Thomas Gerber
email: thomas.gerber@epfl.ch

Jérémie Guilhot
email: jeremie.guilhot@lmpt.univ-tours.fr

Cédric Lecouvey
email: cedric.lecouvey@lmpt.univ-tours.fr

Generalised Howe duality and injectivity of induction: the symplectic case

We study the symplectic Howe duality using two new and independent combinatorial methods: via determinantal formulae on the one hand, and via (bi)crystals on the other hand. The first approach allows us to establish a generalised version where weight multiplicities are replaced by branching coefficients. In turn, this generalised Howe duality is used to prove the injectivity of induction for Levi branchings as previously conjectured by the last two authors.

Introduction

Let n and m be two positive integers and let µ be a partition whose Young diagram is contained in the rectangle n × m with n rows and m columns. Then the conjugate partition µ ′ of µ can be written µ ′ = (µ ′ 1 , . . . , µ ′ m ). A classical result in the representation theory of linear Lie algebras states that the multiplicitiy of the irreducible gl n (C)-module V (λ) in the tensor product

Λ µ ′ n,m = Λ µ ′ 1 (C n ) ⊗ . . . ⊗ Λ µ ′ m (C n )
is equal to the dimension of the µ ′ -weight space in the irreducible gl m (C)-modules V (λ ′ ). There exist numerous proofs of this duality in type A (sometimes referred as the Schur duality in the literature) mostly based on computations on Schur functions or on purely combinatorial arguments using semistandard tableaux [START_REF] Fulton | Representation theory. A first course[END_REF].

For the other classical Lie algebras (or classical Lie groups), there exist similar constructions due to Howe [START_REF] Howe | Perspective on invariant theory , The Schur Lecture[END_REF].

In this paper, we restrict ourselves to the symplectic case (i.e. to the root systems of type C). Then, the Howe duality states that the multiplicity of the irreducible sp 2n (C)-module V (λ) in the tensor product

Λ µ ′ 2n,m = Λ µ ′ 1 (C 2n ) ⊗ • • • ⊗ Λ µ ′ m (C 2n ) (1)
is equal to the dimension of the µ-weight space in the irreducible gl m (C)-module V ( λ). In contrast with the type A case, the conjugate partitions µ ′ and λ ′ are here replaced by the partitions µ and λ defined as the conjugates of the complements of µ and λ in the rectangle n × m.

The goal of this paper is three-fold. Firstly, we give a simple combinatorial proof of the Howe duality in the symplectic case based on the determinantal formulae for Weyl characters (analogue to the Jacobi-Trudi formulae for the Schur polynomials). The tools and computations that we use in our proof generalise those developed in [START_REF] Lecouvey | Branching rules, Kostka-Foulkes polynomials and q-multiplicities in tensor product for the root systems B n , C n and D n[END_REF] and extend naturally to the case where the fundamental gl 2n (C)-modules appearing in the tensor products (1) are replaced by tensor products of simple gl 2n (C)-modules (restricted to sp 2n (C)) or simple sp 2n (C)modules. We prove that the corresponding tensor product multiplicities are equal this time to branching coefficients corresponding to the restriction of the simple sp 2m (C)-modules to a block diagonal subalgebras s of sp 2m (C). This means that we need to consider restrictions to subalgebras s ≃ g 1 ⊕ • • • ⊕ g r where each g i is a Lie algebra isomorphic to sp 2mi or gl mi with m 1 + • • • + m r = m. This generalises the original Howe duality which corresponds to the case r = m and m i = 1 for all i.

Secondly, the previous Schur and Howe dualities can be generalised when µ ′ is replaced by any m-tuple β of nonnegative integers. Then the spaces admit a structure of gl n × gl m -bimodule and of sp 2n × sp 2m -bimodule respectively. It is well known that a lot of information about simple modules associated to a simple Lie algebras is encoded by particular combinatorial structures studied by Lusztig, Kashiwara and Littelmann: their crystal graph. It then makes sense to look for bicrystal structures associated to Λ A n,m and Λ C 2n,m . The second main result of the paper uses the combinatorial duality techniques developed in [START_REF] Gerber | Duality and bicrystals on infinite binary matrices[END_REF] to get a simple bijection between the highest weight vertices in the sp 2ncrystal B(Λ C 2n,m ) associated to Λ C 2n,m and the King tableaux (a particular model of tableaux counting the weight multiplicities in type C m , see [START_REF] King | Weight multiplicities for the classical groups[END_REF]). This is reminiscent of a result by Lee [START_REF] Seung | Crystal structure on King tableaux and semistandard oscillating tableaux[END_REF] expressed in a different combinatorial language. In contrast to [START_REF] Seung | Crystal structure on King tableaux and semistandard oscillating tableaux[END_REF], where a type sp 2n × sp 2m -bicrystal structure is proposed, we then study the action of the type A 2m-1 -crystal operators on B(Λ C 2n,m ). These are indeed the operators which are in connection with the charge statistics defined in [START_REF] Lecouvey | Combinatorics of generalised exponents[END_REF]. In particular, the actions of the A 2m-1 -crystal operators associated to nodes with unbarred label correspond to contraction operations on columns of type C n in Λ C 2n,m whereas the action corresponding to nodes with barred label yield jeu de taquin operations on the positive or negative part of these columns. This leads to an intriguing statistics on tensor products of type C n columns which does not coincide with the intrinsic energy defined from their affine crystal structure.

Finally, our third objective is to use the generalised Howe duality to prove a conjecture by the last two authors. Consider a Levi subalgebra l of sp 2m (C) and ν (1) , ν (2) two dominant weights for l. The conjecture claims that the two sp 2m (C)-modules obtained by induction from ν (1) and ν (2) are isomorphic if and only if ν (1) and ν (2) coincide up to an automorphism of the Dynkin diagram associated to l (or equivalently up to permutation of the components in ν (1) and ν (2) associated to isomorphic simple subalgebras). It was proved in [START_REF] Guilhot | Isomorphic induced modules and Dynkin diagram automorphism[END_REF] under restrictive conditions on ν (1) and ν (2) . Here we prove this conjecture in full generality and obtain in fact a more general result in which l can also be replaced by any direct sum s of subalgebras of type C. The proof uses our generalised Howe duality and some elegant results by Rajan [START_REF] Rajan | On the irreducibility of irreducible characters of simple Lie algebras[END_REF] on the irreducibility of Weyl characters.

Most of the techniques and results developed in this paper can be extended to the orthogonal types. There are nevertheless complications due to the existence of the spin representations and the lack of a natural analogue of King tableaux in the duality context which is relevant for the paper. This will be addressed in [START_REF] Gerber | Generalised Howe duality and injectivity of induction: orthogonal case[END_REF]. We also tried to make the paper more accessible by starting with proofs of some known results using methods that will be central for the generalisations that we propose here and in [START_REF] Gerber | Generalised Howe duality and injectivity of induction: orthogonal case[END_REF].

The paper is organised as follows. In Section 2, we review some well-known results on the combinatorics of root systems and Lie algebras, mainly to set up the notations that we will use. In Section 3, we use the Jacobi-Trudi formula for Schur functions to (re)prove the Schur duality. This allows us to introduce the main tools and methods which will be reinvested in Section 4 where Howe duality is derived similarly from determinantal identities for the Weyl characters of type C n . In Section 5, we use crystals to define a natural combinatorial duality between highest weight vertices in B(Λ C 2n,m ) and King tableaux, thereby giving a bijective proof of the Howe duality. We then study in Section 6 the behavior of B(Λ C 2n,m ) under some crystal operators of type A 2m-1 . Section 7 is devoted to establishing the generalised Howe duality using determinantal techniques similar to that of Section 4. Finally, we prove the generalised version of the conjecture of [START_REF] Guilhot | Isomorphic induced modules and Dynkin diagram automorphism[END_REF] in Section 8.

Generalities and settings

Let g be a Lie algebra with root system R with triangular decomposition

g = α∈R + g α ⊕ h ⊕ α∈R + g -α
so that h is the Cartan subalgebra of g and R + its set of positive roots.

We assume that R is realised in a real Euclidean space V with inner product •, • . Let ∆ be the simple system associated to R. For α ∈ R, let α ∨ = 2α α, α be the coroot associated to α. The Weyl group W of R is the group generated by the orthogonal reflections s α with respect to the hyperplanes

H α := {x ∈ V | x, α ∨ = 0}.
The group W is a finite Coxeter group with distinguished set of generators S = {s α | α ∈ ∆}. The closure of the connected components of the set V \ ∪ α∈R H α are called the Weyl chambers. The fundamental Weyl chamber is defined by

C 0 = {x ∈ V | x, α > 0 for all α ∈ R + }.
For all α ∈ ∆, we define the fundamental weight ω α ∈ V by the relations ω α , β ∨ = δ α,β for all β ∈ ∆.

Let P = ⊕ α∈∆ Zω α be the weight lattice and P + be the set of dominant weights:

P + = {λ ∈ P | λ, α ∨ ≥ 0 for all α ∈ ∆} = P ∩ C 0 .
Example 2.1 (Root system of type A). Let E be the Euclidean space of dimension m with basis (ε 1 , . . . , ε m ) and let V be the quotient of E by the subspace generated by m i=1 ε i . The space V is of dimension m-1 and can be endowed with an euclidean structure as it is isomorphic to the hyperplane of E orthogonal to

ε 1 + • • • + ε m . The set R + = {ε i -ε j | i, j ∈ {1, . . . , m}, i < j}
is a positive root system of type A m-1 . For all i ∈ {1, . . . , m -1} we set α i = ε iε i+1 and we denote by s i the reflection with respect to α i . The simple system associated to R + is

∆ = {α i | i = 1, . . . , m -1}.
The reflection s εi-εj acts on V by permuting the i-th and j-th coordinate. We have

C 0 = {(x 1 , . . . , x m ) ∈ V | x i -x i+1 ≥ 0 for all i}.
For all i ∈ {1, . . . , m -1}, the fundamental weight associated to α i is ω i = ε 1 + . . . + ε i and

P + = {(λ 1 , . . . , λ m ) ∈ Z m | λ i -λ i+1 ≥ 0 for all i}.
The set P + is in bijection with the set of partition of length at most m -1. Indeed

(λ 1 , . . . , λ m ) = (λ 1 -λ m , . . . , λ m-1 -λ m , 0) in V .
Example 2.2 (Root system of type C). Let V be the Euclidean space of dimension m with basis (ε 1 , . . . , ε m ).

The set

R + = {ε i -ε j | i, j ∈ {1, . . . , m}, i < j} ∪ {2ε i | i = 1, . . . , m}
is a positive root system of type C m . The associated simple system is

∆ = {α i | i = 1, . . . , m -1} ∪ {2ε m } where α i = ε i -ε i+1 .
We denote by s i the reflection s αi and by s m the reflection s 2εm . Note that s εi-εj acts on V by permuting the i-th and j-th coordinate and the reflection s 2εi changes the sign of the i-th coordinate. We have

C 0 = {(x 1 , . . . , x m ) ∈ V | x i -x i+1
≥ 0 for all i and x m ≥ 0}.

and

P + = {(λ 1 , . . . , λ m ) ∈ Z m | λ i -λ i+1 ≥ 0 for all i and λ m ≥ 0}.
The set P + is in bijection with the set of partitions of length at most m.

We now turn to the representation theory of g. The representation ring of g is the ring with basis indexed by the isomorphism classes [V ] of irreducible representations V of g over C. The addition is defined such that

[V ] + [V ′ ] = [V ′′ ] whenever V ′′ ≃ V ⊕ V ′ and the multiplication is defined by [V ] × [W ] = [V ⊗ W ].
We will denote it by R(g).

Let Z[P ] be the integral group ring on the abelian group P . We will write e λ for the element associated to λ ∈ P so that we have e λ • e λ ′ = e λ+λ ′ . The Weyl group W of R acts naturally on Z[P ] by setting w • e λ = e w(λ) . We denote by Z[P ] W the set of fix points:

Z[P ] W = {f ∈ Z[P ] | w • f = f }.
Let char be the injective ring homomorphism from the representation ring of g to Z[P ] defined by

char([V ]) = dim(V µ )e µ
where V µ is the µ-weight space in V . For λ ∈ P , let V (λ) be the irreducible module of highest weight λ in g.

Remark 2.3. In this paper, we will deal with modules for Lie algebras of type A or C of various rank. When necessary, we will add a superscript to the notation V (λ) to indicate the type and the rank of the Lie algebra we are working with. For instance, we will write V C (λ) for the irreducible module of highest weight λ in a Lie algebra of type C and V Cm (λ) if we further want to indicate the rank m of the Lie algebra.

Theorem 2.4 ([1, Theorem 23.24]). The ring R(g) is a polynomial ring in the variables (char([V (ω α )])) α∈∆ . The homomorphism char : R(g) → Z[P ] W is an isomorphism.

Let ρ be the half sum of positive roots. Then the Weyl character formula asserts that, for λ ∈ P + , the character of V (λ) is

s λ = a λ+ρ a ρ where a λ = w∈W ε(w)e wλ .
The set {s λ | λ ∈ P + } is a basis of Z[P ] W . The Kostant partition function P is defined by the formula

1 α∈R + (1 -e α ) = β∈Z m P(β)e β .
The dot action of the Weyl group on P is defined by

w • λ = w(λ + ρ) -ρ = t -ρ wt ρ (λ),
where, for all γ ∈ P , t γ denotes the translation by γ.

Theorem 2.5 (Kostant mutliplicity formula). Let λ, µ ∈ P + . Let K λ,µ be the dimension of the µ-weight space in the irreducible representation V (λ) of highest weight λ. We have

K λ,µ = w∈W ε(w)P(w • λ -µ).
Example 2.6. Assume that g = sl n (C). Then the root system R of g is of type A n-1 as in Example 2.1, and we have W = S n . We set x i = e εi ∈ Z[P ]. Since we realised R in the quotient space V we have the relation

x 1 • • • x n = 1 in Z[P ].
As a consequence, one can show that

Z[P ] W ≃ Z[x 1 , . . . , x n ] Sn / x 1 • • • x n -1
where Z[x 1 , . . . , x n ] Sn is the ring of symmetric polynomials in n variables. The representations

V (ω k ) for k = 1, . . . , n -1 are isomorphic to Λ k (C n ), the k-th exterior power of the natural representation C n of g. Then we have char(V (ω k )) = e k (x 1 , . . . , x n ) ∈ Z[P ]
W where e k is k-th elementary symmetric function in n variables:

e i (x 1 , . . . , x n ) = 1≤i1<i2<...<i k ≤n x i1 • • • x i k .

Schur duality in type A

In this section, we prove the Schur duality for sl n (C) using the same methods that we will use to prove the Howe duality in type C. The root system associated to sl n (C) is of type A n-1 and we keep the notations of Example 2.1 and Section 2.

For all integers n, m ∈ Z ≥1 , we denote by P n the set of partitions of the form (λ 1 , . . . , λ n ), and by P n,m the set of partitions in P n such that λ 1 ≤ n (i.e. the Young diagram of a partition in P n,m is included in a rectangle with sides n×m). For any partition λ ∈ P n,m , we denote by (λ 1 , . . . , λ n ) the parts of λ. The conjugate partition is denoted by λ ′ and λ ′ = (λ ′ 1 , . . . , λ ′ m ). Note that λ ∈ P n,m if and only if λ ′ ∈ P m,n . We have seen in Example 2.1 that the set of dominant weights for a root system of type A n-1 is in bijection with P n-1 . We will freely identify those sets.

For N ∈ Z ≥1 and γ, λ ∈ P N we write K AN γ,λ for the dimension of the γ-weight space in the irreducible sl N (C)module V (λ) of highest weight λ. For any sl N (C)-module M , we write [M : V (λ)] for the multiplicity of V (λ) in M . Theorem 3.1 (Schur duality). Let µ ∈ P n,m and let µ ′ = (µ ′ 1 , . . . , µ ′ m ). For all λ ∈ P n-1 , we have

Λ µ ′ 1 (C n ) ⊗ • • • ⊗ Λ µ ′ m (C n ) : V (λ) = K Am λ ′ ,µ ′
where Λ k (C n ) is the k-th exterior power of the natural representation C n of sl n (C).

The character of Λ k (C n ) is e k (x 1 , . . . , x n ) and so the character of the tensor product

Λ µ ′ 1 (C n ) ⊗ . . . ⊗ Λ µ ′ m (C n ) is e µ ′ 1 .
. . e µ ′ m . If we define u λ,µ ′ by the relation

e µ ′ 1 . . . e µ ′ m = λ∈Pn-1,m u λ,µ ′ s λ
the theorem states that u λ,µ ′ = K Am λ ′ ,µ ′ for all λ ∈ P n-1,m . For β = (β 1 , . . . , β m ) ∈ Z m we define the matrix V(β) by

[V(β)] i,j = e βi+j-i
where e k = 0 whenever k / ∈ {0, . . . , n}. We set v β = det(V(β)). More explicitely we have:

v β =
) = x 1 • • • x n = 1 in Z[P ] W we have v β = v β † where β † = (β k , . . . , β 1 , 0, . . . , 0) ∈ Z m . Given β ∈ Z m we set x β = x β1 1 • • • x βm m and e β = e β1 • • • e βm .
We define E to be the linear map

E : Z[P ] → Z[P ] W x β → e β
This map can be extended to the set of formal series Z[[P ]]. Indeed, there are only finitely many β ∈ Z m such that e β = 0. The map E satisfies the following useful lemma.

Lemma 3.3. Let β and γ in Z m be such that there exists an integer 1 ≤ s ≤ m with

β i = 0 for s + 1 ≤ i ≤ m and γ i = 0 for 1 ≤ i ≤ s. Then 1 E(x β+γ ) = E(x β • x γ ) = E(x β ) • E(x γ ).
Proof. The equality E(x β+γ ) = E(x β • x γ ) is clear since we have x β+γ = x β • x γ . On the one hand, we have We set

∆ A m = α∈R + (1 -e α ) = 1≤i<j≤n (1 - x i x j ). Proposition 3.4. For all β ∈ Z m we have E(∆ A m • x β ) = v β .
Proof. Let β = (β 1 , . . . , β m ). We have

x β1 1 x β1+1 1 . . . x β1+m-1 1 x β2-1 2 x β2 2 . . . x β2+m-2 2 . . . . . . . . . . . . x βm-m+1 m x βm-m+2 m . . . x βm m = x β1 1 . . . x βm-m+1 m • 1 x 1 . . . x m-1 1 1 x 2 . . . x m-1 2 . . . . . . . . . . . . 1 x m . . . x m-1 m = x β1 1 . . . x βm-m+1 m • 1≤i<j≤m (x j -x i ) = x β1 1 . . . x βm m • 1≤i<j≤m (1 - x i x j ) = ∆ A m • x β .
By expanding the determinant, we get

E(∆ A m • x β ) = E σ∈Sm x β1-(σ(1)-1) 1 . . . x βm-(σ(m)-m) m = σ∈Sm E(x β1-(σ(1)-1) 1 ) . . . E(x βm-(σ(m)-m) m ) = σ∈Sm e β1-(σ(1)-1) . . . e βm-(σ(m)-m) = v β as required.
Recall the definition of the dot action in Section 2.

Proposition 3.5. Let β ∈ Z m . For all w ∈ W , we have v w•β = ε(w)v β . Further either v β = 0 or there exists a partition γ ∈ P m,n-1 and w ∈ W such that v β = ε(w)v γ .
Proof. We prove the first assertion. To do so, it is enough to show that for all i ∈ {1, . . . , n -1}, we have

v si•β = -v β . The matrix V(s i • β) only differs from V(β) on rows i and i + 1. Let β = (β 1 , . . . , β m ).
We have

[V(s i • β)] i,j = e (si•β)i-i+j = e βi+1-1-i+j = e βi+1-(i+1)+j = [V(β)] i+1,j [V(s i • β)] i+1,j = e (si•β)i+1-(i+1)+j = e βi+1-(i+1)+j = e βi+1-i+j = [V(β)] i,j
It follows that

v si•β = det(V(s i • β)) = -det(V(β)) = -v β .
as required for the first assertion.

We prove the second assertion. Let β ∈ Z m be such that v β = 0. Using the first relation and the fact that the dot action is simply the action of the Weyl group translated by -ρ, we see that there exists w ∈ W such that γ = w • β lies in the chamber t -ρ (C 0 ). This means that for all

α i = ε i -ε i+1 ∈ ∆, we have γ, α i ≥ -ρ, α i = -1.
If there exists α i such that γ, α = -1, then s i • γ = γ and this forces v γ and v β to be 0. This is impossible since we assumed that v β = 0 hence we must have γ, α i ≥ 0 for all i, i.e. γ i ≥ γ i+1 . Next we must have γ 1 ≤ n otherwise the first row of V(γ) is 0 and γ m ≥ 0 otherwise the last row of V(γ) is 0. Finally by Remark 3.2, we can remove all the parts of γ that are equal to n to obtain a partition γ † such that γ † 1 < n and γ † 0 ≥ 0 and v β = ±v γ † , as required.

We are now ready to prove the Schur duality.

Proof of Theorem 3.1. Let µ ∈ P n,m . We have

E(x µ ′ ) = E(∆ A m • 1 ∆ A m • x µ ′ ) = E(∆ A m • β∈Z m P(β)x β+µ ′ ) (by definition of P) = β∈Z m P(β)E(∆ A m • x β+µ ′ ) = β∈Z m P(β)v β+µ ′ (by Proposition 3.4) = γ∈Pm,n-1 w∈W ε(w)P(w • γ -µ ′ )v γ (by Proposition 3.5) = γ∈Pm,n-1 K Am γ,µ ′ s γ ′ (by Theorem 2.5) = λ∈Pn-1,m K Am λ ′ ,µ ′ s λ .
Since E(x µ ′ ) = e µ ′ we get the result.

Howe duality in type C

In this section, we prove the Howe duality for sp 2n (C). The root system associated to sp 2n (C) is of type C n and we keep the notation of Example 2.2 and Section 2. Recall that the set of dominant weights of sp 2n (C) is in bijection with P n . We will freely identify those two sets.

Let V be the Euclidean space of dimension m with basis (ε 1 , . . . , ε m ). We define the involution I on V by

I(β 1 , . . . , β m ) = (-β m , . . . , -β 1 ).
β = I(β ′ ) + 4 • (1, 1, 1, 1, 1) = (3, 2, 2, 1, 0).
In the figure below, we represent the partition β (in green) as the conjugate complement of the partition β (in white) in the rectangle 4 × 5:

The map • will play in the Howe duality a role analogue to the conjugation in the Schur duality. For N ∈ Z ≥1 and γ, λ ∈ P N , we write K CN γ,λ for the dimension of the γ-weight space in the irreducible representation

V (λ) ∈ sp 2N (C) of highest weight λ. Theorem 4.3 (Howe Duality [7]). Let µ ∈ P n,m and let µ ′ = (µ ′ 1 , . . . , µ ′ m ).
For all λ ∈ P n , we have

Λ µ ′ 1 (C 2n ) ⊗ . . . ⊗ Λ µ ′ m (C 2n ) : V (λ) = K Cm λ, µ
where Λ k (C 2n ) is the k-th exterior power of the natural representation C 2n of sp 2n (C).

Following [1, Section 24.2], the character of

Λ k (C 2n ) is e k (x 1 , . . . , x n , x -1 1 , . . . , x -1 n ).
If we define the coefficients u by setting (we omit the variables)

e µ ′ 1 . . . e µ ′ m = λ∈Pn,m u λ,µ ′ s λ then the theorem states that u λ,µ ′ = K Cm λ, µ
for all λ ∈ P n,m .

For β = (β 1 , . . . , β m ) ∈ Z m we define the matrix V(β) by

[V(β)] i,j = e βi-(i+j) -e βi-(i-j)
where e k = 0 whenever k / ∈ {0, . . . , 2n}. We set v β = det(V(β)). More explicitely we have

v β = e β1 -e β1-2 e β1+1 -e β1-3 . . . e β1+m-1 -e β1-m-1 e β2-1 -e β2-3 e β2 -e β2-4 . . . e β2+m-2 -e β2-m-2 . . . . . . . . . . . . e βm-m+1 -e βm-m-1 e βm-m+2 -e βm-m-2 . . . e βm -e βm-2m
.

The Jacobi-Trudi formula in type C n states that for all λ ∈ P n,m , we have

s λ = v λ ′ .
Given β ∈ Z m we set x β = x β1 1 . . . x βn n and e β = e β1 . . . e βn . We then define E to be the linear application:

E : Z[P ] → Z[P ] W x β → e β
It is not hard to check that the map E can be extended to Z[[P ]] and that it satisfies Lemma 3.3.

The involution I on V induces a ring involution Z[P ] that maps e β to e I(β) for all β ∈ P . In particular it maps x i = e εi to x -1 m-i = e -εm-i . We will still denote this involution by I. We set

∆ C m = I α∈R + (1 -e α ) = 1≤i<j≤m 1 - x i x j 1≤i≤j≤m 1 - 1 x i x j . Proposition 4.4. For all β ∈ Z m , we have E(∆ C m • x β ) = v β . Proof. Let β = (β 1 , . . . , β m ). We have det(x βi-i+j i -x βi-i-j i ) = m i=1 (x βi-i+1 i -x βi-i-1 i ) 1 x 1 + x -1 1 . . . x m-1 1 + x -m+1 1 1 x 2 + x -1 2 . . . x m-1 2 + x -m+1 2 . . . . . . . . . . . . 1 x m + x -1 m . . . x m-1 m + x -m+1 m = m i=1 (x βi-i+1 i -x βi-i-1 i ) 1 x 1 + x -1 1 . . . (x 1 + x -1 1 ) m-1 1 x 2 + x -1 2 . . . (x 2 + x -1 2 ) m-1 . . . . . . . . . . . . 1 x m + x -1 m . . . (x m + x -1 m ) m-1 = m i=1 (x βi-i+1 i -x βi-i-1 i ) 1≤i<j≤m (x j + x -1 j -x i -x -1 i ) = m i=1 (x βi-i+1 i -x βi-i-1 i ) 1≤i<j≤m x j (1 - x i x j )(1 - 1 x i x j ) = m i=1 (x βi-i+1 i -x βi-i-1 i )x 2 . . . x m-1 m 1≤i<j≤m (1 - x i x j )(1 - 1 x i x j ) = m i=1 (x βi i -x βi-2 i ) 1≤i<j≤m (1 - x i x j )(1 - 1 x i x j ) = x β m i=1 (1 - 1 x 2 i ) 1≤i<j≤m (1 - x i x j )(1 - 1 x i x j ) = ∆ C m • x β
as required. By expanding the determinant, we get

E(∆ C m • x β ) = E σ∈Sm m i=1 x βi-(σ(i)-i) i -x βi-(σ(i)-i)-2 i = σ∈Sm E m i=1 x βi-(σ(i)-i) i -x βi-(σ(i)-i)-2 i = σ∈Sm m i=1 E x βi-(σ(i)-i) i -x βi-(σ(i)-i)-2 i (by Lemma 3.3) = σ∈Sm m i=1 e βi-(σ(i)-i) -e βi-(σ(i)-i)-2 = v β .
In the proof of the Schur duality, we used the trick

E(x µ ′ ) = E(∆ A m • 1 ∆ A m x µ ′ )
and we expanded the inverse of ∆ A m with the Kostant partition function. Using the same idea here involves some twisted Kostant partition function. Proof. Using the fact that I is an involution we get

1 ∆ C m = I    1 α∈R+ (1 -e α ))    = I( β∈Z m P(β)x β ) = β∈Z m P(β)x I(β) = β∈Z m P(I(β))x β
hence the result.

The next step in our proof of the Schur duality in type A was to show that the determinant v β satisfied the relation v w•β = ε(w)v β for all w ∈ S m . In type C m , we need to modify the dot action to obtain such a relation. More precisely, we are looking for δ ∈ P such that we have for all w ∈ W :

v w•β = ε(w)v β where w • β = w(β + δ) -δ.
It will turn out that there is a unique δ satisfying this relation. The proof is based on the following symmetry. Recall that the m-th fundamental weight in a root system of type C m is ω m = (1, . . . , 1) and that the half sum of positive roots is ρ m = (m, . . . , 1).

Theorem 4.8. Let δ = I(ρ)m • ω m ∈ Z m and define a "dot action" of W on Z m by setting

w • β = w(β + δ) -δ.
We have

(1) For all β ∈ Z m , we have v w•β = ε(w)v β .
(2) If v β = 0, there exists a partition γ ∈ P m,n and w ∈ W such that w

• β = γ and v β = ε(w)v γ .
Proof. We have δ = (-n -1, . . . , -nm). Recall that the Weyl group W is generated by the set {s 1 , . . . , s n-1 , s n } where s i acts on V by permuting the i and the i + 1-th coordinates and s n acts by changing the sign of the n-th coordinate.

(1) To prove this assertion, it is enough to show that

v si•β = -v β for all 1 ≤ i ≤ n -1 and v sn•β = -v β .
Let β = (β 1 , . . . , β m ) ∈ Z m and i ∈ {1, . . . , m -1}. We have

s i • β = s i (β + δ) -δ = (β 1 + δ 1 , . . . , β i+1 + δ i+1 , β i + δ i , . . . , β m + δ m ) -(δ 1 , . . . , δ m ) = (β 1 , . . . , β i+1 + δ i+1 -δ i , β i + δ i -δ i+1 , . . . , β m ) = (β 1 , . . . , β i+1 -1, β i + 1, . . . , β m ).
The matrix V(s i • β) only differs from the matrix V(β) on rows i and i + 1. We have

[V(s i • β)] i,j = e [si•β]i-(i+j) -e [si•β]i-(i-j) = e βi+1-1-(i+j) -e βi+1-1-(i-j) = e βi+1-((i+1)+j) -e βi+1-((i+1)-j) = [V(β)] i+1,j [V(s i • β)] i+1,j = e [si•β]i+1-(i+1+j) -e [si•β]i+1-(i+1-j) = e βi+1-(i+1+j) -e βi+1-(i+1-j) = e βi-(i+j) -e βi-(i-j) = [V(β)] i,j
Hence the result on the determinant.

Next we have

s m • β = s m (β + δ) -δ = (β 1 + δ 1 , . . . , β m-1 + δ m-1 , -β m -δ m ) -(δ 1 , . . . , δ m ) = (β 1 , . . . , β m-1 , -β m -2δ m ) = (β 1 , . . . , β m-1 , -β m + 2n + 2m).
The matrix V(s n • β) only differs from the matrix V(β) on rows m and we have

[V(s m • β)] m,j = e [sn•β]m-(m+j) -e [si•β]m-(m-j) = e -βm+2n+2m-(m+j) -e -βm+2n+2m-(m-j) = e n+(n-βm+m-j) -e n+(n-βm+m+j) = e n-(n-βm+m-j) -e n-(n-βm+m+j) (by the previous proposition) = e βm-m+j -e βm-m-j = -[V(β)] m,j
hence the result on the determinant.

(2) Let β ∈ Z m be such that v β = 0. The "dot action" is simply the action of W translated by -δ. This action is transitive on the set of Weyl chambers centered at δ. The set of Weyl chambers centered at -δ is the set of Weyl chambers centered at 0 translated by -δ. Now the set of Weyl chambers centered at 0 is parametrised by the set of simple system of the form w∆ where w ∈ W . More precisely, the Weyl chamber

w(C 0 ) is w(C 0 ) = {x ∈ V | x, α > 0 for all α ∈ w(∆)}.
We have I ∈ W and

I(∆) = {I(α 1 ), . . . , I(α m-1 ), I(2ε m )} = {α 1 , . . . , α m-1 , -2ε 1 }.
As a consequence the Weyl chamber t -δ (I(C 0 )) is defined by

x ∈ t -δ (I(C 0 )) ⇔ x, α ∨ > -δ, α ∨ for all α ∈ I(∆).
By transitivity of the dot action, there exists w ∈ W such that γ = w • β ∈ t -δ (I(C 0 )). Further, since we assumed that v β = 0, γ cannot lie on the wall of the chamber t -δ (I(C 0 )). The equations above for the root α i ∈ I(∆) and -2ε 1 ∈ I(∆) yield (using

α ∨ i = α i and (-2ε 1 ) ∨ = -ε 1 ) γ, α i > -δ, α i = -δ i + δ i+1 = -1 and γ, -ε 1 > -δ, -ε 1 = -(n + 1)
in other words γ iγ i+1 ≥ 0 and γ 1 ≤ n

which was what we were looking for, since v β = ε(w)v γ .

We are now ready to prove the Howe duality.

Proof of Theorem 4.3. Let µ ∈ P n,m . First we have

E(x µ ′ ) = E ∆ C m • 1 ∆ C m • x µ ′ = E   ∆ C m β∈Z m P(β)x β+µ ′   = β∈Z m P(β)E ∆ C m x β+µ ′ = β∈Z m P(β)v β+µ ′ = γ∈Pm,n w∈W ε(w) P(w • γ -µ ′ )v γ = γ∈Pn,m w∈W ε(w) P(w • λ ′ -µ ′ )s λ .
This shows that

u λ,µ ′ = w∈W ε(w) P(w • λ ′ -µ ′ ).
Next, using the fact that I(δ m,n ) = ρ + n • ω m and the equality P(β) = P(I(β)) we get

w∈W ε(w) P(w • λ ′ -µ ′ ) = w∈W ε(w)P(I(w • λ ′ ) -I(µ ′ )) = w∈W ε(w)P(I(w(λ ′ + δ m,n ) -δ m,n ) -I(µ ′ )) = w∈W ε(w)P(w(I(λ ′ ) + I(δ m,n )) -I(δ m,n ) -I(µ ′ )) = w∈W ε(w)P(w(I(λ ′ ) + ρ + n • ω m ) -ρ -n • ω m -I(µ ′ )) = w∈W ε(w)P(w(I(λ ′ ) + n • ω m + ρ) -(I(µ ′ ) + n • ω m ) -ρ) = w∈W ε(w)P(w( λ + ρ) -µ -ρ) = K Cm λ, µ
as desired.

Combinatorial Howe duality

The goal of this section is to give a bijective proof of Theorem 4.3. This is achieved by establishing in Theorem 5.11 a combinatorial duality between a certain set of tensor products of type C n columns on the one hand, and a set of type C m tableaux called King tableaux on the other hand. To make this duality consistent with the usual convention on the combinatorial objects that we shall need (tensor products of crystals and King tableaux) it will be convenient to use the following realisation of the root system of type C n .

Let n ∈ Z ≥1 and consider the type C n alphabet

C n = n < • • • < 1 < 1 < • • • < n .
This enables us to realise the root system of type C n by setting

α i = ε i -ε i+1 for i = 1, . . . , n -1 α 0 = 2ε 1 .
for the simple roots, where ε i = -ε i for all i = 1, . . . , n, and

ω i = ε n + • • • + ε i+1 for i = 0, . . . , n -1.
A column of height k on C n (also called a column of type C n ) is a subset c of C n of cardinality k, which we represent by the Young tableau of shape (1 k ) filled by the elements of c, increasing from top to bottom. We will write |c| = k. For any column c on C n , and for all i = 1, . . . , n let

N i (c) = {x ∈ c | x ≤ i or x ≥ i} . Definition 5.1. A column c on C n is called n-admissible if N i (c) ≤ n -i + 1 for all i = 1, . . . , n. Example 5.2. The set c = {2, 1, 1, 3} = 2 1 1 3
is a column on C n for all n ≥ 3, and we have |c| = 4. It is not

3-admissible since N 1 (c) = {3, 2, 1, 1} = 4 > 3 -1 + 1. However, c is n-admissible for n ≥ 4.
Let us recall the crystal structure on the set of columns of a given height due to [START_REF] Kashiwara | Crystal graphs for representations of the q-analogue of classical Lie algebras[END_REF]Section 4.3]. First, columns of height 1 realise the crystal of the vector representation of type C n as follows:

n • • • 1 1 • • • n n -1 1 0 1 n -1
where the arrow labeled by i denotes the action of the Kashiwara crystal operator f i . In other terms, since

wt x = n i=1 a i ε i where a i =    1 if x = i -1 if x = i 0 otherwise,
this realises the crystal B(ω n-1 ) of the fundamental representation V (ω n-1 ). To get the crystal structure on any tensor power B(ω n-1 ) ⊗ℓ , we use Kashiwara's tensor product rule. It works as follows. Each vertex

b = x1 ⊗ x2 ⊗ • • • ⊗ x ℓ ∈ B(ω n-1
) ⊗ℓ can be identified with its word w = x 1 • • • x ℓ of length ℓ on C n . Now, label each letter of w in {i + 1, i} (resp. in {i, i + 1}) with a symbol + (resp. -) and ignore the others. Let w i be the word in the symbols + andso obtained. Bracket recursively all possible +-in w i (forgetting the previously bracketed symbols). Then f i w (resp. e i w) is obtained by applying f i (resp. e i ) to the letter of w which contributes as the leftmost unbracketed + (resp. the rightmost unbracketed -). If this symbol does not exist, we set f i w = 0 (resp. e i w = 0). To compute f 0 w (resp. e 0 w), one proceeds similarly but this time by encoding only the letters n in w by a + and the letters n by a -. Let us now fix n, m ∈ Z ≥1 . In the spirit of [2, Section 2.2], we define a "combinatorial Fock space" which will naturally be endowed with a type C n crystal structure.

Definition 5.6. The Kashiwara-Nakashima (KN) Fock space is the set

F n,m = {c 1 ⊗ • • • ⊗ c m | c j
is a column on C n for all j = 1, . . . , m} .

We will be interested in a particular subset of the KN Fock space. Denote by C m,2n the set of compositions

µ ′ = (µ ′ 1 , . . . , µ ′ m ) such that max µ ′ j ; 1 ≤ j ≤ m ≤ 2n. For µ ′ ∈ C m,2n
, consider the set

B µ ′ = c 1 ⊗ • • • ⊗ c m ∈ F n,m | |c j | = µ ′ j for all j = 1, . . . , m .
By choosing to read the m columns of any vertex of B µ ′ first from left to right and next from top bottom, we get an embedding of crystals B µ ′ ֒→ B(ω n-1 ) ⊗|µ| , and B µ ′ realises the crystal of the representation

Λ µ ′ 2n,m = Λ µ ′ 1 (C 2n ) ⊗ • • • ⊗ Λ µ ′ m (C 2n )
of sp 2n (C) (which we have encountered in Section 4). Moreover, we have the decomposition as direct sum of crystals

F n,m = µ ′ ∈Cm,2n B µ ′ .
Note that in the basis (ε n , . . . , ε 1 ), we have wt(b) = (a n , . . . , a 1 ) where, for all i = 1, . . . , n,

a i = # entries i in b -# entries i in b.
Inside B µ ′ , we consider the following two subsets

B hw µ ′ = {b ∈ B µ ′ | e i (b) = 0 for all i = 0, . . . , n -1} and B hw µ ′ ,λ = b ∈ B hw µ ′ | wt(b) = λ
for any λ ∈ P n,m . In other terms, B hw µ ′ is the set of highest weight vertices in B µ ′ . Let us recall briefly how to check that an element b ∈ B µ ′ is in B hw µ ′ . Let w be the word obtained by reading b (as explained above). Define similarly the weight of a word on C n to be the n-tuple whose i-th coordinate is the difference between the number of i's and the number of i's. Then b is a highest weight vertex if and only if the weight of each prefix of w is a partition.

Ḟm,n = {d 1 ⊗ • • • ⊗ d n | d i is a column on C * m for all i = 1, . . . , n} .
Remark 5.9. Unlike F n,m , there is no simple C m -crystal structure on Ḟm,n . This will be discussed in more detail in Section 6.

We are ready to define a duality

F n,m -→ Ḟm,n b -→ b * .
Let µ ′ be a composition as before and b

= c 1 ⊗ • • • ⊗ c m ∈ B µ ′ . For each j = 1, . . . , m, let cj = {1 ≤ x ≤ n | x / ∈ c j } and cj = {1 ≤ x ≤ n | x ∈ c j } = {1, . . . , n} ∩ c j , and set b = c1 ⊗ c1 ⊗ • • • ⊗ cm ⊗ cm .
Then b is a tensor product of 2m columns of type A n-1 . We can now apply the duality * of [START_REF] Gerber | Duality and bicrystals on infinite binary matrices[END_REF] to the element b to get an element b * ∈ Ḟm,n . More precisely, we set .

b * = d 1 ⊗ • • • ⊗ d n where d i = {x ∈ C * m | i ∈ cx } for all i = 1, . . . , n.
We obtain the following product of columns of type A 4 : .

b = c1 ⊗ c1 ⊗ c2 ⊗ c2 =
Finally, we find

b * = 1 2 ⊗ 2 ⊗ 2 ⊗ 1 1 2 2 ⊗ 1 1 .
In particular, [2, Proposition 2.17] ensures that if b ∈ B hw µ ′ , then b * is a semistandard tableau (on the alphabet C * m ), where we identify a tableau with the tensor product of its columns (from left to right with our convention). In fact, we have more. In order to state the following theorem, recall that a tableau with entries in C * m is called a King tableau if (1) it is semistandard, and

(2) if each entry in row j is greater than or equal to j for every index j.

The weight of a tableau t with entries in C * m is the sequence wt(t) = (a m , . . . , a 1 ) where, for all j = 1, . . . , m,

a j = # entries j in b -# entries j in b.
In order to state the main result of this section, recall the involutive map β ′ → β defined in Section 4. Moreover, let us denote by K λ, µ the set of King tableaux of shape λ and weight µ.

Theorem 5.11 (Combinatorial Howe duality). We have a bijection

B hw µ ′ ,λ -→ K λ, µ b -→ b * .
Let us justify the name of the previous theorem. By general crystal theory, the cardinality of B hw µ ′ ,λ equals the multiplicity of V (λ) in Λ µ ′ 2n,m . Moreover, it is known that the cardinality of K λ, µ equals the weight multiplicity K Cm λ, µ , see [START_REF] King | Weight multiplicities for the classical groups[END_REF]. Therefore, the bijection of Theorem 5.11 permits to recover the Howe duality (Theorem 4.3). In fact, we directly obtain the more general version where the heights of the columns in the tensor product are not necessarily decreasing (that is, using compositions instead of partitions). Proof. Let us start by proving that the duality * intertwines shape and weight as claimed. Write wt(b * ) = µ = (µ m , . . . , µ 1 ). From the definition of * , we see that, for all j = 1, . . . , m,

µ j = |c j | -|c j | = # unbarred entries in c j -(n -# barred entries in c j ) = n -|c j | = n -µ ′ j .
Therefore, we have µ = µ. Similarly, if we write b * = d 1 ⊗ • • • ⊗ d n and sh(b * ) = λ = (λ 1 , . . . , λ m ), we have, for all i = 1, . . . , n,

λ ′ i = |d i | = m j=1 1 -1 cj (i) + 1 cj (i) = m - m j=1 1 cj (i) -1 cj (i) = m -# entries i in b -# entries i in b = m -λ i ,
and taking the transpose yields λ = λ.

In (2) In [START_REF] Heo | Combinatorial Howe duality of symplectic type[END_REF], Heo and Kwon studied the type C Howe duality by using a symplectic version of the RSK algorithm.

If |µ| = 1 then the only element of B hw µ ′ is b = n . We have b * = 1 ⊗ • • • ⊗ 1 ⊗ ∅,
Their results also involve King tableaux and other combinatorial objects (the spinor model). In our approach, we expect Theorem 5.11 to be related to a symplectic RSK correspondence arising from a bicrystal structure on tensor products of admissible columns in the spirit of [2, Theorem 2.25], see also Remark 6.14(1) of the upcoming section.

Bicrystals and charge

As mentioned in Remark 5.9, Ḟm,n does not come with a natural type C m crystal structure. In fact, even on the subset of King tableaux, finding such a crystal structure is a challenging problem, see Remark 6.14. However, there is a natural type A 2m-1 crystal structure on Ḟm,n induced from the crystal of the vector representation below

1 1 2 2 • • • m m . 1 1 2 2 m -1 m
To compute the A 2m-1 -crystal structure on Ḟm,n , we use a different reading than the one used in Section 5.

More precisely, we choose this time to read the columns of b ∈ Ḟm,n first from right to left and next from top to bottom. The action of the crystal operators, denoted by ḟj , 1 ≤ j ≤ m, and ḟj , 1 ≤ j ≤ m -1 (resp. ėj and ėj ) on the resulting word is computed by using the same bracketing procedure as in Section 5 (illustrated in Example 5.3), where we encode this time j by + and j byfor ḟj , ėj , and j by + and j + 1 byfor ḟj , ėj (and we ignore the other letters). This is illustrated in the example below. Proof. By definition of the A 1 -crystal structure on Ḟm,n , the element

c * = c * 1 ⊗ • • • ⊗ c * n is a highest weight vertex if and only if for all i = 1, . . . , n, c * i ⊗ • • • ⊗ c
* n has at least as many entries 1 as 1. Now, recall that c is n-admissible if and only if N i (c) ≤ ni + 1 for all i = 1, . . . , n (Definition 5.1). We can compute N i (c) on c * by using the definition of the duality: we get , which is n-admissible. Now, we can compute

N i (c) = # entries 1 in c * i ⊗ • • • ⊗ c * n + (n -i + 1) -(# entries 1 in c * i ⊗ • • • ⊗ c * n ) . Since N i (c) ≤ n -i + 1 we get the desired characterisation.
c * = 1 ⊗ 1 ⊗ 1 ⊗ 1 1 ⊗ 1 ⊗ 1 1
, and we see that c * verifies the expected property. Remark 6.4. Similarly, there is a notion of coadmissibility for columns, which is easily characterised on the dual. More precisely, given a column c of type C n , set, for all i = 1, . . . , n,

M i (c) = x ∈ c | i ≤ x ≤ i .
Then c is called n-coadmissible if M i (c) ≤ i for all i = 1, . . . , n. As in Proposition 6.2, we can prove that c is n-coadmissible if and only if c * is a highest weight vertex with respect to the alternative A 1 -crystal structure where we choose to read the n factors from left to right.

In general, we will now show some interesting relationships between the C n -crystal structure on F n,m and the A 2m-1 -crystal structure on Ḟm,n .

For the next result, recall the type C n plactic relation called contraction of a column defined in [10, Remark (ii) p. 213], which is given by removing a certain pair (k, k) appearing in the column. It is easy to see that contraction has an inverse, which we call dilatation. Moreover, for 1 ≤ j ≤ m define κ j : F n,m → F n,m by setting a = κ j b if and only if a * = ėj b * (i.e. there is an arrow a * j -→ b * ).

Theorem 6.5. For all 1 ≤ j ≤ m, κ j is the contraction of the j-th column.

Proof.

Let b = c 1 ⊗ • • • ⊗ c m ∈ B µ ′ for some µ ′ ∈ C m,2n
. If a * = ėj b * , then a * is obtained from b * by changing a j into a j, say in column k. That is, a is obtained from b by removing the pair (k, k) in column j. In fact, by construction, the contraction of the j-th column is the only C n -crystal isomorphism which removes such a pair. Therefore, it suffices to prove that κ j is a C n -crystal isomorphism to deduce that κ j is the contraction of the j-th column. So we will prove that

κ j e i b = e i κ j b for all 0 ≤ i ≤ n -1. (2)
First of all, it is clear that (2) holds in the following two cases:

• e i does not act on the j-th column of b,

• i / ∈ {k, k -1}.
Let us look at the remaining cases, that is, e i acts on the j-th column of b and

(1) i = k -1. In this case, since we already know that k, k ∈ c j , we are ensured that e i acts non trivially on b if and only if k -1 ∈ c j and k -1 / ∈ c j . This means that we have the configuration

c j = k k-1 k .
One checks that applying κ j e i amounts to (a) changing k into k -1, followed by

(b) removing the pair (k -1, k -1)
On the other hand, applying e i κ j amounts to (a) removing the pair (k, k), followed by

(b) changing k -1 into k
Both of these procedures yield the same result, namely, we get the j-th column

c j = k (
where k -1 and k have been deleted), and we have (2) as expected.

(2) i = k. Similarly, e i acts non trivially on b if and only if k + 1 / ∈ c j and k + 1 ∈ c j , and this means that we have the configuration

c j = k k k+1 .
In this case, one checks that (2) holds and that the resulting column is . We have j = 1, we check that k = 3 and Therefore, we shall consider the A 1 × • • • × A 1 -crystal structure (m factors) on Ḟm,n induced by keeping only arrows of the form j j -→ j . Theorem 6.5 directly implies the following corollary, which can be rephrased by saying that the combinatorial Fock space is endowed with a (C m × A m 1 )-bicrystal structure.

Corollary 6.7. The duality * intertwines the C n -crystal on F n,m and the

A 1 × • • • × A 1 -crystal on Ḟm,n .
We complete Theorem 6.5 by giving a characterisation of the dual κ j of the Kashiwara operators e j (corresponding to the arrows j j -→ j+1 ) in terms of jeu de taquin operators on

F n,m . More precisely, if b = c 1 ⊗ • • • ⊗ c m ∈ B µ ′ , set for all 1 ≤ j ≤ m, c j = {n ≤ x ≤ 1 | x ∈ c j } = {n, . . . , 1} ∩ c j and c j = {n ≤ x ≤ 1 | x / ∈ c j },
so that each c j is the complement of the column cj defined in Section 5. Also, we set

b = c 1 ⊗ c 1 ⊗ • • • ⊗ c m ⊗ c m .
On elements of the form b, consider for 1 ≤ j ≤ m -1 the jeu de taquin operator J j acting on columns j and j + 1 as illustrated in Example 6.8 below. . We want to slide a box from c 1 to c 2 , so we reverse the order of the two columns, which yields the following jeu de taquin operation

4 • 3 5 2 1 → 4 5 3 • 2 1 → 4 5 3 2 • 1 .
Finally, we get

J j (b) = 3 ⊗ 4 3 ⊗ 5 2 1 ⊗ 3 1 .
We extend the definition of J j to F n,m by setting J j (b) = J j (b). We are ready to state the desired result, which is a symplectic analogue of [2, Theorem 2.9]. Theorem 6.9. For all 1 ≤ j ≤ m -1, we have κ j = J j .

Proof. We use a similar argument to the proof of Theorem 6.5. Namely, we first notice that if a * = e j b * , then a * is obtained from b * by changing a j + 1 into a j, say in column k. By definition of * , this means that. a is obtained from b by moving an entry k from column j to column j + 1. Then we use the fact that J j is the only map that verifies this property and that commutes with the C n -crystal operators e 1 , . . . , e n-1 , and show by case analysis that this also holds for κ j . .

We get

e 1 b * = 1 1 2 ⊗ 1 ⊗ 1 2 ⊗ 1 2 ⊗ 1 1
, so k = 2 and κ 1 b = .

The jeu de taquin operation corresponding to J 1 is

• 3 5 2 1 → 5 3 • 2 1 → 5 3 2 • 1 ,
and we check that this yields J 1 (b) = κ 1 (b).

At this point, we make a small digression and explain briefly the relationship with the charge statistic defined in [START_REF] Lecouvey | Combinatorics of generalised exponents[END_REF]Theorem 6.13] to compute q-weight zero multiplicities. (2(mj) + 1)

ε j (b * low ) 2 + m-1 j=1 2(m -j) ε j (b * low ) 2
We can in fact construct a statistic D directly on B µ ′ such that

D(b) = ch(b * ).
To do this, starting from

b = c 1 ⊗ • • • ⊗ c m ∈ B µ ′ . let b dil = d 1 ⊗ • • • ⊗ d m
where, for all j = 1, . . . , m, d j is the type C n column obtained by dilating c j recursively as much as possible. Then d j can be contracted a certain number of times, say δ j , until it becomes admissible. At this point, we have by Theorem 6.5

(b dil ) * = b * low ,
and ε j (b * low ) = δ j . In fact, we can easily compute δ j . Let h denote the height of the admissible column corresponding to c j (i.e. obtained from c j by applying recursively as many contractions as possible). Proposition 6.13. We have then δ j = nh. Moreover, δ j is even.

Proof. Note that since |c j | = n, the difference of the sizes nh is even (since each contraction deletes 2 entries). Therefore, it suffices to show that δ j = nh. Denote c adm j the admissible columns corresponding to c j , so that h = |c adm j |. Since contraction is a C n -crystal isomorphism, we can compute δ j by considering the highest weight column associated to c adm j in the C n -crystal. Obviously, this column also has height h and is admissible (since c adm j is admissible), so it must be the column {n, n -1, . . . , nh + 1}. It is straightforward to see that it can be dilated at most nh times, resulting in the column {n, n -1, . . . , nh + 1, nh, . . . , 1, 1, . . . , n -h} of height 2nh (which is the highest weight column associated to d j ), that is, δ j = nh.

It remains to express ε j (b * low ) directly on b dil . By Theorem 6.9, this equals the number γ j of successive possible application of J j to the pair (d j , d j+1 ). In other terms, if the minimal skew tableau associated to (d j , d j+1 ) has skew shape ν/(1 ℓ ), we have γ j = ℓ. To sum up, we set

D(b) = m j=1 (2(m -j) + 1) δ j 2 + m-1 j=1 2(m -j) γ j 2 .
This can be seen as an analogue of the energy statistic for type C, naturally generalising [2, Theorem 2.51].

We end this section by three important remarks.

Remark 6.14.

(1) As mentioned in Remark 6.10, since the operators κ j do not commute with e 0 , we cannot establish a (C n × A 2m-1 )-bicrystal structure on B µ ′ . In another direction, we expect to obtain a (C n × C m )-bicrystal structure by considering appropriate jeu de taquin operations and contraction on columns. In this case, it would be interesting to compare the resulting bicrystal structure to that obtained by Lee in [START_REF] Seung | Crystal structure on King tableaux and semistandard oscillating tableaux[END_REF].

(2) Let us consider elements of B hw µ ′ that are products of admissible columns. Combining Theorem 5.11 and Proposition 6.2, these are in duality with King tableaux of weight µ which are highest weight vertices in the A m 1 -crystal. This can be seen as a combinatorial version the new duality which will be proved in Theorem 7.5, in the special case r = m, m = (1, . . . , 1), X = (A, . . . , A) and µ = ((1 µ ′ 1 ), . . . , (1 µ ′ m )) (columns of height µ ′ j ).

(3) The energy function D defined previously does not coincide with the intrinsic energy function on type C

(1) n tensor products of column Kirillov-Reshetikhin crystals. This reflects the fact that the q-weight multiplicities do not coincide with the one-dimensional sums beyond type A. Nevertheless, this suggests that other interesting statistics could exist on these particular affine crystals.

Branching coefficients and a new duality

The aim of this section is to extend the results of Section 4 to the case where the fundamental gl 2n (C)modules appearing in the tensor products in Theorem 4.3 are replaced by tensor products of simple gl 2n (C) or simple sp 2n (C)-modules. More precisely, let X be a sequence (X 1 , . . . , X r ) of symbols in {A, C} r and let µ = (µ (1) , . . . , µ (r) ) be a sequence of partitions such that µ (j) ∈ P n×mj . Then one can associate to (X, µ) the tensor product

V X1 (µ (1) ) ⊗ V X2 (µ (2) ) ⊗ • • • ⊗ V Xr (µ (r) )
where V C (δ) (δ ∈ P n ) denotes the irreducible sp 2n (C)-module of highest weight δ and V A (δ) denotes the restriction to sp 2n (C) of the irreducible gl 2n (C)-module of highest weight δ. We show that the mutliplicity of the highest weight module of weight λ in the tensor product above is a branching coefficient of the form

[V Cm ( λ) : V X * m * ( µ)]
where m = m j and V X * m * ( µ) is an irreducible highest weight module for a block diagonal subalgebra of sp 2m (C) that depends on X, µ and m.

Remark 7.1. The weights of gl 2n (C) are in bijection with non-increasing sequences of integers in Z n . If two such sequences δ, δ ′ differ by a multiple of (1, . . . , 1) ∈ Z n , then the corresponding Schur functions s δ and s δ ′ will differ by a power of x 1 • • • x 2n . Since the characters of V A (δ) and V A (δ ′ ) are the specialisation of s δ and s δ ′ at (x 1 , . . . , x n , x -1 1 , . . . , x -1 n ), these two characters will be equal. As a consequence, it is enough in the tensor products defined above to restrict ourself to partitions.

Our first task in this section is to defined the module V X * m * ( µ). We start by extending the definition of the map • to r-tuples of partitions. Let µ = (µ (1) , . . . , µ (r) ) be a sequence of partitions such that µ (j) ∈ P n×mj . We set µ = µ (r) , . . . , µ

R

C [a,b] = {ε i -ε j | a ≤ i < j ≤ b} ∪ {ε i + ε j | a ≤ i < j ≤ b} ∪ {2ε i | a ≤ i ≤ b}.
In other words, we have

g (X,k) = g 1 ⊕ • • • ⊕ g r where g j = gl kj (C) if X j = A sp 2kj (C) if X j = C Let P (X,k) +
be the set of dominant weights of g (X,k) . A weight ν in P (X,k) + is a sequence ν = (ν (1) , . . . , ν (r) ) of partitions such that ν (j) ∈ P kj . We define V X k (ν) to be the g (X,k) -module of highest weight ν. Let us now recall how to compute the branching coefficient [V (κ) : V X k (ν)] where κ ∈ P m . To do so, we need the partition function P (X,k) defined by the expansion

α∈R + \R (X ,k) 1 1 -x α = β∈Z m P (X,k) (β)x β .
The following proposition is a consequence of the Weyl character formula. Recall that W m is the Weyl group of type C m and that ρ m is the half sum of positive roots, that is ρ m = (m, . . . , 1). Proposition 7.4 ([4, Theorem 8.2.1]). With the previous notation, we have

V (κ) : V X k (ν) = w∈Wm ε(w)P (X ,k) (w(κ + ρ m ) -(ν + ρ m )).
We are now ready to state the main theorem of this section. For any tuple a = (a 1 , . . . , a k ) we write a * for the inverse tuple (a k , . . . , a 1 ).

Theorem 7.5. Let µ = (µ (1) , . . . , µ (r) ) be a sequence of partitions such that µ (j) ∈ P n×mj . Let X = (X 1 , . . . , X r ) be a sequence of symbols in {A, C} and m = (m 1 , . . . , m r ) ∈ Z r ≥1 . For all λ ∈ P n , we have

V X1 (µ (1) ) ⊗ • • • ⊗ V Xr (µ (r) ) : V (λ) = V ( λ) : V X * m * ( µ) .
The rest of this section is devoted to the proof of the theorem. From now on and until the end of the section, we fix µ, X and m as in the theorem. Further, we set m = m i and we define the integers M 0 , . . . , M r by the relations M 0 = 0 and M j = j i=1 M i .

For any j = 1, . . . , r, and any partition δ ∈ P n write • s C δ for the character of the irreducible module V (δ) of highest weight δ in sp 2n (C)

• s A δ for the function s δ (x 1 , . . . , x n , 1 xn , . . . , 1 x1 ) where s δ is the Schur function of type A in 2n variables.

The character of V X1 (µ (1)

) ⊗ • • • ⊗ V Xr (µ (r) ) is s (X,m) µ := r j=1 s Xj µ (j) ∈ char(sp 2n ).
Thus, if we define the coefficients m 1) , . . . , β (r) ) be an r-tuple of partitions such that β (j) = (β

(X,m) λ,µ = V ( λ) : V X * m * ( µ) for all λ ∈ P n . Let △ (X,m) = α∈R (X ,m) (1 -x [α] ) where x [α] = xi xj if α = ε i -ε j where i < j 1 xixj if α = ε i + ε j where i ≤ j Let β = (β (
(j) 1 , . . . , β (j)
mj ) ∈ P mj . Then since m i = m, β can be seen as an element of Z m . We can then define x β and the map E that sends x β to e β as in Section 4.

Lemma 7.6. We have s

(X,m) µ = E(△ (X,m) x µ ′ ). Proof. We have R (X,m) = r j=1 R Xj [Mj-1+1,Mj ] and △ (X,m) x µ ′ = r j=1     α∈R X j [M j-1 +1,M j ] (1 -x [α] )     x µ ′ .
Let µ ′ = (µ (1) ′ , . . . , µ (r) ′ ) and µ (j) ′ = (µ (j) 1 ′ , . . . , µ (j) mj ′ ) ∈ P mj . For all j = 1, . . . , r, we set

x µ (j) ′ = x µ (j) 1 ′ Mj-1+1 • • • x µ (j) m j ′ Mj
Then we obtain

△ (X,m) x µ ′ = r j=1     α∈R X j [M j-1 +1,M j ] (1 -x [α] )x µ (j) ′     .
Observe that the variables appearing in each parenthesised expression are separated. Therefore, we can apply Lemma 3.3 recursively and get

E(△ (X,m) x µ ′ ) = r j=1 E     α∈R X j [M j-1 +1,M j ] (1 -x [α] )x µ (j) ′     .
But for any j = 1, . . . , r, we have by Propositon 4.4 and its analogue for the ordinary Jacobi-Trudi formula

s C µ (j) = E     α∈R X j [M j-1 +1,M j ] (1 -x [α] )x µ (j) ′     when X j = C and s A µ (j) = E     α∈R X j [M j-1 +1,M j ] (1 -x [α] )x µ (j) ′     when X j = A.
Finally, we get s

(X,m) µ = E(△ (X,m) x µ ′ ) as desired.
Define the partition function P (X,m) by

α∈R + \R (X ,m) 1 1 -x [α] = β∈Z m P (X,m) (β)x β . Lemma 7.7. We have P (X,m) (β) = P (X * ,m * ) (I(β)) for any β ∈ Z m .
Proof. The involution I sends the set R (X,m) to R (X * ,m * ) . Therefore we have

I   α∈R + \R (X ,m) 1 1 -x [α]   = α∈R + \R (X * ,m * ) 1 1 -x α = β∈Z m P (X * ,m * ) (β)x β which gives β∈Z m P (X,m) (β)x β = α∈R + \R (X,m) 1 1 -x [α] = β∈Z m P (X * ,m * ) (I(β))x β
hence the result. We conclude using Theorem 4.8.

We are now ready to complete the proof of the main theorem. 8 Injectivity of the induction functor Theorem 7.5 permits to express tensor multiplicities of sp 2n (C)-modules in terms of branching coefficients in irreducible sp 2m -modules. We started from the tensor product of sp 2n (C)-modules V X1 (µ (1) ) ⊗ V X2 (µ (2) ) ⊗ • • • ⊗ V Xr (µ (r) ) associated to the sequence µ = (µ (1) , . . . , µ (r) ) of partitions such that µ (j) ∈ P n×mj and to the sequence X = (X 1 , . . . , X r ) of symbols in {A, C}. This determined the dominant weight µ = I(µ ′ ) + n • ω m of the subalgebra g (X * ,m * ) of sp 2m (C) where m = m i and m = (m 1 , . . . , m r ).

Conversely, we can start from a dominant weight µ of g (X * ,m * ) ⊂ sp 2m (C) and realise the associated branching coefficient as a tensor product multiplicity for sp 2n (C)-modules. But here, one has to keep in mind that the datum of µ and g (X * ,m * ) does not determine the integer n. One can only say that n is at least n µ , the greatest part in the partitions µ (j) where µ = ( µ (1) , . . . , µ (r) ). This means that for any integer n ≥ n µ , we have an Now we need a result by Rajan [START_REF] Rajan | On the irreducibility of irreducible characters of simple Lie algebras[END_REF] on the irreducibility of the characters for sp 2n . In fact, we do not need Rajan's result in its full generality and we will only state a weaker version, sufficient for our purposes. Recall that ρ n = (n, n -1, . . . , 1) and set ρ n = (2n -1, 2n -3, . . . , 1).

For any partition δ ∈ P n , let

• s C δ be the character of the irreducible module V (δ) of highest weight δ in sp 2n ,

• s δ (X) be the usual Schur function associated to δ (i.e. the character of the irreducible module V (δ) of highest weight δ in gl n ) thus a symmetric polynomial in the set of variables x 1 , . . . , x n ,

• s δ (X ±1 ) be the Schur function s δ (X ±1 ) = s δ (x 1 , . . . , x n , 1 xn , . . . , 1 x1 ) where s δ is the Schur function in 2n variables.

Theorem 8.5. Let δ ∈ P n be a dominant weight for sp 2n regarded as an element of Z n . Assume that the coordinates of δ + ρ n are relatively prime (i.e. they have no trivial common divisor). Then the character s C δ is irreducible in C Wn [x ±1 1 , . . . , x ±1 n ].

Remark 8.6. In fact the irreducibility property proved in [START_REF] Rajan | On the irreducibility of irreducible characters of simple Lie algebras[END_REF] is more general. Let d(δ) be the greatest common divisor for the coordinates of δ + ρ n on the basis of the fundamental weights ω 1 , . . . , ω n-1 , ω n . Equivalently, d(δ) is the gcd of the coordinates of δ + ρ n in the usual basis of Z n2 . Define also d(δ) as the greatest common divisor for the coordinates of δ + ρ n on the weight Q-basis 2ω 1 , . . . , 2ω n-1 , ω n (with the convention d(δ) = 1 as soon as we have a non integer coordinate). For any weight β, recall that a β = w∈W ε(w)x w(β+ρn) . Then set D(δ) = gcd(a d(δ)ρn , a d(δ)ρn )

where the greatest common divisor is here considered in C Wn [x ±1 1 , . . . , x ±1 n ]. It is proved in [START_REF] Rajan | On the irreducibility of irreducible characters of simple Lie algebras[END_REF] that

S C δ := a δ+ρn D(δ)
is irreducible as soon as δ + ρ n is not a multiple of ρ n or ρ n . By our assumption in the previous theorem, the coordinates of δ +ρ n are relatively prime, therefore d(δ) = 1 and δ +ρ n is not a multiple of ρ n . Now, observe that a ρn divides a kρn for any integer k ≥ 1 because a kρn aρ n = s C (k-1)ρn . This implies that D(δ) = gcd(a ρn , a d(δ)ρn ) = a ρn and s C δ = S C δ is irreducible, as claimed.

Rajan's irreducibility result [START_REF] Rajan | On the irreducibility of irreducible characters of simple Lie algebras[END_REF] holds for any finite root systems. It can be used to prove the second result by Rajan that we shall need. Again, it holds for any root system but we shall only need it for the Schur functions.

Theorem 8.7. Let λ (1) , . . . , λ (r) and µ (1) , . . . , µ (r) be two sequences of partitions in P n such that

s λ (1) (X) × • • • × s λ (r) (X) = s µ (1) (X) × • • • × s µ (r) (X).
Then we have the multiset equality {λ (1) , . . . , λ (r) } = {µ (1) , . . . , µ (r) }.

One can observe that this result on Schur functions easily implies its analogue on Weyl characters thanks to Lemma 8.4 and the simple observation head(s C ν ) = s ν (X) for all ν ∈ P n .

  = β=(β1,...,βm)∈Z m ≥0 max(β)≤n Λ β n,m and Λ C 2n,m = β=(β1,...,βm)∈Z m ≥0 max(β)≤2n Λ β 2n,m

E

  (x β+γ ) = e β1 • • • e βs • e γs+1 • • • e γm by the hypothesis on β and γ. On the other hand, the same hypothesis implies that E(x β ) = e β1 • • • e βs and E(x γ ) = e γs+1 • • • e γm since e βi = 1 for any s + 1 ≤ i ≤ m and e γi = 1 for any 1 ≤ i ≤ s. This yields the desired equality.

Definition 4 . 1 .Example 4 . 2 .

 4142 Let β ∈ P n,m . We define the partition β ∈ P m,n by β := I(β ′ ) + n • (1, . . . , 1). The partition β is obtained by taking the conjugate of the complement of β in the rectangle n × m. Note that the map • depends on the integers n and m. It sends a weight of C n to a weight of C m . Let β = (5, 4, 2, 1) ∈ 4 × 5. Then we have β ′ = (4, 3, 2, 2, 1) and

  Definition 4.5. ([12, Lemma 3.3.1]) The element ∆ C m is invertible in the ring of formal Laurent series in the variables (x ±1 i ) 1≤i≤n and we can define a partition function associated to ∆ C m by setting 1 ∆ C m = β∈Z m P(β)x β . Lemma 4.6. We have P(β) = P(I(β)) where P is the usual Kostant partition function.

Proposition 4 . 7 .

 47 We have e n+k = e n-k for all k ∈ Z where e k = 0 if k / ∈ {1, . . . , 2n}.

Example 5 . 3 . 3 3

 533 Let n = 2 and w = 12 21221 12 . Then w 1 = --+ + --+ -+ and we get the bracketing w 1 = --(+(+-)-)(+-)+. This gives f 1 w = 12 21221 11 and e 1 w = 11 21221 12 . One can now interpret any column c = x1 x ℓ of height ℓ as the element x1⊗ • • • ⊗ x ℓ ∈ B(ω n-1 ) ⊗ℓ In particular, for all i = 0, . . . , n -1, the column c = n n-iis a highest weight vertex of weight ω n-i-1 , and therefore generates the crystal B(ω n-i-1 ) of the fundamental representation V (ω n-i-1 ). Note that the shape of c is given by the partition (1 i+1 ), that is, the coordinates of wt(c) = ω n-i-1 in the basis (ε n , . . . , ε 1 ). The following result is due to [8, Section 4.5], see also[START_REF] Lecouvey | Combinatorics of crystal graphs for the root systems of types A n , B n , C n , D n and G 2 . In Combinatorical aspect of integrable systems[END_REF] Proposition 4.2.1] for the reformulation using Definition 5.1.Theorem 5.4. The vertices of B(ω n-i-1 ) are the n-admissible columns.Example 5.5. Take n = 3 and i = 1. We realise the fundamental crystal B(ω 1 ) by n-admissible columns of height 2 as follows. Note that only is not n-admissible.

Example 5 . 7 . 4 ∈Definition 5 . 8 .

 57458 Let n = 4, m = 3, µ ′ = (2, 3, 1) and b = B µ ′ , so that w = 4321 1 4. The prefixes of w are 4, 43 , 432 , 4321 , 4321 1, 4321 1 4 with respective weights (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 0), (1, 1, 1, 1), (1, 1, 1, 0), (2, 1, 1, 0), which are all partitions, therefore b is a highest weight vertex. More precisely, b ∈ B hw (2,3,1),(2,1,1,0) . Consider now the alternative alphabet C * m = 1 < 1 < • • • < m < m . Similarly to Definition 5.6, we can consider columns on C * m and construct another combinatorial Fock space. The King Fock space is the set

Example 5 . 10 .

 510 Let n = 5 and b = c 1 ⊗ c 2 =

Example 5 . 12 . 3 ,

 5123 Let us go back to Example 5.7, where we had b ∈ B hw µ ′ ,λ with µ ′ = (2, 3, 1) and λ = (2which is a King tableau of shape (4, 3, 1) = λ and weight (3, 1, 2) = µ.

  which is the only King tableau of weight µ = (n -1). Fix r ∈ Z ≥1 and assume that the claim holds for all a ∈ B ν with |ν| = r. Let b ∈ B µ ′ with |µ| = r + 1. Let x be the bottommost entry of the rightmost column of b, and let a be the element obtained by deleting x from b. Assume first that a and b have the same number of non-trivial columns, say k. If x = j ≥ 1, then by definition, b * is obtained by adding a k in the j-th column of a * . By the induction hypothesis applied to a = a 1 ⊗ • • • ⊗ a n , one sees that Condition (2) holds for b * unless j = 1 and a 1 = {1, 2, . . . , k}. This is equivalent to saying that the contribution in ε 1 in wt(b) is negative. If x = j ≤ 1, then b * is obtained by adding a k in the j-th column of a * . One sees that Condition (2) holds for b * if and only if it holds for a * , and that e 0 (b) = 0 if and only if e 0 (a) = 0, and we conclude using the induction hypothesis applied to a. Finally, if a has k non-trivial columns and b has k + 1 non-trivial columns, one uses similar arguments to validate the induction step in this case too. Remark 5.13. We conclude this section by mentioning two related recent results. (1) In [15, Theorem 2.7], Lee constructed a weight-preserving bijection between certain semistandard oscillating tableaux on the one hand and certain King tableaux on the other hand. In fact, the highest weight vertices of Theorem 5.11 correspond to certain semistandard oscillating tableaux (as illustrated in Example 5.7). Therefore, Theorem 4.3 gives a simple proof of [15, Theorem 2.7] just based on the combinatorics of the columns of type C n .

Example 6 . 1 . 2 . 2 . 1 2Proposition 6 . 2 .

 6122162 Take n = 3, m = 2 and b * = Let us detail the computation of ḟ1 b * . Reading b * yields the word w 1 = 121 121 2. Looking only at 1 and 2, we get w 1 = -+ +-and the bracketing yields w 1 = -+ (+-). Thus we get ḟ1 b * = Similarly, one checks that ė1 b * = and that ė1 b * = ė2 b * = ḟ1 b * = 0.In the case m = 1, the duality * intertwines the following two important properties. Let c be a column on C n . Then c is n-admissible if and only if c * is a highest weight vertex in the A 1 -crystal.

Example 6 . 3 .

 63 Let n = 6 and c =

Example 6 . 6 .

 66 We illustrate Case (1) of the previous proof by taking n = 4, m = 1 and b = c 1 =

Example 6 . 8 .so that b = c 1 ⊗ c 1 ⊗ c 2 ⊗ c 2 1 .

 6811221 Let n = 5, m = 2 and b = Let us perform one jeu de taquin operation between c 1 =

Remark 6 . 10 . 1 . 7 . 6 . 11 .

 61017611 Unlike the operators κ j , we do not have that κ j commute with the type C n Kashiwara operator e 0 . For instance, if n = 1, m = 2, and b = ∅ ⊗ 1 , one checks that κ 1 e 0 b = 0 but e 0 κ 1 b = 1 ⊗ 1 Therefore, we do not get an analogue of Corollary 6.Example Let n = 5, m = 2, j = 1 and

  Let b ∈ B hw µ ′ such that b * is a King tableau (Theorem 5.11) of weight µ = 0. In particular, we have µ ′ = (m n ), that is all the columns in µ have height n. Denote b * low the lowest weight vertex corresponding to b * in the A 1 × • • • × A 1 -crystal. In other terms, b * low is obtained from b * by applying all possible ḟj to b * recursively (they commute). Definition 6.12. The charge of b * is the nonnegative integer ch(b * ) = m j=1

  above character in the basis of irreducible characters in type C s (X,m)

Proposition 7 . 8 .P

 78 For all λ ∈ P n , we havem (X,m) λ,µ = w∈Wm ε(w) P (X,m) (w(λ ′ + δ m,n ) -(µ ′ + δ m,n ).Proof. By definition of ∆ C m (see Section 4) we have△ (X,m) (X,m) (β)x β+µ ′ ) = β∈Z m P (X,m) (β)E(∆ m x β+µ ′ ) = β∈Z m P (X,m) (β)v β+µ ′ .

  ) P (X,m) (w(λ ′ + δ m,n ) -(µ ′ + δ m,n )) = w∈Wm ε(w)P (X * ,m * ) (I(w(λ ′ + δ m,n )) -I(µ ′ + δ m,n )) = w∈Wm ε(w)P (X * ,m * ) (I(w(λ ′ + δ m,n )) -I(µ ′ )n • ω mρ m ) .By setting w ′ = IwI in the last sum and using Proposition 7.4, this yieldsm (X,a) λ,µ = w ′ ∈Wm ε(w ′ )P (X * ,m * ) (w ′ (I(λ ′ ) + nω m ) + ρ m ) -(I(µ ′ ) + nω m )ρ m ) = w ′ ∈Wm ε(w ′ )P (X * ,m * ) w ′ ( λ + ρ m ) -( µρ m ) = V ( λ) : V X * m * (µ)as required.

Lemma 8 . 3 .

 83 Consider P and Q in Z[x ±1 1 , . . . , x ±1n ] such that Q divides P . Then max(Q) ≤ max(P ).

For any β = (β 1

 1 , . . . , β n ) ∈ Z n , set |β| = β 1 + • • • + β n . Given P = β∈Z n c β x β (all but a finite number of coefficients c β are equal to zero) in Z Wn [x ±1 1 , . . . , x ±1 n ], define head(P ) = β∈Z n |β|=|max(P )| c β x β .We have the following easy lemma. Lemma 8.4. Let P 1 , . . . , P r be a sequence of polynomials in Z[x ±1 1 , . . . , x ±1 n ]. Then head(P 1 × • • • × P r ) = head(P 1 ) × • • • × head(P r ).

  If β = (n, . . . , n, β k , . . . , β 1 ) ∈ Z m then since e n (x 1 , . . . , x n

	e β1	e β1+1	. . . e β1+m-1
	e β2-1 . . .	e β2 . . .	. . . e β2+m-2 . . . . . .	.
	e βm-m+1 e βm-m+2 . . .	e βm
	The well-known Jacobi-Trudi formula [1, Appendix A] tells us that for all λ ∈ P n-1 , we have s λ = v λ ′ .
	Remark 3.2.

  particular, if b ∈ B hw µ ′ then λ is a partition. In fact, b ∈ B hw µ ′ is a highest weight vertex for the parabolic A n-1 -action if and only if b * is semistandard on C * n by [2, Proposition 2.17]. The only thing that remains to be proved is that e 0 (b) = 0 if and only if b * satisfies the Condition (2) defining King tableaux. We prove this by induction on |µ|.

The map E is not a morphism of algebras, the equality only holds when the variables are separated in a given monomial.

For any β = (β 1 , . . . , βn) ∈ Z, we indeed have β = n i=1 a i ω i where a i = β i -β i+1 for 1 ≤ i ≤ n -1 and βn = an.

where for each 1 ≤ j ≤ r, the partition µ (j) is defined with respect to the pair (n × m j ) and lies in P mj×n . Note that the definition of this map depends on the pair (n, m). Remark 7.2. (1) Let µ ∈ P n,m and write µ ′ = (µ ′ 1 , . . . , µ ′ m ). Let m = (m 1 , . . . , m r ) be such that m j = m. If we see µ as an r-tuple of partitions µ = (µ (1) , . . . , µ (r) ) as follows µ ′ 1 , . . . , µ ′ m1 µ (1) , µ ′ m1+1 , . . . , µ ′ m2 µ (2) , . . . , µ ′ m-mr+1 , . . . , µ ′ mr µ (r) then the partition µ computed with respect to the pair (n, m) is equal, as an element of Z m , to the partition µ computed with respect to the pair (n, m).

(2) Let µ = (µ (1) , . . . , µ (r) ) be such that µ (j) ∈ P n,mj and let µ ′ = (µ (1) ′ , . . . , µ (r) ′ ). Let m = (m 1 , . . . , m r ) and m = m j . We have

where ω m = (1, . . . , 1) and the equality is an equality in Z m .

Example 7.3. Consider the sequence of partitions µ = (µ (1) , µ (2) , µ (3) ) = , , ∈ P 3,2 × P 3,2 × P 3,3 .

Let n = 3 and m = (2, 2, 3). We compute the image of µ under the map • associated to (n, m) by first taking the complement of the partitions of µ (in their respective rectangles), which yields

and then by taking the conjugate of each green partition and reversing the order of the triple, which yields

Finally, note that m = 7 and that

which is indeed equal to µ as an element of Z 7 .

Next we need to define the module V X * m * ( µ). To simplify the exposition, we will explain how to construct the module V X k (ν) associated to

• a sequence X = (X 1 , . . . , X r ) of symbols in {A, C};

• a sequence ν = (ν (1) , . . . , ν (r) ) of partitions such that ν (j) ∈ P kj .

We set m = k i and we define the integers K 0 , . . . , K r by setting

The algebra g (X,k) is defined to be the subalgebra of sp 2k associated to the root system

where we have set for any pair of integers (a, b):

r-tuple of partitions µ[n] = (µ (1) [n], . . . , µ (r) [n]) such that µ[n] = µ (where the map • is defined with respect to the pair (n, m)). Observe in particular that for any j = 1, . . . , r, we have

, µ (j) [n µ ] for any integer a ≥ 0

or equivalently, µ (j) [n + 1] is obtained by adding a part m j to µ (j) [n] for any n ≥ n µ .

Example 8.1. Consider the following sequence of partition:

so that n µ = 3. We have seen in Example 7.3 that

Now setting n = 5 and adding 2 parts of respective size 2, 2 and 3 to µ [START_REF] Gerber | Generalised Howe duality and injectivity of induction: orthogonal case[END_REF] we get

Then taking the complements in the rectangles of respective size 3 × 5, 2 × 5 and 2 × 5 we obtain , , ,

and taking the conjugate of each green partition and reversing, we see that we obtain µ[5] = µ (here µ [START_REF] Guilhot | Isomorphic induced modules and Dynkin diagram automorphism[END_REF] is computed with respect to the pair (5, (

Then, one can apply Theorem 7.5 and get for all n ≥ n µ and all partitions λ ∈ P n the equality

In the relation above, the partition λ is defined from the action of the map • corresponding to the pair (n, m) and the r-partition µ is defined similarly with respect to the pair (n, m) (in both cases, the same n as in µ[n]).

In the following, we fix two dominant weights µ, ν of the algebra g (X * ,m * ) ⊂ sp 2m . From the definition, we see that µ and ν are sequences of partitions in which the j-th components lies in P mr-j+1 . We set µ = ( µ (1) , . . . , µ (r) ) and ν = ( ν (1) , . . . , ν (r) ).

Recall that V X * m * ( µ) and V X * m * ( ν) are the associated highest weight g (X * ,m * ) -modules. We consider the following problem.

Problem 8.2. Assume that the induction of V X * m * ( µ) and V X * m * ( ν) to sp 2m are isomorphic. What can we say about µ and ν?

We are going to prove that in the case where all the components in X * = (X r , . . . , X 1 ) are of type C or when X * is a parabolic Dynkin subdiagram (i.e. X 1 is the unique component of type C) then the sequences µ and ν coincide up to permutations of their parts. Further, in the second case we will show that µ (r) = ν (r) . In particular, this proves the main conjecture of [START_REF] Guilhot | Isomorphic induced modules and Dynkin diagram automorphism[END_REF] in type C: in the parabolic case, the two previous induced modules are isomorphic if and only if the dominant weights µ, ν coincide up to an automorphism of the underlying parabolic Dynkin diagram. In [START_REF] Guilhot | Isomorphic induced modules and Dynkin diagram automorphism[END_REF], this conjecture was proved in any finite types but only when µ and ν are far enough from the walls of Weyl chambers.

To do this, recall first that the character ring of type C n can be regarded as the ring Z Wn [x ±1 1 , . . . , x ±1 n ] of Laurent polynomials fixed by permutations of the variables x i and the inversions x i → 1

xi . Also, we have a total order on the monomials in Z[x ±1 1 , . . . , x ±1 n ] defined by, for all β, γ ∈ Z n ,

where ≤ lex is the lexicographic order on Z n . This enables us to define, for any P ∈ Z[x ±1 1 , . . . , x ±1 n ] the monomial max(P ) as the maximal monomial appearing in P . Given A and B in Z Wn [x ±1 1 , . . . , x ±1 n ], we have max(AB) = max(A) × max(B). This implies the following useful lemma.

Corollary 8.8. Consider λ (1) , . . . , λ (r) and µ (1) , . . . , µ (r) two sequences of partitions in P n such that

Then we have the multiset equality {λ (1) , . . . , λ (r) } = {µ (1) , . . . , µ (r) }.

Now recall the branching formula for the decomposition of a Schur function s ν (X ±1 ) on the basis of the irreducible characters for sp 2n .

Theorem 8.9 ([1, Appendix A]). Let ν ∈ P n . We have

where P

(1,1) n

is the subset of P n of partitions which can be tiled in vertical dominoes and c ν λ,δ is the usual Littlewood-Richardson coefficient.

Remark 8.10. Observe that s ν (X ±1 ) = s C ν as soon as ν has at least two rows since c ν λ,(1,1) > 0 for any λ obtained from ν by removing one box in two different rows. The previous decomposition can then be written

where the coefficients a ν,λ belong to N and is the dominant order on partitions. Assume that

where ν ♭ is the partition obtained by decreasing by 1 the two lowest nonzero parts of ν. Since x ν > x ν ♭ , this contradicts Lemma 8.3. This shows that s

). Now let us come back to Problem 8.2 and assume that the induction of V X * m * ( µ) and V X * m * ( ν) to sp 2m are isomorphic. This can be rewritten, thanks to our main theorem in the last section as:

for all n ≥ max(n µ , n ν ).

Assume first that all the components of X are of type C. We get

where for any j = 1, . . . , r, the partitions µ (j) [n] and ν (j) [n] have at most m j -columns. Here we can apply Corollary 8.8 and deduce the multiset equality {µ (1) [n], . . . , µ (r) [n]} = {ν (1) [n], . . . , ν (r) [n]}.

Alternatively, we can choose n so that each partition µ (j) [n] and ν (j) [n] starts with two parts equal to m j . Then, the two first components in µ (j) [n] + ρ n or ν (j) [n] + ρ n are equal to m j + n and m j + n -1, respectively. In particular, this implies that the coordinates of µ (j) [n] + ρ n and ν (j) [n] + ρ n are relatively prime. We thus get by Theorem 8.5 that each character s C µ (j) [n] or s C ν (j) [n] are irreducible. Thus we recover the multiset equality {µ (1) [n], . . . , µ (r) [n]} = {ν (1) [n], . . . , ν (r) [n]}. It implies that the r-partitions µ and ν coincide up to permutation of their partitions.

Next, assume X = (C, A, . . . , A). This time we get

Write N = µ (1) [n] + • • • + µ (r) [n] the sum of all parts in the partitions µ (j) [n], j = 1, . . . , r (that is the rank of the r-partition µ[n]). For any partition κ ∈ P n , we have max(s C κ ) = max(s κ (X)) = x κ and head(s C κ ) = s κ (X). Then Lemma 8.4 implies that for any partition ν of rank N , the coordinates of

on the irreducible character s C ν coincides with the coordinates of

on the Schur function s ν (X). Equation (5) then gives

and we obtain {µ (1) [n], . . . , µ (r) [n]} = {ν (1) [n], . . . , ν (r) [n]} by Theorem 8.7.

Having in hand the multiset equality {µ (1) [n], . . . , µ (r) [n]} = {ν (1) [n], . . . , ν (r) [n]} and equation ( 5), we deduce that

(1) Either µ (1) [n] = ν (1) [n] and {µ (2) [n], . . . , µ (r) [n]} = {ν (2) [n], . . . , ν (r) [n]},

(2) or µ (1) [n] = ν (p) [n], µ (q) [n] = ν (1) [n] with p > 1 and q > 1 and {µ (2) [n], . . . , µ (r) [n]}\{µ (q) [n]} = {ν (2) [n], . . . , ν (r) [n]} \ {ν (p) [n]}.

In Case (1), since r) [n], . . . , µ (1) [n]) µ = ( µ (1) , . . . , µ (r) )

we get that µ (r) = ν (r) and { µ (1) , . . . , µ (r-1) } = { ν (1) , . . . , ν (r-1) }.

In Case [START_REF] Gerber | Duality and bicrystals on infinite binary matrices[END_REF], by simplifying the identical factors in (5), we get

As before, we can choose the integer n so that s C µ [START_REF] Fulton | Representation theory. A first course[END_REF] [n] and s C ν [START_REF] Fulton | Representation theory. A first course[END_REF] [n] = s C µ (q) [n] are irreducible polynomials, and the partitions µ (1) [n] and ν (1) [n] have their two first parts equal to m 1 . This implies that s C µ [START_REF] Fulton | Representation theory. A first course[END_REF] [n] divides s µ [START_REF] Fulton | Representation theory. A first course[END_REF] [n] (X ±1 ). But, by using Remak 8.10, this is only possible when s C µ [START_REF] Fulton | Representation theory. A first course[END_REF] [n] = s µ [START_REF] Fulton | Representation theory. A first course[END_REF] [n] (X ±1 ), that is when µ (1) [n] is a row. We thus get a contradiction since µ (1) [n] has at least two parts equal to m 1 .

Finally, we have proved the following theorem. Theorem 8.11. Assume that the induction of V X * m * ( µ) and V X * m * ( ν) to sp 2m are isomorphic. Then the following holds.

(1) The r-partitions µ and ν coincide up to permutation of their partitions when all the components in X are of type C

(2) We have µ (r) = ν (r) and µ and ν coincide up to permutation of their other partitions when X is a parabolic Dynkin subdiagram (i.e. X 1 is the unique component equal to C)