Horn(p,q)

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Horn(p,q)

Paul-Emile Paradan

To cite this version:

Paul-Emile Paradan. Horn(p,q). 2020. �hal-02867105�

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Hornpp, qq

Paul-Emile Paradan ^∗ June 15, 2020

Abstract

In this article, we obtain a recursive description of the Horn cone Hornpp, qq with respect to the integers p and q, as in the classical Horn’s conjecture.

1 Introduction 2

1.1 Horn’s conjecture . . . . 3

1.2 Holomorphic Horn cone Horn _hol pp, qq . . . . 4

1.3 Statement of the main result . . . . 5

1.4 Examples . . . . 6

1.5 Outline of the article . . . . 7

2 The K ³ -manifold K _C ˆ K _C ˆ E 7 2.1 Admissible elements . . . . 9

2.2 Ressayre’s data . . . . 10

2.3 Cohomological characterization of Ressayre’s data . . . . 11

2.4 Convex cone ∆pK _C ˆ K _C ˆ Eq : equations of the facets . . . 12

2.5 Remark on the saturation property . . . . 13

3 Saturated semigroups 14 3.1 The semigroup Q ^Z pp, qq . . . . 15

3.2 The semigroups Horn ^Z pp, qq and S ^Z pp, qq . . . . 16

3.3 Final remarks . . . . 18

∗

IMAG, Univ Montpellier, CNRS, email : [email protected]

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4 Convex cone Spp, qq 18

4.1 Admissible elements for pGL _p ˆ GL _q q ² ˆ C ^p b C ^q . . . . 18

4.2 Admissible elements ˘pγ ₀ ⁰ , γ ₀ ⁰ , γ ₀ ⁰ q . . . . 19

4.3 Admissible element pw 1 γ _s ^r , w 2 γ _s ^r , w 3 γ _s ^r q . . . . 20

4.4 Schubert classes . . . . 20

4.5 Cohomological conditions . . . . 21

5 Computation of the Euler class Eulp V _s ^r q 23 5.1 Polynomial representations . . . . 23

5.2 Duality I . . . . 24

5.3 Duality II . . . . 24

5.4 Morphism φ _m,n : geometric definition . . . . 25

5.5 Cauchy formula . . . . 25

6 Convex cone Spp, qq : equations of the facets 28 6.1 r “ 0 and s “ q . . . . 28

6.2 r “ p and s “ 0 . . . . 28

6.3 0 ă r ă p and s “ 0 . . . . 28

6.4 0 ă r ă p and s “ q . . . . 29

6.5 r “ 0 and 0 ă s ă q . . . . 29

6.6 r “ p and 0 ă s ă q . . . . 29

6.7 0 ă r ă p and 0 ă s ă q . . . . 30

6.8 Summary . . . . 31

6.9 Proof of the main result . . . . 31

1 Introduction

When G is Lie group, a natural problem is to understand how the sum of two adjoint orbits decomposes into a union of adjoint orbits. Let g be the the Lie algebra of G and let g {G be the set of adjoint orbits. The Horn cone is defined as follows

HornpGq “ tp O, O ¹ , O ² q P p g {Gq ³ , O ² Ă O ` O ¹ u.

Consider the case where G is a compact connected Lie group. Let T Ă G

be a maximal torus with Lie algebra t. The set g {G admits a canonical

identification with a Weyl chamber t _ě0 Ă t. In this setting, the Horn cone

HornpGq Ă p t _ě0 q ³ has been at the center of numerous studies [13, 15, 16,

4, 2, 3, 14, 27] that we summarize by the following theorem. We refer the

reader to the survey articles [5, 18] for details.

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Theorem 1.1 If G is a compact connected Lie group, HornpGq is a poly- hedral convex cone and one can parametrize the equation of its facets by cohomological means.

1.1 Horn’s conjecture

When G is the unitary group U pnq, the convex polyhedral cone ¹ Hornpnq has a nice feature which was predicted by A. Horn in the 60s : it admits a recursive description relative to the integer n ě 1 [13].

Denote the set of cardinality r subsets I “ ti ₁ ă i ₂ ă ¨ ¨ ¨ ă i _r u of rns “ t1, . . . , nu by P _r ⁿ . To each I P P _r ⁿ we associate a weakly decreasing sequence of non-negative integers

(1) λpI q “ pλ ₁ ě λ ₂ ¨ ¨ ¨ ě λ _r q P Z ^r _ě0 where λ _a “ n ´ r ` a ´ i _a for a P rrs.

Let d : R ⁿ Ñ upnq be the map that sends X “ px 1 , . . . , x n q to the diagonal matrix d X “ Diagpix 1 , . . . , ix n q. The map d induces a one to one correspondence between C _n “ tpx ₁ ě ¨ ¨ ¨ ě x _n qu Ă R ⁿ and the set of U pnq-adjoint orbits. If X “ px 1 , . . . , x n q P R ⁿ and I Ă rns, we define

| X | I “ ř

iPI x i and | X | “ ř n i“1 x i . Definition 1.2 Let n ě 1.

Hornpnq “ tpA, B, Cq P p C _n q ³ , U pnqd C Ă U pnqd A ` U pnqd B u.

The following Horn’s conjecture [13] was settled in the affirmative by combining the work of A. Klyachko [15] with the work of A. Knutson and T. Tao [16] on the “saturation” problem. We refer the reader to Fulton’s survey article [10] for details.

Theorem 1.3 (Horn’s conjecture) An element pA, B, Cq P p C _n q ³ belongs to Hornpnq if and only if the following conditions holds

• | A | ` | B | “ | C |,

• @r P rn ´ 1s, @I, J, K P P _r ⁿ , we have

| A | I ` | B | J ď | C | K if pλpI q, λpJ q, λpKqq P Hornprq.

1

We note HornpU pnqq simply by Hornpnq.

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1.2 Holomorphic Horn cone Horn _hol pp, qq

Let p ě q ě 1. We begin by recalling the definition of the holomorphic Horn cone Horn _hol pp, qq associated with the pseudo-unitary group U pp, qq.

The Lie group U pp, qq Ă GL _p`q pCq is defined by the relations gId _p,q g ^˚ “ Id p,q , where Id p,q is the diagonal matrice DiagpId p , ´Id q q. The Lie algebra u pp, qq of U pp, qq admits the following invariant convex cone

Cpp, qq “ X P upp, qq, ImpTrpgXg ^´1 Id _p,q qq ě 0, @g P U pp, qq ( . Let us consider

C _p,q “ tx P R ^p ˆ R ^q , x 1 ě ¨ ¨ ¨ ě x p ą x p`1 ě ¨ ¨ ¨ ě x p`q u Ă C _p ˆ C _q and the map d : R ^p ˆ R ^q Ñ u pp, qq. A well-know result says that for any U pp, qq-orbit O contained in the interior of Cpp, qq, there exists a unique X P C _p,q such that O “ U pp, qqd X (see [28, 21]). In other words, the map d realizes a one to one map between C _p,q and the set of U pp, qq-orbits in the interior of the invariant convex cone Cpp, qq. The holomorphic Horn cone is then defined as follows :

Horn _hol pp, qq “ pA, B, Cq P p C _p,q q ³ , Upp, qqd _C Ă U pp, qqd _A ` U pp, qqd _B ( . In a companion paper [23], we have proved that Horn _hol pp, qq is a closed convex cone of p C _p,q q ³ , and we have explained a way to compute it. In order to detail this result, we need some additional notations. For any n ě 1, we consider the semigroup ^ ^` _n “ tpλ 1 ě ¨ ¨ ¨ ě λ n qu Ă Z ⁿ . If λ “ pλ ¹ , λ ² q P ^ ^` _p ˆ ^ ^` _q , then V _λ :“ V _λ ^Uppq

1

b V _λ ^Upqq

2

denotes the irreducible representation of U ppq ˆ U pqq with highest weight λ. We denote by M _p,q the vector space of p ˆ q complex matrices, and by SympM p,q q the symmetric algebra of M p,q .

If H is a representation of U ppq ˆ U pqq, we denote by rV _ν : Hs the multiplicity of V ν in H.

Definition 1.4 1. Horn ^Z pp, qq is the semigroup of p^ ^` _p ˆ ^ ^` _q q ³ defined by the conditions:

pλ, µ, νq P Horn ^Z pp, qq ðñ rV ν : V λ b V µ b SympM p,q qs ‰ 0.

2. Hornpp, qq is the convex cone of pC _p ˆ C _q q ³ defined as the closure of Q ^ą0 ¨ Horn ^Z pp, qq.

The following result is proved in [23].

Theorem 1.5 We have

Horn _hol pp, qq “ Hornpp, qq č

p C _p,q q ³ .

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1.3 Statement of the main result

We now explain the main purpose of this paper that concerns a recursive description of the convex polyhedral cones Hornpp, qq as in Horn’s conjecture.

We need another notations.

1. If A “ pA ¹ , A ² q P R ^p ˆ R ^q and I “ I ¹ ˆ I ² Ă rps ˆ rqs, we define

| A | I “ | A ¹ | I

¹

` | A ² | I

²

and | A | “ | A ¹ | ` | A ² |.

2. If I “ I ¹ ˆ I ² Ă rns ˆ rms then λpIq “ pλpI ¹ q, λpI ² qq P ^ ^` _n ˆ ^ ^` _m . 3. Let 1 _n “ p1, . . . , 1q P Z ⁿ .

The main result of this paper is the following theorem.

Theorem 1.6 Let p ě q ě 1. An element pA, B, Cq P p C _p ˆ C _q q ³ belongs to Hornpp, qq if and only if the following conditions holds:

• | A | ` | B | “ | C | .

• | A ¹ | ` | B ¹ | ď | C ¹ | .

• For any r P rp´1s, for any I ¹ , J ¹ , K ¹ P P _r ^p , we have :

| A ¹ | I

¹

` | B ¹ | J

¹

ď | C ¹ | K

¹

if pλpI ¹ q, λpJ ¹ q, λpK ¹ qq P Hornprq.

| A ¹ | I

¹

` | B ¹ | J

¹

ě | C ¹ | K

¹

if pλpI ¹ q, λpJ ¹ q, λpK ¹ q ` pq `p´rq1 r q P Hornprq.

• For any s P rq´1s, for any I ² , J ² , K ² P P _s ^q , we have :

| A ² | I

²

` | B ² | J

²

ě | C ² | K

²

if pλpI ² q, λpJ ² q, λpK ² q ` pq´sq1 _s q P Hornpsq.

| A ² | I

²

` | B ² | J

²

ď | C ² | K

²

if pλpI ² q, λpJ ² q, λpK ² q ´ p1 s q P Hornpsq.

• For any pr, sq P rp´1s ˆ rq´1s with r ě s, for any I, J, K P P _r ^p ˆ P _s ^q , we have

| A | I ` | B | J ď | C | K if `

λpIq, λpJ q, λpKq`p0, pr´pq1 s q ˘

P Hornpr, sq.

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1.4 Examples

The convex cones Hornp1, 1q, Hornp2, 1q and Hornp2, 2q admit the following descriptions.

Proposition 1.7 An element pA, B, Cq P pR ˆ Rq ³ belongs to Hornp1, 1q if and only if the following conditions holds:

a ₁ ` a ₂ ` b ₁ ` b ₂ “ c ₁ ` c ₂ a ₁ ` b ₁ ď c ₁

Proposition 1.8 An element pA, B, Cq P p C ₂ ˆ Rq ³ belongs to Hornp2, 1q if and only if the following conditions holds:

a 1 ` a 2 ` a 3 ` b 1 ` b 2 ` b 3 “ c 1 ` c 2 ` c 3

a 1 ` a 2 ` b 1 ` b 2 ď c 1 ` c 2

a 2 ` b 2 ď c 2

a ₂ ` b ₁ ď c ₁ a ₁ ` b ₂ ď c ₁ a 1 ` b 1 ě c 2

Proposition 1.9 An element pA, B, Cq P p C ₂ ˆ C ₂ q ³ belongs to Hornp2, 2q if and only if the following conditions holds:

a 1 ` a 2 ` a 3 ` a 4 ` b 1 ` b 2 ` b 3 ` b 4 “ c 1 ` c 2 ` c 3 ` c 4

a ₁ ` a ₂ ` b ₁ ` b ₂ ď c ₁ ` c ₂ a ₂ ` b ₂ ď c ₂ a ₂ ` b ₁ ď c ₁ a 1 ` b 2 ď c 1

a ₃ ` b ₃ ě c ₃ a 3 ` b 4 ě c 4

a ₄ ` b ₃ ě c ₄

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a 2 ` a 4 ` b 2 ` b 4 ď c 1 ` c 4

a 2 ` a 4 ` b 2 ` b 4 ď c 2 ` c 3

a ₂ ` a ₄ ` b ₁ ` b ₄ ď c ₁ ` c ₃ a 1 ` a 4 ` b 2 ` b 4 ď c 1 ` c 3

a 2 ` a 4 ` b 2 ` b 3 ď c 1 ` c 3

a ₂ ` a ₃ ` b ₂ ` b ₄ ď c ₁ ` c ₃ 1.5 Outline of the article

The recursive description of Hornpp, qq is obtained by studying the Hamil- tonian action of pU ppq ˆ U pqqq ³ on the manifold ² pGL p ˆ GL q q ² ˆ C ^p b C ^q . Let Spp, qq Ă p C _p ˆ C _q q ³ be the corresponding Kirwan polyhedron.

In §2, we study the general framework of a Hamiltonian action of a compact Lie group K ³ on pK _C ˆ K _C q ² ˆ E : here E is a K-module such that the coordinate ring CrEs does not admit non-constant invariant vec- tors. We explain how to parameterize the facets of the Kirwan polyhedron

∆ppK _C ˆ K _C q ² ˆ Eq in terms of Ressayre’s data [22]. This parametrization requires two steps : determination of the admissible elements which are the potential vectors orthogonal to the facets, and computation of cohomological conditions on flag varieties.

In §3, we check that the semigroup Horn ^Z pp, qq is saturated. It is a direct consequence of the Darksen-Weyman saturation theorem [7].

In §4, we determine the admissible elements relative to the action of pU ppqˆU pqqq ³ on pGL p ˆGL q q ² ˆC ^p bC ^q , and we detailed the cohomological conditions in this particular case. The formulas need the computation of certain Euler classes which we carry over to §5.

In §6, we calculate (recursively) the facets of the Kirwan polyhedron Spp, qq. In the last subsection, we complete the proof of our main result.

Acknowledgements

I wish to thank Mich` ele Vergne for our discussions on this subject and for pointing to my attention the Derksen-Weyman saturation theorem.

2 The K ³ -manifold K _C ˆ K _C ˆ E

In this section, we briefly recall the result of §6 of [22] concerning the parametrization of the facets of Kirwan polyhedrons in terms of Ressayre’s

2

We use the notation GL

n

for the Lie group GLp C

ⁿ

q.

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data.

Let K be a compact connected Lie group with complexification K _C . Let T Ă K be a maximal torus with Lie algebra t. We consider the lattice ^ :“

1 2π kerpexp : t Ñ T q and the dual lattice ^ ^˚ Ă t ^˚ defined by ^ ^˚ “ homp^, Zq.

We remark that iη is a differential of a character of T if and only if η P ^ ^˚ . The Q -vector space generated by the lattice ^ ^˚ is denoted by t ^˚

Q : the vectors belonging to t ^˚

Q are designed as rational. Let t ^˚ _ě0 be a Weyl chamber. The set

^ ^˚ _` :“ ^ ^˚ X t ^˚ _ě0 parametrizes the irreducible representations of K: for any µ P ^ ^˚ _` , we denote by V _µ the irreducible representation of K with highest weight µ.

When K acts linearly on a vector space H, we denote by H ^K the sub- space of invariant vectors under the K-action.

Let E be a K-module such that CrEs ^K “ C : hence the coordinate ring CrEs has finite K-multiplicities. We consider the following K ˆ K ˆ K action on the affine variety K _C ˆ K _C ˆ E :

pk ₁ , k ₂ , k ₃ q ¨ px, y, vq “ pk ₁ xk ^´1 ₃ , k ₂ yk ₃ ^´1 , k ₃ vq.

The coordinate ring CrK _C ˆ K _C ˆ Es, viewed as a K ³ -module, admits the following decomposition

CrK _C ˆ K _C ˆ Es “ ÿ

λ,µ,νP^

^˚_`

m _E pλ, µ, νq V _λ ¹ b V _µ ² b V _ν ³ ,

where m E pλ, µ, ν q “ dimrV _λ b V µ b V ν b SympEqs ^K . Definition 2.1 We define the following sets :

• The semigroup ∆ ^Z pK _C ˆ K _C ˆ Eq Ă p^ ^˚ _` q ³ is defined as follows:

pλ, µ, νq P ∆ ^Z pK _C ˆ K _C ˆ Eq ðñ m _E pλ, µ, νq ‰ 0.

• The convex cone ∆pK _C ˆ K _C ˆ Eq Ă p t ^˚ _ě0 q ³ is the closure of Q ^ą0 ¨ ∆ ^Z pK _C ˆ K _C ˆ Eq.

Let us explain why the complex K ³ -manifold N “ K _C ˆ K _C ˆ E admits a symplectic structure Ω _N compatible with the complex structure, and a moment map Φ : N Ñ p k ^˚ q ³ associated to the action of K ³ on pN, Ω _N q.

Let h E be a K-invariant hermitian structure on E. We equip E with the 2-form Ω _E “ ´Imph _E q. The moment map Φ _E relative to the action of K on the symplectic vector space pE, Ω _E q is defined by

xΦ _E pvq, Xy “ ¹ ₂ Ω _E pXv, vq, @v P V, @X P k.

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The hypothesis CrEs ^K “ C implies that Φ E is a proper map.

There is a diffeomorphism of the cotangent bundle T ^˚ K with K _C defined as follows. We identify T ^˚ K with K ˆ k ^˚ by means of left-translation and then with K ˆ k by means of an invariant inner product on k. The map ϕ : K ˆ k Ñ K _C given by ϕpk, X q “ ke ^iX is a diffeomorphism. If we use ϕ to transport the canonical symplectic 2-form of T ^˚ K to K _C , then the resulting 2-form Ω K

_C

on K _C is compatible with the complex structure (see [12], §3).

Finally, the K ³ -manifold K _C ˆ K _C ˆ E » T ^˚ K ˆ T ^˚ K ˆ E carries the symplectic 2-form Ω N :“ Ω K

_C

ˆ Ω K

_C

ˆ Ω E which is compatible with the complex structure. The moment map relative to the K ³ -action on pN, Ω _N q is the proper map Φ “ Φ ₁ ‘ Φ ₂ ‘ Φ ₃ : T ^˚ K ˆ T ^˚ K ˆ E Ñ k ^˚ ‘ k ^˚ ‘ k ^˚ defined by

(2) Φpg 1 , ξ 1 , g 2 , ξ 2 , vq “ p´g 1 ξ 1 , ´g 2 ξ 2 , ξ 1 ` ξ 2 ` Φ _E pvqq.

By definition, the Kirwan polyhedron ∆pT ^˚ K ˆ T ^˚ K ˆ Eq is the inter- section of the image of Φ with pt ^˚ _ě0 q ³ . The following result is classical (see Theorem 4.9 in [25]).

Proposition 2.2 The Kirwan polyhedron ∆pT ^˚ K ˆ T ^˚ K ˆ Eq is equal to

∆pK _C ˆ K _C ˆ Eq.

2.1 Admissible elements

Definition 2.3 When a Lie group G acts on a manifold N , the stabilizer subgroup of n P N is denoted by G n “ tg P G, gn “ nu, and its Lie algebra by g _n . Let us define dim _G p X q “ min nPX dimp g _n q for any subset X Ă N .

We start by introducing the notion of admissible elements. The group hompU p1q, T q admits a natural identification with the lattice ^ :“ _2π ¹ kerpexp : t Ñ T q. A vector γ P t is called rational if it belongs to the Q -vector space t _Q generated by ^.

We consider the K ³ -action on N :“ T ^˚ K ˆ T ^˚ K ˆ E.

Definition 2.4 A non-zero element pγ 1 , γ 2 , γ 3 q P t ³ is called admissible if the elements γ _i are rational and if dim _K

3

pN ^pγ

¹

^,γ

²

^,γ

³

^q q ´ dim _K

3

pNq P t0, 1u.

Let R be the set of roots for pK, T q, and let W “ N pT q{T be the Weyl group. The set of weights for the T -action on E is denoted R _E . If γ P t, we denote by p R Y R _E q X γ ^K the subsets of weight vanishing against γ.

If w “ pw ₁ , w ₂ , w ₃ q P W ³ and γ P t, we write γ _w “ pw ₁ γ, w ₂ γ, w ₃ γq. We

start with the following lemma whose proof is left to the reader.

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Lemma 2.5 1. N ^pγ

¹

^,γ

²

^,γ

³

^q ‰ H if and only if γ 1 , γ 2 P W γ 3 . 2. dim _K

³

pN q “ dim T pk ˆ Eq “ dimptq ´ dimpVectpR Y R _E qq.

3. dim _K

3

pN ^γ

^w

q “ dim _T p k ^γ ˆ E ^γ q “ dimp t q ´ dimpVectpp R Y R _E q X γ ^K qq.

The following result is a direct consequence of the previous lemma.

Lemma 2.6 The admissible elements relative to the K ³ -action on T ^˚ K ˆ T ^˚ K ˆ E are of the form γ _w where w P W ³ and γ is a non-zero rational element satisfying VectpR Y R _E q X γ ^K “ VectppR Y R _E q X γ ^K q.

2.2 Ressayre’s data

Definition 2.7 1. Consider the linear action ρ : G Ñ GL _C pV q of a compact Lie group on a complex vector space V . For any pη, aq P gˆR , we define the vector subspace V ^η“a “ tv P V, dρpηqv “ iavu. Thus, for any η P g, we have the decomposition V “ V ^ηą0 ‘ V ^η“0 ‘ V ^ηă0 where V ^ηą0 “ ř

aą0 V ^η“a , and V ^ηă0 “ ř

aă0 V ^η“a .

2. The real number Tr _η pV ^ηą0 q is defined as the sum ř

aą0 a dimpV ^η“a q.

We consider an admissible element γ _w “ pw ₁ γ, w ₂ γ, w ₃ γq. The sub- manifold fixed by γ w is N ^γ

^w

“ w 1 K ^γ

C w ₃ ^´1 ˆ w 2 K ^γ

C w ^´1 ₃ ˆ E ^w

³

^γ . There is a canonical isomorphism of the manifold N ^γ

^w

equipped with the action of w ₁ K ^γ w ₁ ^´1 ˆw ₂ K ^γ w ₂ ^´1 ˆw ₃ K ^γ w ₃ ^´1 with the manifold K ^γ

C ˆK ^γ

C ˆE ^γ equipped with the action of K ^γ ˆ K ^γ ˆ K ^γ . The tangent bundle pTN | N

^γw

q ^γ

^w

^ą0 is isomorphic to N ^γ

^w

ˆ k ^γą0

C ˆ k ^γą0

C ˆ E ^γą0 .

The choice of positive roots R ^` induces a decomposition k _C “ n ‘ t _C ‘ n, where n “ ř

αPR

^`

p k b Cq α . We consider the map ρ ^γ,w : K ^γ

C ˆK ^γ

C ˆE ^γ ÝÑ hom

´

n ^w

¹

^γą0 ˆ n ^w

²

^γą0 ˆ n ^w

³

^γą0 , k ^γą0

C ˆ k ^γą0

C ˆ E ^γą0

¯

defined by the relation

ρ ^γ,w px, y, v q : pX, Y, Zq ÞÑ ppw ₁ xq ^´1 X´w ₃ ^´1 Z ; pw ₂ yq ^´1 Y ´w ^´1 ₃ Z ; pw ₃ ^´1 Zq¨vq, for any px, y, vq P K ^γ

C ˆ K ^γ

C ˆ E ^γ .

Definition 2.8 pγ, wq P t ˆ W ³ is a Ressayre’s data if

1. γ w is admissible,

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2. Dpx, y, vq such that ρ ^γ,w px, y, vq is bijective.

Remark 2.9 In [22], the Ressayre’s data were called regular infinitesimal B-Ressayre’s pairs.

Since the linear map ρ ^γ,w px, y, vq commutes with the γ-actions, we obtain the following necessary conditions.

Lemma 2.10 If pγ, wq P t ˆ W ³ is a Ressayre’s data, then

• Relation (A) : ř ₃

i“1 dimpn ^w

ⁱ

^γą0 q “ 2 dimpk ^γą0

C q ` dimpE ^γą0 q.

• Relation (B) : ř ₃

i“1 Tr w

i

γ p n ^w

ⁱ

^γą0 q “ 2 Tr γ p k ^γą0

C q ` Tr γ pE ^γą0 q.

2.3 Cohomological characterization of Ressayre’s data Let γ P t be a rational element. We denote by B Ă K _C the Borel subgroup with Lie algebra b “ t _C ‘ n. Consider the parabolic subgroup P _γ Ă K _C defined by

(3) P _γ “ tg P K _C , lim

tÑ8 expp´itγqg exppitγq existsu.

We work with the projective variety F _γ :“ K _C {P _γ . We associate to any w P W , the Schubert cell

X ^o _w,γ :“ Brws Ă F _γ ,

and the Schubert variety X _w,γ :“ X ^o _w,γ . If W ^γ denotes the subgroup of W that fixes γ, we see that the Schubert cell X ^o _w,γ and the Schubert variety X _w,γ depends only of the class of w in W {W ^γ .

We consider the cohomology ³ ring H ^˚ pF _γ , Zq of F _γ . If Y is an irreducible closed subvariety of F _γ , we denote by rY s P H ²ⁿ

^Y

p F ˜ _γ , Zq its cycle class in cohomology : here n _Y “ codimpY q. Recall that the cohomology class rpts associated to a singleton tptu Ă F _γ is a basis of H

^max

pF _γ , Zq.

To an oriented real vector bundle E Ñ N of rank r, we can associate its Euler class Eulp E q P H ^r pN, Zq. When E Ñ N is a complex vector bundle, then EulpE _R q corresponds to the top Chern class c p pEq. Here p is the complex rank of E, and E _R means E viewed as a real vector bundle oriented by its complex structure (see [6], §21).

The isomorphism E ^γą0 » E{E ^γď0 shows that E ^γą0 can be viewed as a P γ -module. Let E ^γą0 “ K _C ˆ P

γ

E ^γą0 be the corresponding complex vector

3

Here, we use singular cohomology with integer coefficients.

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bundle on F _γ . In the following proposition, we denote simply by EulpE ^γą0 q the Euler class Eulp E ^γą0

R q P H ^˚ p F _γ , Zq.

The following characterization of Ressayre’s data was obtained in [22],

§6.

Proposition 2.11 pγ, wq P t ˆ W ³ is a Ressayre’s data if and only if 1. γ w is admissible,

2. Relation (B) holds,

3. The following relation holds in H ^˚ pF _γ , Zq :

(4) r X _w

₁

_,γ s ¨ r X _w

₂

_,γ s ¨ r X _w

₃

_,γ s ¨ EulpE ^γą0 q “ krpts, k ě 1.

Remark 2.12 Notice that relation (A) is equivalent to ř ₃

i“1 codimp X _w

_i

_,γ q`

dimpE ^γą0 q “ dimp F _γ q, hence relation (A) follows from (4).

2.4 Convex cone ∆pK _C ˆ K _C ˆ Eq : equations of the facets The following result is proved in [22], §6.

Theorem 2.13 Let E be a K-module such that CrEs ^K “ C . An element pξ ₁ , ξ ₂ , ξ ₃ q P p t ^˚ _ě0 q ³ belongs to ∆pK _C ˆ K _C ˆ Eq if and only

(5) xξ 1 , w 1 γy ` xξ 2 , w 2 γ y ` xξ 3 , w 3 γy ě 0

for any pγ, wq P t ˆ W ³ that is a Ressayre’s data, that is to say satisfying the following properties:

a) γ is a non-zero rational element.

b) VectpR Y R _E q X γ ^K “ VectppR Y R _E q X γ ^K q.

c) r X _w

₁

_,γ s ¨ r X _w

₂

_,γ s ¨ r X _w

₃

_,γ s ¨ EulpE ^γą0 q “ krpts, k ě 1 in H ^˚ p F _γ , Zq.

d) Relation (B) holds : ř ₃

i“1 Tr w

i

γ p n ^w

ⁱ

^γą0 q “ 2 Tr γ p k ^γą0

C q ` Tr γ pE ^γą0 q.

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2.5 Remark on the saturation property

The semigroup ∆ ^Z pK _C ˆ K _C ˆ Eq Ă p^ ^˚ _` q ³ is called saturated if for any θ P p^ ^˚ _` q ³ and any N ě 1 we have N θ P ∆ ^Z pK _C ˆ K _C ˆ Eq only if θ P

∆ ^Z pK _C ˆ K _C ˆ Eq.

Proposition 2.14 1. We have

Q ^ą0 ¨ ∆ ^Z pK _C ˆ K _C ˆ Eq “ ∆pK _C ˆ K _C ˆ Eq X p t ^˚

Q q ³ . 2. The semigroup ∆ ^Z pK _C ˆ K _C ˆ Eq is saturated if and only if

∆ ^Z pK _C ˆ K _C ˆ Eq “ ∆pK _C ˆ K _C ˆ Eq X p^ ^˚ _` q ³ .

Proof. Let us prove the first point. The inclusion Q ^ą0 ¨ ∆ ^Z pK _C ˆ K _C ˆ Eq Ă ∆pK _C ˆ K _C ˆ Eq X p t ^˚

Q q ³ follows from the definitions. Let us explain why the opposite inclusion is a consequence of the rQ, Rs “ 0 theorem.

We consider the proper moment map Φ : K _C ˆ K _C ˆ E Ñ p k ^˚ q ³ . For any µ “ pµ ₁ , µ ₂ , µ ₃ q P p^ ^˚ _` q ³ , we denote by m _E pµq the multiplicity of V _µ “ V µ ^K,1

1

b V µ ^K,2

2

bV µ ^K,3

3

in the coordinate ring CrK _C ˆ K _C ˆ Es, and we consider the reduced space

M _µ :“ Φ ^´1 pKµ ₁ ˆ Kµ ₂ ˆ Kµ ₂ q{K ˆ K ˆ K.

that is equipped with the line bundle

L _µ “ Φ ^´1 pKµ 1 ˆ Kµ 2 ˆ Kµ 2 q ˆ K

µ1

ˆK

µ2

ˆK

µ2

pC ´µ

1

b C ´µ

2

b C ´µ

3

q . Suppose that M _µ is non-empty. Then M _µ is a complex-projective va- riety, a projective embedding being given by the Kodaira map M _µ Ñ PpH ⁰ p M _µ , L ^bk _µ qq for all sufficiently large k (see Theorem 2.17 in [24]). More- over, the rQ, Rs theorem says that for all k ě 1, we have m _E pkµq “ dim H ⁰ pM _µ , L ^bk _µ q: hence m E pkµq ‰ 0 for k sufficiently large.

Let ξ P ∆pK _C ˆ K _C ˆ Eq X p t ^˚

Q q ³ : let N ě 1 such that µ o :“ N ξ P p^ ^˚ _` q ³ . By definition, the reduced space M _µ

_o

is non-empty. So there exists, k _o ě 1 such that m E pk o µ o q ‰ 0, i.e. k o µ o P ∆ ^Z pK _C ˆ K _C ˆ Eq. We have proved that ξ “ _k

_o

¹ _N k o µ P Q ^ą0 ¨ ∆ ^Z pK _C ˆ K _C ˆ Eq.

The first point is settled and the second one is an immediate consequence

of the first one. l

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3 Saturated semigroups

For any n ě 1, we consider the semigroup ^ ^` _n “ tpλ ₁ ě ¨ ¨ ¨ ě λ _n qu Ă Z ⁿ that parametrizes the irreducible representations of the unitary group U pnq.

When λ P ^ ^` _n , the notation λ ě 0 (resp. λ ď 0) means that λ n ě 0 (resp. λ ₁ ď 0), and lengthpλq is the number of non-zero coordinates λ _i . To λ “ pλ 1 ě ¨ ¨ ¨ ě λ n q P ^ ^` _n , we associate λ ^˚ “ p´λ n ě ¨ ¨ ¨ ě ´λ 1 q P ^ ^` _n : the representation V _λ ^Upnq

˚

is then the dual of V _λ ^Upnq .

Let start by recalling the properties of the semigroup Horn ^Z pnq of p^ ^` _n q ³ defined by the relations

pλ, µ, ν q P Horn ^Z pnq ðñ

”

V _ν Ûpnq : V _λ Ûpnq b V _µ Ûpnq ı

‰ 0.

First, the convex cone of p C _n q ³ defined as the closure of Q ^ą0 ¨ Horn ^Z pnq corresponds to Hornpnq (see Definition 1.2). Moreover, thanks to the sat- uration Theorem of A. Knutson and T. Tao [16], we know that an ele- ment pλ, µ, νq P p^ ^` _n q ³ belongs to the semigroup Horn ^Z pnq if and only if pλ, µ, νq P Hornpnq (see Proposition 2.14).

In the rest of this section we work with the compact Lie group K “ U ppq ˆ U pqq, so that K _C “ GL p ˆ GL q . If λ “ pλ ¹ , λ ² q P ^ ^` _p ˆ ^ ^` _q , then V _λ :“ V _λ ^Uppq

1

b V _λ ^Upqq

2

denotes the irreducible representation of U ppq ˆ U pqq with highest weight λ.

Recall that for any p, q ě 1, M p,q denotes the vector space of p ˆ q complex matrices.

The purpose of this section is the study of the following semigroups of p^ ^` _p ˆ ^ ^` _q q ³ .

Definition 3.1 Let pλ, µ, νq P p^ ^` _p ˆ ^ ^` _q q ³ .

• The semigroup Horn ^Z pp, qq is defined by the conditions:

pλ, µ, νq P Horn ^Z pp, qq ðñ rV _ν : V _λ b V _µ b SympM _p,q qs ‰ 0.

• The semigroup Q ^Z pp, qq is defined by the conditions:

pλ, µ, νq P Q ^Z pp, qq ðñ

$

’ &

’ % λ ď 0, µ ď 0,

rV _λ b V _µ b V _ν b SympM _p,q qs ^UppqˆUpqq ‰ 0.

• The semigroup S ^Z pp, qq is defined by the conditions:

pλ, µ, νq P S ^Z pp, qq ðñ rV _λ b V _µ b V _ν b SympC ^p b C ^q qs ^UppqˆUpqq ‰ 0.

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Figure 1: Quiver Q with dimension vector v p,q

3.1 The semigroup Q ^Z p p, q q

Let p, q ě 1. We consider the quiver Q of Figure 1, with dimension vector v p,q “ pp, p, p, q, q, qq. The vector space

ReppQ, v p,q q “ pM p,p ˆ M q,q q ² ˆ M p,q

admits a natural action of the algebraic group GLpQ, v _p,q q “ pGL _p ˆ GL _q q ³ that we recall. Take g “ pg ₁ , g ₂ , g ₃ q P GLpQ, v _p,q q with g _i “ pg _i ¹ , g _i ² q P GL p ˆ GL q and pX 1 , X 2 , Y q P ReppQ, vq where X i “ pX _i ¹ , X _i ² q P M p,p ˆM q,q

and Y P M _p,q . Then g ¨ X “ px ₁ , y ₂ , yq where x _i “ pg ¹ _i X _i ¹ pg ¹ ₃ q ^´1 , g _i ² X _i ² pg ₃ ² q ^´1 q and y “ g ¹ ₃ Y pg ₃ ² q ^´1 .

We consider the multipicity map m : p^ ^` _p ˆ ^ ^` _q q ³ Ñ N defined by mpλ, µ, νq “ dim rV _λ b V _µ b V _ν b SympM _p,q qs ^GL

^p

^ˆGL

^q

.

Lemma 3.2 The coordinate ring CrReppQ, v p,q qs, viewed as GLpQ, v p,q q- module, admits the following decomposition

CrReppQ, v _p,q qs “ ÿ

λď0,µď0,ν

mpλ, µ, νq V _λ ¹ b V _µ ² b V _ν ³ .

Proof. It is due to the fact that the GLpQ, v _p,q q-module CrpM _p,p ˆM _q,q q ² s admits the decomposition CrpM _p,p ˆM _q,q q ² s “ ř

λď0,µď0 V _λ ¹ bV _µ ² bV _λ ³

˚

bV _µ ³

˚

. l

The previous lemma shows that Q ^Z pp, qq corresponds to the semigroup of highest weights associated to the action of the group GLpQ, v _p,q q on the coordinate ring CrReppQ, v p,q qs.

Proposition 3.3 The semigroup Q ^Z pp, qq is saturated.

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Figure 2: Quiver Q r with dimension vector v r p,q

Proof. This is a direct consequence of the Derksen-Weyman saturation theorem [7], which asserts that, for a quiver without cycles, the semigroup of weights of semi-invariants is saturated. Indeed, augment the quiver pQ, v p,q q to the quiver p Q, r r v _p,q q (see Figure 2). Then, using the Cauchy formula for the decomposition of b ^n´1 _k“1 CrpC ^k q ^˚ b C ^k`1 s under the action of ś _n

k“1 GL k , one sees that there is a bijective morphism between the semigroup of weights of semi-invariants of the coordinate ring CrRepp Q, r r v _p,q qs under the action of p ś _p

k“1 GL _k q ³ ˆ p ś _q

`“1 GL _` q ³ and the semigroup Q ^Z pp, qq. l 3.2 The semigroups Horn ^Z pp, qq and S ^Z pp, qq

We use the involution Θ : p^ ^` _p ˆ ^ ^` _q q ³ Ñ p^ ^` _p ˆ ^ ^` _q q ³ that sends pλ, µ, νq to ˆˆ λ ¹

pλ ² q ^˚

˙ ,

ˆ µ ¹ pµ ² q ^˚

˙ ,

ˆ pν ¹ q ^˚ ν ²

˙˙

.

Let us denote by 1 P ^ ^` _p ˆ ^ ^` _q the vector p1, . . . , 1, 1, . . . , 1q. The main purpose of this section is the following result.

Proposition 3.4 1. For any pλ, µ, νq P Horn ^Z pp, qq, there exists k o ě 0 such that pλ ´ k1, µ ´ k1, ν ^˚ ` 2k1q P Q ^Z pp, qq, @k ě k o .

2. pλ, µ, νq P Horn ^Z pp, qq if and only if Θpλ, µ, νq P S ^Z pp, qq.

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3. The semigroups Horn ^Z pp, qq and S ^Z pp, qq are saturated.

Proof. Let pλ, µ, νq P Horn ^Z pp, qq : it means that rV _λ b V _µ b V _ν

^˚

b SympM _p,q qs ^GL

^p

^ˆGL

^q

‰ 0 and so pλ, µ, ν ^˚ q P Q ^Z pp, qq if λ ď 0 and µ ď 0. We notice that pλ ` k1, µ ` k1, ν ` 2k1q P Horn ^Z pp, qq,

@k P Z . Let k o “ supp|λ 1 |, |µ 1 |q: we see that λ ´ k1 ď 0 and µ ´ k1 ď 0 if k ě k _o , and consequently pλ ´ k1, µ ´ k1, ν ^˚ ` 2k1q P Q ^Z pp, qq if k ě k _o . The first point is settled.

Cauchy formulas give the decompositions SympM p,q q “ SympC ^p b pC ^q q ^˚ q “

ÿ

lengthpaqďinfpp,qq

aě0

V a ^GL

^p

b V _a ^GL

˚ ^q

,

SympC ^p b C ^q q “

ÿ

lengthpaqďinfpp,qq

aě0

V a ^GL

^p

b V a ^GL

^q

.

For the second point, we have to compare the following cases :

• pλ, µ, νq P Horn ^Z pp, qq if there exists a ě 0 with lengthpaq ď infpp, qq such that both conditions hold

“ V _λ

¹

b V _µ

¹

b V _pν

1

q

^˚

b V _a ‰ _GL

_p

‰ 0

“ V _λ

²

b V _µ

²

b V _pν

²

_q

^˚

b V _a

^˚

‰ _GL

_q

‰ 0.

• pλ, µ, νq P S ^Z pp, qq if there exists a ě 0 with lengthpaq ď infpp, qq such that both conditions hold

“ V _λ

¹

b V _µ

¹

b V _ν

¹

b V a

‰ _GL

_p

‰ 0

“ V _λ

²

b V _µ

²

b V _ν

²

b V _a ‰ GL

q

‰ 0.

It is then immediate to conclude that pλ, µ, νq P Horn ^Z pp, qq if and only if Θpλ, µ, νq P S ^Z pp, qq.

Let us check that Horn ^Z pp, qq is saturated. Let pλ, µ, νq such that N pλ, µ, νq P Horn ^Z pp, qq for some N ě 1. If we use the first point, we know that there exists k _o ě 0, such that

N pλ, µ, ν ^˚ q ´ kp1, 1, ´2 ¨ 1q P Q ^Z pp, qq, @k ě k o .

Take k “ N k o ě k o : we obtain that N pλ ´ k o 1, µ ´ k o 1, ν ^˚ ` 2k o 1q P

Q ^Z pp, qq. It follows that pλ ´ k o 1, µ ´ k o 1, ν ^˚ ` 2k o 1q P Q ^Z pp, qq because

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the semigroup Q ^Z pp, qq is saturated. Finally, the last relation implies that pλ, µ, νq P Horn ^Z pp, qq.

We have verified that the semigroup Horn ^Z pp, qq is saturated, and the second point also allows us to conclude that the semigroup S ^Z pp, qq is satu- rated. l

3.3 Final remarks

We have seen that the semigroups Horn ^Z pp, qq, S ^Z pp, qq and Q ^Z pp, qq are all related. Thus, the associated convex cones Hornpp, qq “ Q ^ą0 ¨ Horn ^Z pp, qq, Spp, qq “ Q ^ą0 ¨ S ^Z pp, qq and Qpp, qq “ Q ^ą0 ¨ Q ^Z pp, qq are also interdependent.

The calculation of one entails those of the others. In this paper, we obtain a recursive description of the Horn cone Hornpp, qq through the calculation of Spp, qq.

Let Q o be a quiver without cycle which is equipped with a dimension vector v o . In [1], V. Baldoni, M. Vergne and M. Walter have proposed a recursive description of the cone generated by the highest weights associated to the action of the group GLpQ o , v o q on the coordinate ring CrReppQ o , v o qs.

By applying their result to the quiver pQ, v p,q q in Figure 1, this should also allow a recursive description of the Horn cone Hornpp, qq.

4 Convex cone S p p, q q

In this section, we apply the results of §2 to the case where K _C “ GL _p ˆ GL q and the K _C -module is E “ C ^p b C ^q . The coordinate ring CrpGL p ˆ GL _q q ² ˆ C ^p b C ^q s, viewed as a pGL _p ˆ GL _q q ³ -module, admits the following decomposition

CrpGL p ˆ GL q q ² ˆ C ^p b C ^q s “

ÿ

λ,µ,νP^

^`p

ˆ^

^`q

mpλ, µ, νq V _λ ¹ b V _µ ² b V _ν ³ ,

where mpλ, µ, νq “ dimrV _λ b V µ b V ν b SympC ^p b C ^q qs ^GL

^p

^ˆGL

^q

.

We see that the semigroup ∆ ^Z ppGL _p ˆ GL _q q ² ˆ C ^p b C ^q q corresponds to S ^Z pp, qq. Hence we will denote by Spp, qq the convex cone ∆ppGL p ˆ GL q q ² ˆ C ^p b C ^q q : it corresponds to the Kirwan polyhedron relative to the Hamiltonian action of Uppq ˆ U pqq on pGL _p ˆ GL _q q ² ˆ C ^p b C ^q .

4.1 Admissible elements for pGL _p ˆ GL _q q ² ˆ C ^p b C ^q

Let T » U p1q ^p ˆ U p1q ^q be the maximal torus of K “ U ppq ˆ U pqq formed by

the diagonal matrices. The Lie algebra t admits a canonical identification

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with R ^p ˆR ^q through the map d and the Weyl group of pK, T q is isomorphic to W “ S _p ˆ S _q .

The set of roots for pK, T q is R “ t _i ´ _j , 1 ď i ‰ j ď puY t ¹ _k ´ ¹ _l , 1 ď k ‰ l ď qu. The set of weights for the T -action on E “ C ^p b C ^q is R _E “ t _i ` ¹ _k , 1 ď i ď p, 1 ď k ď qu. Let us denote R _o “ R Y R _E . We first notice that R ^K _o “ R γ ₀ ⁰ , with γ ⁰ ₀ “ 1 _p ‘ ´1 _q P t.

Definition 4.1 For any pr, sq P t0, . . . , pu ˆ t0, . . . , qu, we define γ _s ^r “ γ ^r ‘ γ s P t where γ ^r “ p0, . . . , 0

lo omo on

r times

, 1, . . . , 1q and γ s “ p´1, . . . , ´1, 0, . . . , 0 lo omo on

s times

q.

Lemma 4.2 Let γ P t be a non-zero rational element such that VectpR o q X γ ^K “ Vectp R _o X γ ^K q. There exists pr, sq R tp0, 0q , pp, qqu, w P W and pa, bq P Q ^ě0 ˆ Q such that γ “ apwγ _s ^r q ` bγ ₀ ⁰ .

Proof. For any t P R , we define γptq “ ÿ

1ďiďp

γ

i

“t

e _i ´ ÿ

1ďkďq

γ

k

“´t

e ¹ _k .

We notice that γ ptq is orthogonal to R _o X γ ^K . If Vectp R _o q X γ ^K “ VectpR o X γ ^K q holds we get that γptq P pVectpR o q X γ ^K q ^K “ Rγ ⁰ ₀ ` Rγ for all t P R . Take t o such that γpt o q ‰ 0. Two situations holds.

1. γpt o q P Rγ ⁰ ₀ . This case only occurs if γ P Qγ ₀ ⁰ .

2. γpt o q R Rγ ⁰ ₀ . Then there exists pr, sq R tp0, 0q , pp, qqu, w P W and px, yq P Q ´ t0u ˆ Q such that γpt o q “ wγ _s ^r and γpt o q “ xγ ` yγ ₀ ⁰ : hence γ “ ¹ _x pwγ ^r _s q ´ ^y _x γ ₀ ⁰ . If ¹ _x ą 0, the proof is completed. If ¹ _x ă 0, we use that ´γ _s ^r “ w _o γ _q´s ^p´r ´ γ ₀ ⁰ for some w _o P W in order to come back to the previous case. l

In order to describe the facets of the convex cone Spp, qq, we must con- sider the following admissible elements:

• ˘pγ ₀ ⁰ , γ ₀ ⁰ , γ ₀ ⁰ q,

• pw 1 γ _s ^r , w 2 γ ^r _s , w 3 γ _s ^r q where w 1 , w 2 , w 3 P W and pr, sq R tp0, 0q , pp, qqu.

4.2 Admissible elements ˘pγ ₀ ⁰ , γ ₀ ⁰ , γ ₀ ⁰ q

The admissible elements ˘pγ ₀ ⁰ , γ ₀ ⁰ , γ ₀ ⁰ q act on pGL _p ˆ GL _q q ² ˆ C ^p b C ^q trivially. Hence ˘pγ ₀ ⁰ , γ ₀ ⁰ , γ ₀ ⁰ q are Ressayre’s data, and inequalities (5) are

˘ `

xA, γ ₀ ⁰ y ` xB, γ ₀ ⁰ y ` xC, γ ₀ ⁰ y ˘

ě 0. In other words,

(6) | A ¹ | ` | B ¹ | ` | C ¹ | “ | A ² | ` | B ² | ` | C ² | .

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4.3 Admissible element pw ₁ γ _s ^r , w ₂ γ _s ^r , w ₃ γ _s ^r q

Recall the relations that a Ressayre’s data pγ, w 1 , w 2 , w 3 q must satisfy (see Lemma 2.10):

Relation pAq :

3 ÿ

i“1

dimp n ^w

ⁱ

^γą0 q “ 2 dimp k ^γą0

C q ` dimppC ^p b C ^q q ^γą0 q, Relation pBq :

3 ÿ

i“1

Tr _w

_i

_γ p n ^w

ⁱ

^γą0 q “ 2 Tr _γ p k ^γą0

C q ` Tr _γ ppC ^p b C ^q q ^γą0 q.

It is immediate to see that for pγ _s ^r , w ₁ , w ₂ , w ₃ q, Relations (A) and (B) are equivalent.

We associate to pw 1 , w 2 , w 3 q P p S _p ˆ S _q q ³ the following subsets :

• Those of cardinal r : I ¹ “ w ¹ ₁ pt1, . . . , ruq, J ¹ “ w ₂ ¹ pt1, . . . , ruq, and K ¹ “w ¹ ₃ pt1, . . . , ruq.

• Those of cardinal s : I ² “ w ₁ ² ptq ´ s ` 1, . . . , quq, J ² “ w ² ₂ ptq ´s ` 1, . . . , quq, and K ² “w ² ₃ ptq ´s ` 1, . . . , quq.

Inequality (5) becomes |A ¹ | pI

¹

q

^c

`|B ¹ | pJ

¹

q

^c

`|C ¹ | pK

¹

q

^c

ě |A ² | pI

²

q

^c

`|B ² | pJ

²

q

^c

`

|C ² | pK

²

q

^c

which is equivalent to

(7) | A ¹ | I

¹

` | B ¹ | J

¹

` | C ¹ | K

¹

ď | A ² | I

²

` | B ² | J

²

` | C ² | K

²

, thanks to (6).

4.4 Schubert classes

For any m, n ě 0, let Gpm, nq denote the Grassmannian of complex m- dimensional linear subspaces of C ^m`n . The singular cohomology of Gpm, nq with integers coefficients is denoted H ^˚ pGpm, nq, Zq.

Let m, n ě 1. When a partition λ is included in a m ˆ n rectangle, we write λ Ă m ˆ n : n ě λ ₁ ě ¨ ¨ ¨ ě λ _m ě 0.

Denote the set of cardinality m subsets I “ ti 1 ă i 2 ă ¨ ¨ ¨ ă i m u of rm ` ns “ t1, . . . , m ` nu by P _m ^m`n . To each I P P _m ^m`n we associate λpIq “ pλ ₁ ě λ ₂ ¨ ¨ ¨ ě λ _m q Ă m ˆ n where λ _a “ n ` a ´ i _a for a P rms. The map I ÞÑ λpIq is one to one map between P _m ^m`n and the set of partitions of size m ˆ n. The inverse map is denoted by λ Ă m ˆ n ÞÑ Ipλq P P _m ^m`n .

We work with the flag 0 Ă C Ă C ² Ă ¨ ¨ ¨ Ă C ^n`m´1 Ă C ^n`m . For any partition λ Ă m ˆ n, we define the Schubert cell

X ^o _λ “ tF P G m,n , C ^k´1 X F ‰ C ^k X F if and only if k P Ipλqu.

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and the Schubert variety X _λ “ X ^o _λ . When the partition is pk, 0, . . . , 0q with 1 ď k ď n, the corresponding Schubert variety is denoted by X _k .

Let B _n`m Ă GL _n`m be the Borel subgroup formed by the upper-triangular matrices. The following facts are well-known (see e.g. [11, 8, 20]) :

1. X ^o _λ “ B m`n ¨ Vectpe i , i P Ipλqq, 2. G m,n “ Ť

λĂmˆn X ^o _λ , 3. codimp X ^o _λ q “ |λ| “ ř

k λ _k .

Since the Schubert cells define a complex cellular decomposition of the Grassmannian, an immediate consequence is that the fundamental class of the Schubert varieties, the Schubert classes σ _λ “ r X _λ s P H ^2|λ| pG m,n , Zq, where λ Ă m ˆ n, form a basis of the cohomology with integers coefficients :

H ^˚ pGpm, nq, Zq “ à

λĂmˆn

Z σ _λ .

When λ “ pk, 0, . . . , 0q, we denote by σ k P H ^2k pG m,n , Zq the correspond- ing Schubert class.

We finish this section by recalling that the Grassmannian G m,n admits the following complex vector bundles :

1. A canonical vector bundle of rank m : E m,n .

2. A vector bundle of rank n, denoted E ^K m,n , such that E m,n ‘ E ^K m,n is a trivial bundle of rank m ` n.

4.5 Cohomological conditions

Theorem 2.13 tells us that an element pA, B, Cq P pC _p ˆ C _q q ³ belongs to Spp, qq “ ∆ppGL p ˆ GL q q ² ˆ C ^p b C ^q q if and only (6) holds and (7) holds for any pw ₁ , w ₂ , w ₃ q P p S _p ˆ S _q q ³ and any couple pr, sq P t0, . . . , puˆt0, . . . , qu´

tpp, qq, p0, 0qu, satisfying the relation (8) r X _w

₁

_,γ

^r

s

s ¨ r X _w

₂

_,γ

^r

s

s ¨ r X _w

₃

_,γ

^r

s

s ¨ Eulp V _s ^r q “ krpts, k ě 1 in H ^˚ p F _γ

^r

s

, Zq.

Let us detailed (8). We fix pr, sq R tp0, 0q , pp, qqu. The parabolic sub- group P γ

_s^r

Ă GL p ˆ GL q associated to γ _s ^r by (3) is equal to P γ

^r

ˆ P γ

s

where

P _γ

^r

“

ˆ GL r ˚ 0 GL p´r

˙

Ă GL _p and P _γ

_s

“

ˆ GL q´s ˚ 0 GL s

˙

Ă GL _q .

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Let C ^r Ă C ^p and C ^q´s Ă C ^q denote respectively the subspaces Vectpe _i , 1 ď i ď rq and Vectpe _j , 1 ď j ď q ´ sq. We use on C ^p and C ^q the canonical bilinear forms px, yq ÞÑ ř

k x _k y _k . Then pC ^r q ^K Ă C ^p and pC ^q´s q ^K Ă C ^q are respectively the subspaces Vectpe i , r ` 1 ď i ď pq and Vectpe _j , q ´ s ` 1 ď j ď qq.

The flag variety F _γ

r

s

“ GL _p {P _γ

^r

ˆ GL _q {P _γ

_s

admits a canonical identifi- cation with Gpr, p ´ rq ˆ Gpq ´ s, sq through the map

prgs, rhsq P GL _p {P _γ

^r

ˆGL _q {P _γ

_s

ÞÝÑ `

gpC ^r q, hpC ^q´s q ˘

P Gpr, p´rqˆGpq´s, sq.

Let B _p Ă GL _p and B _q Ă GL _q be the Borel subgroups formed by the upper-triangular matrices. For any w “ pw ¹ , w ² q P S _p ˆ S _q , we consider the Schubert cell X ^o _w,γ

r

s

“ B p rw ¹ s ˆ B q rw ² s Ă Gpp ´ r, rq ˆ Gps, q ´ sq and the Schubert variety

X _w,γ

^r

s

“ B p rw ¹ s ˆ B q rw ² s “ X _µ

1

ˆ X _µ

2

where µ ¹ “ λpw ¹ t1, . . . , ruq Ă rˆp´r and µ ² “ λpw ² t1, . . . , q´suq Ă q´sˆs.

If we associate to pw ₁ , w ₂ , w ₃ q P p S _p ˆ S _q q ³ , the subsets

• I ¹ “ w ₁ ¹ pt1, . . . , ruq, J ¹ “w ¹ ₂ pt1, . . . , ruq, and K ¹ “w ¹ ₃ pt1, . . . , ruq,

• I ² “ w ² ₁ ptq ´ s ` 1, . . . , quq, J ² “ w ₂ ² ptq ´ s ` 1, . . . , quq, and K ² “ w ₃ ² ptq ´s ` 1, . . . , quq,

the term r X _w

₁

_,γ

_s^r

s ¨ r X _w

₂

_,γ

_s^r

s ¨ r X _w

₃

_,γ

_s^r

s P H ^˚ pGpr, p ´ rq ˆ Gpq ´ s, sq, Zq is then equal to the tensor product of the cohomology classes ⁴

σ _λpI

¹

_q ¨ σ _λpJ

¹

_q ¨ σ _λpK

¹

_q P H ^˚ pGpr, p ´ rq, Zq, σ _λppI

2

q

^c

q ¨ σ _λppJ

2

q

^c

q ¨ σ _λppK

2

q

^c

q P H ^˚ pGpq ´ s, sq, Zq.

The subspace pC ^p b C ^q q ^γ

^s^r

^ą0 is equal to pC ^r q ^K b pC ^q´s q ^K . Hence the vector bundle V _s ^r is equal to the tensor product E ^K r,p´r bE ^K _q´s,s . Let Eulp V _s ^r q P H ^2pp´rqs pGpr, p ´ rq ˆ Gpq ´ s, sq, Zq be its Euler class.

Finally, the cohomological condition (8) says that the product

` σ _λpI

1

q b σ _λppI

2

q

^c

q

˘ ¨ `

σ _λpJ

1

q b σ _λppJ

2

q

^c

q

˘ ¨ `

σ _λpK

1

q b σ _λppK

2

q

^c

q

˘ ¨ Eulp V _s ^r q is a non zero multiple of rpts P H ^max pGpr, p´rqˆGpq ´s, sq, Zq.

4

X

^c

denotes the complement of X .

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5 Computation of the Euler class Eul p V _s ^r q

Before giving a formula for the Euler class Eulp V _s ^r q, we need to recall some well-known facts.

5.1 Polynomial representations

We are concerned here with the polynomial representations of GL _m . When a representation π _V : GL _m Ñ GLpV q is polynomial, its character is an invariant polynomial χ V P CrM m,m s. We denote then by s V the restriction of χ _V to the diagonal matrices.

Let R _` pGL _m q denotes the polynomial representation ring of GL _m , and let Ź

m “ Zrx 1 , . . . , x m s ^S

^m

be the ring of symmetric polynomials, with integral coefficients, in m variables. The map V P R _` pGL _m q ÞÑ s _V P Ź

m is a ring isomorphism.

For any partition λ of length m, we associate the irreducible polynomial representation ⁵ V _λ ^GL

^m

of the group GL _m and the Schur polynomial s _λ :“

s _V

_λ

P Ź

m . Recall that the Schur polynomials s _λ determine a Z -basis of Ź

m . We recall the following classical fact (for a proof see §3.2.2 in [20]).

Theorem 5.1 The map φ _m,n : Ź

m ÝÑ H ^˚ pG m,n , Zq defined by the rela- tions

φ m,n ps λ q “

#

σ _λ if λ ₁ ď n, 0 if λ 1 ą n.

is a ring morphism.

Remark 5.2 Since V P R ` pGL m q ÞÑ s V P Ź

m is a ring isomorphism, we will also denote by φ m,n : R _` pGL m q ÝÑ H ^˚ pG m,n , Zq the ring morphism V ÞÑ φ _m,n ps _V q.

If k ě 1, we denote by 1 ^k the partition p1, . . . , 1, 0, . . . , 0q where there are k-times 1.

Example 5.3

φ _m,n pSym ^k pC ^m qq “

#

σ k if 1 ď k ď n, 0 if k ą n.

φ _m,n p ľ k

C ^m q “

#

σ ₁

k

if 1 ď k ď m,

0 if k ą m.

5

When the group is understood, we use the notation V

λ

.

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5.2 Duality I

We associate to a partition λ Ă m ˆ n it’s complementary partition p λ Ă m ˆ n : p λ _k “ n ´ λ _m`1´k , 1 ď k ď m. Recall that

• If λ “ λpI q then p λ “ λp I r q where I r “ tn ` m ` 1 ´ i ; i P I u.

• If V _λ is the irreducible polynomial representation of GL m associated to λ Ă m ˆ n, then pV _λ q ^˚ “ V

λ p b det ^´n .

The cohomology class rpts P H ^2nm pG m,n , Zq of top degree associated to a singleton tptu is a basis of H ^2nm pG m,n , Zq.

We recall the following classical fact (for a proof see §3.2.2 in [20]).

Proposition 5.4 Let λ ¹ , λ Ă m ˆ n be two partitions such that |λ| ` |λ ¹ | “ nm. Then, the following relations hold in H ^˚ pG m,n , Zq :

σ _λ ¨ σ _λ

¹

“

#

rpts if λ ¹ “ p λ, 0 if λ ¹ ‰ p λ.

The next corollary follows from Theorem 5.1 and Proposition 5.4.

Corollary 5.5 Let λ 1 , λ 2 , λ 3 Ă m ˆ n. The following assertions are equiv- alent :

• σ _λ

₁

¨ σ _λ

₂

¨ σ _λ

₃

“ krpts, k ě 1 in H ^˚ pG m,n , Zq.

• ” V _p _λ

3

: V λ

1

b V λ

2

ı

‰ 0.

• “

V _λ

₁

b V _λ

₂

b V _λ

₃

b det ^´n ‰ _GL

_m

‰ 0.

5.3 Duality II

Taking the transpose of the Young diagram defines a bijective map λ Ă m ˆ n ÞÝÑ λ ^_ Ă n ˆ m. The following lemma is useful in our computations.

Lemma 5.6 If the partition λ Ă m ˆ n is equal to λpIq then λ ^_ “ λpI ^_ q where I ^_ “ pI Ą ^c q.

The canonical bilinear form on C ^n`m permits to define the map δ : G n,m ÝÑ G m,n that sends F to F ^K . Let δ ^˚ : H ^˚ pG m,n q Ñ H ^˚ pG n,m q denotes the pullback map in cohomology.

Lemma 5.7 For any partition λ Ă m ˆ n, we have δ ^˚ pσ _λ q “ σ _λ

^_

.

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5.4 Morphism φ _m,n : geometric definition

To a polynomial representation V P R ` pGL m q, we associated the polyno- mial map π _V : M m,m ÝÑ EndpV q and the invariant polynomial χ _V pXq :“

Tr _V pπ _V pXqq.

Let E m,n Ñ G m,n be the canonical vector bundle of rank m. Let Ω m,n P A ² pG m,n , EndpE m,n qq be its curvature. The Chern-Weil homomorphism as- sociates to the invariant polynomial χ _V the closed form χ _V ` _i

2π Ω _m,n ˘

of even degree on G m,n . We denote by H ^˚ pG m,n q the de Rham cohomology of G m,n . We have a natural (injective) morphism H ^˚ pG m,n , Zq Ñ H ^˚ pG m,n q.

Here is a geometric definition of the map φ _m,n [26].

Theorem 5.8 For any V P R _` pGL _m q, φ _m,n pV q P H ^˚ pG m,n q is the class defined by the closed form χ V

` _i

2π Ω m,n

˘ .

If V Ñ N is a complex vector bundle, we denote by c _k p V q its k-Chern class. In the next lemma, we recall the computation of the Chern classes of the vector bundles E m,n and E ^K _m,n .

Lemma 5.9 The following relations holds in H ^˚ pG m,n q.

c _k pE m,n q “

#

σ ₁

k

if 1 ď k ď m,

0 if k ą m.

c _k pE ^K _m,n q “

#

σ _k if 1 ď k ď n, 0 if k ą n.

Proof : If k ą m “ rankpE m,n q, then c k pE m,n q “ 0. If 1 ď k ď m, then c k pE m,n q “ φ m,n p Ź _k

C ^m q “ σ ₁

^k

. For the second point, let us use the isomorphism δ : G m,n Ñ G n,m . We see that the vector bundle E ^K m,n is isomorphic to δ ^´1 pE n,m q. Then c _k pE ^K _m,n q “ δ ^˚ pc _k pE n,m qq “ δ ^˚ pσ ₁

k

q “ σ _k for any 1 ď k ď n. l

5.5 Cauchy formula

We fix some integers m, n, m ¹ , n ¹ ě 1.

We consider the vector bundles E m,n Ñ G m,n and E m

¹

,n

¹

Ñ G m

¹

,n

¹

. We can form the bundles E m,n b E m

¹

,n

¹

and E ^K _m,n b E ^K _m

1

,n

¹

on G m,n ˆG m

¹

,n

¹

Horn(p,q)

HAL Id: hal-02867105

https://hal.archives-ouvertes.fr/hal-02867105

Preprint submitted on 15 Jun 2020

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Horn(p,q)

Paul-Emile Paradan

To cite this version:

Paul-Emile Paradan. Horn(p,q). 2020. �hal-02867105�

Hornpp, qq

Paul-Emile Paradan ∗ June 15, 2020

Abstract

In this article, we obtain a recursive description of the Horn cone Hornpp, qq with respect to the integers p and q, as in the classical Horn’s conjecture.

Contents

1 Introduction 2

1.1 Horn’s conjecture . . . . 3

1.2 Holomorphic Horn cone Horn hol pp, qq . . . . 4

1.3 Statement of the main result . . . . 5

1.4 Examples . . . . 6

1.5 Outline of the article . . . . 7

2 The K 3 -manifold K C ˆ K C ˆ E 7 2.1 Admissible elements . . . . 9

2.2 Ressayre’s data . . . . 10

2.3 Cohomological characterization of Ressayre’s data . . . . 11

2.4 Convex cone ∆pK C ˆ K C ˆ Eq : equations of the facets . . . 12

2.5 Remark on the saturation property . . . . 13

3 Saturated semigroups 14 3.1 The semigroup Q Z pp, qq . . . . 15

3.2 The semigroups Horn Z pp, qq and S Z pp, qq . . . . 16

3.3 Final remarks . . . . 18

IMAG, Univ Montpellier, CNRS, email : [email protected]

4 Convex cone Spp, qq 18

4.1 Admissible elements for pGL p ˆ GL q q 2 ˆ C p b C q . . . . 18

4.2 Admissible elements ˘pγ 0 0 , γ 0 0 , γ 0 0 q . . . . 19

4.3 Admissible element pw 1 γ s r , w 2 γ s r , w 3 γ s r q . . . . 20

4.4 Schubert classes . . . . 20

4.5 Cohomological conditions . . . . 21

5 Computation of the Euler class Eulp V s r q 23 5.1 Polynomial representations . . . . 23

5.2 Duality I . . . . 24

5.3 Duality II . . . . 24

5.4 Morphism φ m,n : geometric definition . . . . 25

5.5 Cauchy formula . . . . 25

6 Convex cone Spp, qq : equations of the facets 28 6.1 r “ 0 and s “ q . . . . 28

6.2 r “ p and s “ 0 . . . . 28

6.3 0 ă r ă p and s “ 0 . . . . 28

6.4 0 ă r ă p and s “ q . . . . 29

6.5 r “ 0 and 0 ă s ă q . . . . 29

6.6 r “ p and 0 ă s ă q . . . . 29

6.7 0 ă r ă p and 0 ă s ă q . . . . 30

6.8 Summary . . . . 31

6.9 Proof of the main result . . . . 31

1 Introduction

When G is Lie group, a natural problem is to understand how the sum of two adjoint orbits decomposes into a union of adjoint orbits. Let g be the the Lie algebra of G and let g {G be the set of adjoint orbits. The Horn cone is defined as follows

HornpGq “ tp O, O 1 , O 2 q P p g {Gq 3 , O 2 Ă O ` O 1 u.

Consider the case where G is a compact connected Lie group. Let T Ă G

be a maximal torus with Lie algebra t. The set g {G admits a canonical

identification with a Weyl chamber t ě0 Ă t. In this setting, the Horn cone

HornpGq Ă p t ě0 q 3 has been at the center of numerous studies [13, 15, 16,

4, 2, 3, 14, 27] that we summarize by the following theorem. We refer the

reader to the survey articles [5, 18] for details.

Theorem 1.1 If G is a compact connected Lie group, HornpGq is a poly- hedral convex cone and one can parametrize the equation of its facets by cohomological means.

1.1 Horn’s conjecture

When G is the unitary group U pnq, the convex polyhedral cone 1 Hornpnq has a nice feature which was predicted by A. Horn in the 60s : it admits a recursive description relative to the integer n ě 1 [13].

Denote the set of cardinality r subsets I “ ti 1 ă i 2 ă ¨ ¨ ¨ ă i r u of rns “ t1, . . . , nu by P r n . To each I P P r n we associate a weakly decreasing sequence of non-negative integers

(1) λpI q “ pλ 1 ě λ 2 ¨ ¨ ¨ ě λ r q P Z r ě0 where λ a “ n ´ r ` a ´ i a for a P rrs.

| X | I “ ř

iPI x i and | X | “ ř n i“1 x i . Definition 1.2 Let n ě 1.

Hornpnq “ tpA, B, Cq P p C n q 3 , U pnqd C Ă U pnqd A ` U pnqd B u.

The following Horn’s conjecture [13] was settled in the affirmative by combining the work of A. Klyachko [15] with the work of A. Knutson and T. Tao [16] on the “saturation” problem. We refer the reader to Fulton’s survey article [10] for details.

Theorem 1.3 (Horn’s conjecture) An element pA, B, Cq P p C n q 3 belongs to Hornpnq if and only if the following conditions holds

• | A | ` | B | “ | C |,

• @r P rn ´ 1s, @I, J, K P P r n , we have

| A | I ` | B | J ď | C | K if pλpI q, λpJ q, λpKqq P Hornprq.

We note HornpU pnqq simply by Hornpnq.

1.2 Holomorphic Horn cone Horn hol pp, qq

Let p ě q ě 1. We begin by recalling the definition of the holomorphic Horn cone Horn hol pp, qq associated with the pseudo-unitary group U pp, qq.

The Lie group U pp, qq Ă GL p`q pCq is defined by the relations gId p,q g ˚ “ Id p,q , where Id p,q is the diagonal matrice DiagpId p , ´Id q q. The Lie algebra u pp, qq of U pp, qq admits the following invariant convex cone

Cpp, qq “ X P upp, qq, ImpTrpgXg ´1 Id p,q qq ě 0, @g P U pp, qq ( . Let us consider

b V λ Upqq

denotes the irreducible representation of U ppq ˆ U pqq with highest weight λ. We denote by M p,q the vector space of p ˆ q complex matrices, and by SympM p,q q the symmetric algebra of M p,q .

If H is a representation of U ppq ˆ U pqq, we denote by rV ν : Hs the multiplicity of V ν in H.

Paul-Emile Paradan ^∗ June 15, 2020

1.2 Holomorphic Horn cone Horn _hol pp, qq . . . . 4

2 The K ³ -manifold K _C ˆ K _C ˆ E 7 2.1 Admissible elements . . . . 9

2.4 Convex cone ∆pK _C ˆ K _C ˆ Eq : equations of the facets . . . 12

3 Saturated semigroups 14 3.1 The semigroup Q ^Z pp, qq . . . . 15

3.2 The semigroups Horn ^Z pp, qq and S ^Z pp, qq . . . . 16

4.1 Admissible elements for pGL _p ˆ GL _q q ² ˆ C ^p b C ^q . . . . 18

4.2 Admissible elements ˘pγ ₀ ⁰ , γ ₀ ⁰ , γ ₀ ⁰ q . . . . 19

4.3 Admissible element pw 1 γ _s ^r , w 2 γ _s ^r , w 3 γ _s ^r q . . . . 20

5 Computation of the Euler class Eulp V _s ^r q 23 5.1 Polynomial representations . . . . 23

5.4 Morphism φ _m,n : geometric definition . . . . 25

HornpGq “ tp O, O ¹ , O ² q P p g {Gq ³ , O ² Ă O ` O ¹ u.

identification with a Weyl chamber t _ě0 Ă t. In this setting, the Horn cone

HornpGq Ă p t _ě0 q ³ has been at the center of numerous studies [13, 15, 16,

When G is the unitary group U pnq, the convex polyhedral cone ¹ Hornpnq has a nice feature which was predicted by A. Horn in the 60s : it admits a recursive description relative to the integer n ě 1 [13].

Denote the set of cardinality r subsets I “ ti ₁ ă i ₂ ă ¨ ¨ ¨ ă i _r u of rns “ t1, . . . , nu by P _r ⁿ . To each I P P _r ⁿ we associate a weakly decreasing sequence of non-negative integers

(1) λpI q “ pλ ₁ ě λ ₂ ¨ ¨ ¨ ě λ _r q P Z ^r _ě0 where λ _a “ n ´ r ` a ´ i _a for a P rrs.

Hornpnq “ tpA, B, Cq P p C _n q ³ , U pnqd C Ă U pnqd A ` U pnqd B u.

Theorem 1.3 (Horn’s conjecture) An element pA, B, Cq P p C _n q ³ belongs to Hornpnq if and only if the following conditions holds

• @r P rn ´ 1s, @I, J, K P P _r ⁿ , we have

1.2 Holomorphic Horn cone Horn _hol pp, qq

Let p ě q ě 1. We begin by recalling the definition of the holomorphic Horn cone Horn _hol pp, qq associated with the pseudo-unitary group U pp, qq.

The Lie group U pp, qq Ă GL _p`q pCq is defined by the relations gId _p,q g ^˚ “ Id p,q , where Id p,q is the diagonal matrice DiagpId p , ´Id q q. The Lie algebra u pp, qq of U pp, qq admits the following invariant convex cone

Cpp, qq “ X P upp, qq, ImpTrpgXg ^´1 Id _p,q qq ě 0, @g P U pp, qq ( . Let us consider

b V _λ ^Upqq

denotes the irreducible representation of U ppq ˆ U pqq with highest weight λ. We denote by M _p,q the vector space of p ˆ q complex matrices, and by SympM p,q q the symmetric algebra of M p,q .

If H is a representation of U ppq ˆ U pqq, we denote by rV _ν : Hs the multiplicity of V ν in H.

Definition 1.4 1. Horn ^Z pp, qq is the semigroup of p^ ^` _p ˆ ^ ^` _q q ³ defined by the conditions:

pλ, µ, νq P Horn ^Z pp, qq ðñ rV ν : V λ b V µ b SympM p,q qs ‰ 0.

2. Hornpp, qq is the convex cone of pC _p ˆ C _q q ³ defined as the closure of Q ^ą0 ¨ Horn ^Z pp, qq.

Horn _hol pp, qq “ Hornpp, qq č

p C _p,q q ³ .

1. If A “ pA ¹ , A ² q P R ^p ˆ R ^q and I “ I ¹ ˆ I ² Ă rps ˆ rqs, we define

| A | I “ | A ¹ | I

` | A ² | I

and | A | “ | A ¹ | ` | A ² |.

2. If I “ I ¹ ˆ I ² Ă rns ˆ rms then λpIq “ pλpI ¹ q, λpI ² qq P ^ ^` _n ˆ ^ ^` _m . 3. Let 1 _n “ p1, . . . , 1q P Z ⁿ .

Theorem 1.6 Let p ě q ě 1. An element pA, B, Cq P p C _p ˆ C _q q ³ belongs to Hornpp, qq if and only if the following conditions holds:

• | A ¹ | ` | B ¹ | ď | C ¹ | .

• For any r P rp´1s, for any I ¹ , J ¹ , K ¹ P P _r ^p , we have :

| A ¹ | I

` | B ¹ | J

ď | C ¹ | K

if pλpI ¹ q, λpJ ¹ q, λpK ¹ qq P Hornprq.

| A ¹ | I

` | B ¹ | J

ě | C ¹ | K

if pλpI ¹ q, λpJ ¹ q, λpK ¹ q ` pq `p´rq1 r q P Hornprq.

• For any s P rq´1s, for any I ² , J ² , K ² P P _s ^q , we have :

| A ² | I

` | B ² | J

ě | C ² | K

if pλpI ² q, λpJ ² q, λpK ² q ` pq´sq1 _s q P Hornpsq.

| A ² | I

` | B ² | J

ď | C ² | K

if pλpI ² q, λpJ ² q, λpK ² q ´ p1 s q P Hornpsq.

• For any pr, sq P rp´1s ˆ rq´1s with r ě s, for any I, J, K P P _r ^p ˆ P _s ^q , we have

Proposition 1.7 An element pA, B, Cq P pR ˆ Rq ³ belongs to Hornp1, 1q if and only if the following conditions holds:

a ₁ ` a ₂ ` b ₁ ` b ₂ “ c ₁ ` c ₂ a ₁ ` b ₁ ď c ₁

Proposition 1.8 An element pA, B, Cq P p C ₂ ˆ Rq ³ belongs to Hornp2, 1q if and only if the following conditions holds:

a ₂ ` b ₁ ď c ₁ a ₁ ` b ₂ ď c ₁ a 1 ` b 1 ě c 2

Proposition 1.9 An element pA, B, Cq P p C ₂ ˆ C ₂ q ³ belongs to Hornp2, 2q if and only if the following conditions holds:

a ₁ ` a ₂ ` b ₁ ` b ₂ ď c ₁ ` c ₂ a ₂ ` b ₂ ď c ₂ a ₂ ` b ₁ ď c ₁ a 1 ` b 2 ď c 1

a ₃ ` b ₃ ě c ₃ a 3 ` b 4 ě c 4

a ₄ ` b ₃ ě c ₄

a ₂ ` a ₄ ` b ₁ ` b ₄ ď c ₁ ` c ₃ a 1 ` a 4 ` b 2 ` b 4 ď c 1 ` c 3

a ₂ ` a ₃ ` b ₂ ` b ₄ ď c ₁ ` c ₃ 1.5 Outline of the article

The recursive description of Hornpp, qq is obtained by studying the Hamil- tonian action of pU ppq ˆ U pqqq ³ on the manifold ² pGL p ˆ GL q q ² ˆ C ^p b C ^q . Let Spp, qq Ă p C _p ˆ C _q q ³ be the corresponding Kirwan polyhedron.

In §2, we study the general framework of a Hamiltonian action of a compact Lie group K ³ on pK _C ˆ K _C q ² ˆ E : here E is a K-module such that the coordinate ring CrEs does not admit non-constant invariant vec- tors. We explain how to parameterize the facets of the Kirwan polyhedron

∆ppK _C ˆ K _C q ² ˆ Eq in terms of Ressayre’s data [22]. This parametrization requires two steps : determination of the admissible elements which are the potential vectors orthogonal to the facets, and computation of cohomological conditions on flag varieties.

In §3, we check that the semigroup Horn ^Z pp, qq is saturated. It is a direct consequence of the Darksen-Weyman saturation theorem [7].

In §4, we determine the admissible elements relative to the action of pU ppqˆU pqqq ³ on pGL p ˆGL q q ² ˆC ^p bC ^q , and we detailed the cohomological conditions in this particular case. The formulas need the computation of certain Euler classes which we carry over to §5.