_REF] Lai | On-chip methane sensing by near-ir absorption signatures in a photonic crystal slot waveguide[END_REF] using a 300 µm long SPCW absorption spectrometer which was demonstrated experimentally in the near-infrared wavelength range [START_REF] Lai | Photonic crystal slot waveguide absorption spectrometer for on-chip near-infrared spectroscopy of xylene in water[END_REF][START_REF] Lai | On-chip methane sensing by near-ir absorption signatures in a photonic crystal slot waveguide[END_REF]. Moreover, a new type of integrated optical sensor that performed sensing in mid-infrared spectrum range was designed in Ref. [START_REF] Turduev | Mid-infrared t-shaped photonic crystal waveguide for optical refractive index sensing[END_REF]. In the proposed configuration, a strong light confinement and interaction were obtained by engineering the structural parameters of square-lattice photonic crystal incorporated with a T-shaped airslot. The direct interaction between the slot waveguide mode and the analyte infiltrated into the slot gave rise to an average refractive index sensitivity of 530 nm/RIU within the 1.10-1.25 index range and an average sensitivity of 390 nm/RIU between n = 1.00 and n = 1.30.

In this context, we design in this paper a mid-IR refractive index sensor with a higher sensitivity and a lower insertion loss level through numerical electromagnetic simulation. The device is made of an SPCW, which is formed by uniformly etching a rectangular nano-scale slot with air along the center of the line defect of a photonic crystal waveguide. An approach based on a resonant coupling mechanism is used to improve the coupling efficiency of the device [START_REF] Scullion | High efficiency interface for coupling into slotted photonic crystal waveguides[END_REF]. For the set of optimized geometrical parameters, an SPCW with an average on-chip coupling loss of -0.3 dB over a 350 nm per interface is designed. Using plane wave (PW) calculations of photonic bands and finite-difference time-domain (FDTD) simulations, both in 3D, the optical sensitivity of the proposed sensor is investigated in both the spectral and spatial domains by considering the infiltration of liquids. The sensitivity of the SPCW waveguide sensor is enhanced by fine-tuning the radius of the first row of the air holes near the air slot.

A high-sensitivity liquid sensing device based on the optimized SPCW is then proposed and a sensitivity of ∼ 1150 (nm/RIU ) is obtained at λ = 3.6 µm.

With respect to previous works, we bring here a viable design of Mid-IR slot photonic crystal waveguide sensors with beyond 1000 nm/RIU sensitivities and below -0.3 dB insertion loss.

Design of the slot waveguide configuration

To demonstrate mid-IR sensing, a slab photonic crystal waveguide (PCW) structure has been chosen, consisting of a two-dimensional hexagonal PhC lattice of the air holes etched in a silicon membrane. The slotted PCW was formed by omitting the central row of air holes in the ΓK direction and replacing it with a narrow air slot (see Fig. 1). A width of the air slot W slot = 0.33a, a membrane thickness of h = 0.48a, and a radius of the air hole of r o = 0.29a, where a is the lattice constant, were considered, respectively. The refractive index of the silicon membrane at the mid-infrared range was set to n Si = 3.42. The lattice constant was set to a = 1060 nm, fixing the cut-off wavelength of the guided mode to about 3.6 µm, as shown hereafter. Three-dimensional plane wave expansion (3D-PWE) calculations were achieved to analyze the dispersion properties of the SPCW. This analysis was conducted by choosing a suitable supercell using the MPB software [START_REF] Johnson | Block-iterative frequency-domain methods for maxwells equations in a planewave basis[END_REF].

As described in previous works, if compared with standard W1 waveguides, the effect of the slot is to shift every guided modes of the photonic crystal waveguide towards higher frequencies and to pull the fundamental mode into the gap [START_REF] Di Falco | Photonic crystal slotted slab waveguides[END_REF]. We recovered these classical trends and observed in the studied configuration the appearance of two optical modes remaining in the bulk photonic crystal bandgap, the so-called true-slot mode and the Ey-odd mode, respecti- vely (see Fig. 2). All the design steps described hereafter in this article were thus achieved by considering the first one only due to the field asymmetry of the Ey-odd mode one (see Fig. 2). As also visible in Fig. 2, the true-slot mode concentrates a non-negligible part of the propagating energy in the slot, which evidently is a good point for refractive index sensing purpose. Last, we also observe its slow light properties close to the edge of the Brillouin zone, graphically visible by its flat-band dispersion curve above ka/2π = 0.45. In terms of lightmatter interaction, slow light propagation, i.e. large group index propagation, was reported to reinforce most of physical effects [START_REF] Y.-N. Zhang | Optimizing the slow light properties of slotted photonic crystal waveguide and its application in a high-sensitivity gas sensing system[END_REF][START_REF] Caer | Dispersion engineering of wide slot photonic crystal waveguides by bragg-like corrugation of the slot[END_REF][START_REF] Caer | Enhanced localization of light in slow wave slot photonic crystal waveguides[END_REF][START_REF] Mortensen | Liquid-infiltrated photonic crystals : enhanced light-matter interactions for lab-on-a-chip applications[END_REF], showing that the chosen configuration provided a suitable starting point for sensing purpose.

Overall, these preliminary calculations thus provided a firm starting basis for the investigation of the device sensing properties.

Investigation of the sensing properties of the selected geometry

Improved coupling efficiency of the SPCW for refractive index sensing

As mentioned earlier, our main goal was to design an SPCW sensor with high sensitivity. However, the ability to efficiently couple light into the slow light regime is a critical factor that may affect the performances of SPCW for practical applications. Recently, an effective method was proposed by adding suitable resonant couplers on both ends of the SPCW to improve the light coupling efficiency from a conventional strip waveguide into a SPCW (see Fig. 3(a)) [START_REF] Scullion | High efficiency interface for coupling into slotted photonic crystal waveguides[END_REF].

110

Based on this structure, low coupling losses were realized experimentally. These coupling losses were as low as 1.5 dB per interface over a bandwidth of 78 nm around λ = 1550 nm [START_REF] Scullion | High efficiency interface for coupling into slotted photonic crystal waveguides[END_REF]. Furthermore, this method proved to be robust in terms of fabrication. Consequently, we have adopted a similar strategy while transposing it to Mid-IR wavelengths.

115

The performance of the resonant coupler depends on several geometrical parameters, namely, the distance between the PhC edge and the edge of the slot (S), the width of the access waveguide (W ), and the radius of the holes (R). A sketch of the geometry highlighting these different parameters is shown in Fig. 3(a). It was found that the distance slot (S) had the strongest effect on the transmission spectrum [START_REF] Scullion | High efficiency interface for coupling into slotted photonic crystal waveguides[END_REF]. The properties of the SPCW with the resonant coupler were numerically simulated using a three-dimensional finite difference time-domain (3D-FDTD) method [START_REF] Taflove | Computational electrodynamics[END_REF] using the MEEP software with subpixel smoothing for increased accuracy [START_REF] Oskooi | Meep : A flexible free-software package for electromagnetic simulations by the fdtd method[END_REF]. In order to prevent reflections, the simulation area was surrounded by a one-spatial unit thick perfectly matched layers absorbing out-going fields from the simulated region. In our 3D-FDTD calculation, the spatial resolution was set to 20 (namely, with a grid spacing of a/20, where a is the lattice constant). The S distance was then varied with 150 nm steps from S = 2200 to S = 3500 nm. The defect radius was varied in 50 nm steps from R = 400 to R = 550 nm, and the access width was varied in 200 nm steps from W = 2000 to W = 6500 nm. The best found parameters of the resonant coupler were finally : R = 550 nm, S = 3050 nm and W = 5200 nm. effect [START_REF] Scullion | High efficiency interface for coupling into slotted photonic crystal waveguides[END_REF]. A high transmission spectrum with low level ripples was evidently achieved.

Sensing characteristics of a first SPCW configuration

The sensing properties of the SPCW structure were investigated by monitoring the evolution of the cut-off wavelength position resulting from the filling of the air holes by various refractive index liquids. To obtain a sufficient degree of consistency with a realistic structure, the transmission characteristics of the SPCW sensor were directly computed with 3D-FDTD. To study the performance of the proposed sensor, four liquids, namely, pentane,

Improved sensing characteristics of the SPCW structure

As previously stated, the field of the slot mode of SPCW structure is very intense at the center of the slot. As a result, the modification of the first row of the air holes adjacent to the slot can lead to even strong light-matter interaction.

In order to obtain higher sensitivities, we thus further optimized the proposed structure by enhancing the localization of the field in the SPCW structure through the tuning of the diameter of the first row of air holes (r 1 ) of the structure. Figure 5(a) shows the evolution of the dispersion curve for (r 1 ) ranging from 0.29a to 0.34a, with a step of 0.01, while keeping other structural parameters unchanged with respect to those of the initial structure. It is seen that as the hole radius (r 1 ) is increased, the dispersion curve shifts toward higher fre-quencies. This shift occurs due to the increase of the low dielectric permittivity material of the waveguide. Using a similar approach, the refractive index sensitivities of the modified SPCW structures were studied through 3D-FDTD simulations. Figure 6(a) shows the variation in the cut-off wavelength shift of the SPCW structure as a function of the refractive index of the infiltrated liquids for different (r 1 ) values.

As visible, a linear relationship between the cut-off wavelength shift and the refractive of the cladding liquids is obtained and a slight optimization of the device sensitivity can be obtained by this approach. The best result was obtained for r 1 = 360 nm. The highest shift for a cut-off wavelength of 510 nm was obtained based on this (r 1 ) value.

Conclusion

In this article, we have explored the possibility of transposing concepts derived from telecom-range silicon photon devices based on photonic crystal slot waveguides for the realization of refractive index sensors around λ = 3.6 m.

An advantage of this approach is that the fabrication of periodic structures is simpler in the mid-infrared wavelength in view of the relaxed lithography constraints. The approach proposed in this article, based on a particular configuration, thus opens the way to a whole range of photonic crystal structures whose transposition from the telecom domain to the mid-IR domain is effective in terms of possible exploitation of physical effects. We have considered completely liberated silicon structures, i.e. free standing membrane ones, and explored the sensitivity of slot photonic crystal slow light true slot modes to cladding refractive index variation. A specific effort was devoted to the engineering of a resonant coupling stage to sustain the practical viability of photonic crystal silicon refractive index sensing structures for applications. Moreover, all numerical calculations were carried out using realistic 3D electromagnetic simulation.

A high-sensitivity beyond 1000 nm/RIU was demonstrated at λ = 3.6 m (S λ = 1150 nm/RIU in the best optimized case) for a device of 22.3 µm total length. These very encouraging results also show that the whole range of photonic crystal cavities toolbox can be envisaged for the realization of refractive index sensors in the mid-IR.

Figure 1 :

 1 Figure 1: Schematic of the slotted photonic crystal waveguide (SPCW) where r is the radius of the air holes, a is the lattice constant, W slot is the width of the air slot, h is the thickness of slab, and r 1 is the radius of the first row of the air holes.

Figure 2 :

 2 Figure 2: (a) The dispersion diagram of the SPCW, where the blue line indicates the Eyodd mode, the red line indicates the slot mode, and the dotted line indicates the light line. (b) The Ey electric field component distribution of the Ey-odd mode and the slot mode at kx = 0.5 (2π/a).

Figure 3 :

 3 Figure 3: (a). Structure of SPCW with resonant structure added on both ends of the waveguide. The optimized parameters are : R = 550 nm, S = 3050 nm and W = 5200 nm. (b) Transmission spectrum of SPCW with (blue line) and without (dotted red line) a resonant coupling structure. The average coupling efficiency per interface in the 3350-3700 nm wavelength windows is -0.3 dB for the optimized structure.

Figure 3 (

 3 Figure 3(b) shows the transmission spectrum of the SPCW structure after a

C 5 H 18 (

 518 12 (n clad = 1.3570) ; ethanol, C 2 H 5 OH (n clad = 1.3811) ; octane, C 8 H nclad = 1.4170) ; and ethylene glycol, C 2 H 6 O 2 (n clad = 1.4418) were considered. The refractive index of each liquid was selected for the mid-IR wavelength range around λ = 3.6 m. Figure 4(a) shows the evolution of the TE-like true slot mode transmission spectrum as a function of different refractive indices ranging from n = 1.00 to n = 1.50. As it can be anticipated, with the increase of RI of the liquids the transmission spectrum is shifted to longer wavelengths if compared with the situation of the SPCW structure immersed in air. Figure 4(b) shows the quantitative cut-off wavelength shift evolution versus the cover refractive index. A nearly linear shift is observed. A cut-off wavelength shift of ∼ 440 nm is observed with a change in the local refractive index from n clad = 1.00 (air) to n clad = 1.4418 (ethylene glycol). The SPCW structure sensitivity (S λ = ∂λ/∂n) was further evaluated and estimated to ∼ 995 nm/RIU . In comparison with a silicon-based PhC waveguide with optimized radius of the first rows of air holes nearest to the line-defect, which had a potential sensitivity of 282.4 nm/RIU at telecommunication wavelength [25], a comparatively much higher sensitivity of 995 nm/RIU is obtained in the present work.

Figure 4 :

 4 Figure 4: (a) Transmission spectra for TE-like polarization of the slotted photonic crystal waveguide studied in section 2 immersed into liquids with different refractive index ranging from n = 1.0 to n = 1.50 in 0.1 step. (b) The cut-off wavelength shifts for different refractive index values, giving rise to a sensitivity around 995 nm/RIU .

Figure 5 (

 5 b) illustrates the related variation in the edge frequency. As can be seen, the edge frequency linearly increases with the increasing (r 1 ) values.

Figure 5 :

 5 Figure 5: (a) TE-like band dispersion diagram for the slotted photonic crystal waveguide (SPCW) with different radiuses of the first row of the air holes ranging from r 1 = 0.29a to r 1 = 0.34a in 0.01 step. (b) The SPCW band edge frequency versus the hole radius r 1 .

Figure 6 (Figure 6 :

 66 Figure 6: (a) Variation of the cut-off wavelength shift as a function of the refractive indices for the different radiuses of the first row of the air holes (r 1 ). (b) Variation of the sensitivity as a function of the radius of the first row of air holes for different refractive indices of the liquids.


Amira Bouabid
email: amira.bouabid@gmail.com

Sophie Lepreux

Christophe Kolski

Clémentine Havrez
email: clementine.havrez@univ-valenciennes.fr

Context-sensitive and Collaborative application for Distributed User Interfaces on tabletops

Keywords: Context, Model, Distribution, Collaboration, Interactive tabletop, Tangible object, RFID. ACM Classification H.5.2, H.5.3. Information Interfaces and Presentation (e.g.HCI): User Interfaces, Group and Organization Interfaces

This paper focuses on collaborative work on interactive tabletops. To optimize the travel time of team members (remote workplace, telecommuting, and so on), collaborative work is now often remotely done. This brings many user interfaces issues between distributed platforms of each member. In the domain of context-sensitive user interfaces, which aims at an adaptation to the users, the platforms and the environment, context models have been proposed in the literature. We propose, in this paper, a context model for distributed applications centered on collaboration and interactive tabletops. The proposed model is validated by a distributed application, which is developed on two interactive tabletops with tangible interaction; these tabletops are equipped with RFID technology. This application, which has educational purposes, highlights the collaborative aspect and exchanges between remote users. The paper ends with a conclusion and several perspectives.

INTRODUCTION

With the development of remote work on various platforms, user interfaces have evolved. In the 2000s, an application was intended for a single end user and worked on a single platform in a single environment. Nomadic applications engendered researches about the adaptation of applications to the various types of platform (e.g. PDA, Smartphone).

Thus, the consideration of the interaction context became essential for their adaptation. The introduction of interaction surfaces with a more important size has led to applications available to multiple users interacting in a single environment and on the same support [START_REF] Lepreux | From Centralized interactive tabletops to Distributed Surfaces: the Tangiget concept[END_REF]. The improvement of the network capacities has brought new remote collaborative applications based on a variety of platforms [START_REF] Kubicki | Context Model for Distributed UI on Interactive Table(s). 2nd Workshop on Distributed User Interfaces: Collaboration and Usability, DUI 2012[END_REF]. This led to treat the distribution of user interfaces. Collaboration became possible between users in the same interaction context or in different contexts.

This paper focuses on collaborative interactions on interactive tabletops. Section 2 presents the state of the art on collaboration. Some types of collaboration are presented with an example for each type. Section 3 is dedicated to present our proposition. It is a context model which supports collaboration between remote or collocated users. In section 4, a case study on a pedagogical application adapted from [START_REF] Kubicki | Evaluation of an interactive table with tangible objects: Application with children in a classroom[END_REF] is shown on interactive tabletops with a distributed tangible interaction. The paper ends with a conclusion and research perspectives.

COLLABORATION AND INTERACTIVE TABLETOPS

The high quality networks coupled with the arrival of efficient systems increases the remote collaborative work. Many researches are carried out in the field of CSCW since [START_REF] Johansen | Groupware-Computer Support for Business Teams[END_REF]. Some researches dealing with collaboration are focused on many domains, such as education [START_REF] Large | The web as a classroom resource: Reactions from the users[END_REF], information retrieval on the web [START_REF] Isenberg | Collaborative Brushing and Linking for Co-located Visual Analytics of Document[END_REF][START_REF] Hansen | Collaborative Information Retrieval in an Information-Intensive Domain[END_REF], and so on. In this paper, we particularly focus on collaboration via interactive tabletops according to two types of collaboration: (1) collaboration where the team is in the same workspace (co-located) and (2) remote collaboration (Distributed).

The technology of interactive tabletops allows sharing information and tasks between team members located face to face or side by side thanks to their large surface. It is therefore particularly suitable for the co-located collaboration. Interactive tabletops provide the ability to handle multiple real and / or virtual objects depending on the capture technology used.

The DTLens system [4] uses an interactive tabletop in a multi-user environment. It is based on a zoom-in-context that enables group exploration of spatial data with multiple Paste the appropriate copyright/license statement here. ACM now supports three different publication options:

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Most of the works, which aim to realize distributed collaborative systems, seek to find a good remotely data presentation to perform a task or a common work between the different members of a geographically dispersed team [START_REF] Brennan | Towards a Multi-Analyst, Collaborative Framework for Visual Analytics[END_REF]. Balakrishnan and colleagues are looking for solutions to solve complex problems by sharing remote data visualizations [START_REF] Balakrishnan | Pitfalls of information access with visualizations in remote collaborative analysis[END_REF]. They conclude that the more members have access to a complete visualization of their data, the more they can be fruitful and provide solutions to problems. Another research encourages task sharing and data personalization in a collaborative framework: Brennan and colleagues adopt an approach that requires the explicit sharing and merging views of data during distributed analysis [START_REF] Brennan | Towards a Multi-Analyst, Collaborative Framework for Visual Analytics[END_REF]. In this way team members can first work separately on sub-tasks, and then gather all the ideas in a common view that takes charge of the construction of the common sense. The Keel system [START_REF] Keel | Collaborative Visual Analytics: Inferring from the Spatia Organization and Collaborative Use of Information[END_REF], which is dedicated to analysts of distributed data, offers a solution for groupwork. Each member works in his/her personal workspace and offers the data considered as appropriated in the common workspace. This method allows the implicit sharing of information.

Collaboration takes place in a collaborative environment. This environment can support two types of interaction: synchronous or asynchronous [START_REF] Yang | Does context matter? The impact of use context on mobile internet adoption[END_REF]. The types depend on the goal of the application and the manner in which several collaborators interact. [START_REF] Yarosh | Developing a Media Space for Remote synchronous Parent-Child Interaction[END_REF] mentions a type of interaction called semi-synchronization. We face this situation when it is possible for some users to see and interact with the work of other absent users. This is possible thanks to traceability of past activities.

In the case of the tangible interaction, the concept of Tangiget was proposed in [START_REF] Lepreux | From Centralized interactive tabletops to Distributed Surfaces: the Tangiget concept[END_REF]. Tangigets are tangible objects which support the interaction on one or several tabletops. They are divided into six categories: (1) Control objects of the application, (2) Context objects of the application, (3) Control of the interface objects, (4) Communication objects, (5) Coordination objects (6) Creation objects. Among these categories, we note that some Tangigets support communication and coordination, two activities required during collaboration. They are exploited in our proposition.

PROPOSITION

Our proposition is based on the work of Calvary and colleagues [START_REF] Calvary | Adaptation des Interfaces Homme-Machine à leur contexte d'usage[END_REF]: they define the context as a triplet <user, platform, environment>. This model was adapted by [START_REF] Kubicki | Context Model for Distributed UI on Interactive Table(s). 2nd Workshop on Distributed User Interfaces: Collaboration and Usability, DUI 2012[END_REF] to focus on interactions on interactive tabletops in a distributed context. Our goal is to incorporate in this previous context model [START_REF] Kubicki | Context Model for Distributed UI on Interactive Table(s). 2nd Workshop on Distributed User Interfaces: Collaboration and Usability, DUI 2012[END_REF] the notions linked with collaboration presented above and the different types of Tangigets used for collaboration between remote tables.

Main evolutions are shown in bold in the Figure 1. Concerning the user part, the most significant changes are the adding of a reflexive relationship between users and of a Collaboration class. This class contains the properties of the collaboration between users. It sets the privileges granted to each user according to his or her role in the application and the social link between the users who collaborate. Another class named Type of collaboration is added. This class specifies the type of collaboration between users. An Interact on relationship is added between user and collaborative environment in which he/she interacts. Concerning the environment part, a relationship named realised on is created between tasks and collaborative environment. Another relationship named takes place on specifies the environment in which the collaboration is performed. Concerning the platform part, the several categories of Tangigets were introduced in the model through six new classes. They are connected either to the Collaboration class or to the local tasks.

CASE STUDY: COLLABORATIVE APPLICATION

The "learning color" application (previously proposed and described in [START_REF] Kubicki | Evaluation of an interactive table with tangible objects: Application with children in a classroom[END_REF]) is developed with a collaborative goal, following a scenario inspired from [START_REF] Kubicki | Context Model for Distributed UI on Interactive Table(s). 2nd Workshop on Distributed User Interfaces: Collaboration and Usability, DUI 2012[END_REF] in which a child and parents interact in a distributed manner. This distribution is achieved through a multi-agent system developed with JADE [START_REF] Lebrun | Combination between Multi-Agent System and Tangigets for DUI Design on several Tabletops[END_REF]. The child is doing his or her color learning exercise on one tabletop. Parents are in another room where they also have a tabletop. The two tabletops are connected. There could be several users on each table. The scenario which aims to generate collaboration is: "The child has a difficulty; he or she wants to seek assistance from their parents who are distant but also possess an interactive tabletop. Parents wish to let the child work independently but also want to control the work done by the child without disturbing him or her with their presence."

A remote collaboration between users is possible. The child can ask parents for help. Parents can have a visualization of child labor. The picture shown in Figure 2 shows the distributed application on both TangiSense tabletops equipped with RFID technology (developed by the RFidées Company, http://www.rfidees.fr/). On one side (child), we see tangible objects placed by the child and their virtual feedback. On the right side, parents have a view of objects put by the child. Parents in this way can monitor the progress of the game on the tabletop without having to move.

In Figure 3, we present the object diagram of the Request for assistance collaborative use case. We see in the figure the Tangigets use for the realization of a distributed task between a child and his father.

CONCLUSION AND PERSPECTIVES

In this paper, we focused on the concept of collaboration between users on interactive tabletops involving distributed user interfaces. Two types of collaboration were considered: co-located collaboration and distributed collaboration. Based on this notion of collaboration exploiting a distribution of user interfaces, a context model has been proposed, built on the distributed interfaces and integrating collaboration features on tangible inter connected tabletops. A case study involving two RFID interactive tabletops with tangible objects implements this model. As research perspectives we aim to diversify the types of distributed platforms and to implement variants of the tangigets proposed in [START_REF] Lepreux | From Centralized interactive tabletops to Distributed Surfaces: the Tangiget concept[END_REF].

  BOUABID A., LEPREUX S., KOLSKI C., HAVREZ C. (2014). Context-sensitive and Collaborative application for Distributed User Interfaces on tabletops. DUI'2014, 4th Workshop on Distributed User Interfaces and Multimodal Interaction, Toulouse, France, july.

Figure 1 .

 1 Figure 1. Context model focused on collaboration properties

Figure 2 .

 2 Figure 2. Picture of the distributed application showing the child's view on the left side (with tangible objects) and the parents' view on the right side (with virtual objects as feedback).

ACKNOWLEDGMENTS

This research was partially financed by the French National Research Agency (ANR IMAGIT project ANR-10-CORD-017, partners LIG & RFIdees). We would like to thank the Region Nord-Pas de Calais, France for its support, as well as E. Adam, S. Kubicki, Y. Lebrun & R. Mandiau.


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Vincent Pelgrin, Yuchen Wang, Jonathan Peltier, Carlos Alonso-Ramos, Laurent Vivien, et al.. Boosting the SiN nonlinear photonic platform with transition metal dichalcogenide monolayers. Optics Letters, 2022, 47 (4), pp.734-737. 10.1364/OL.440462. hal-04456786