Structure coefficients of the Hecke algebra of $(S_{2n},B_n)$

The Hecke algebra of the pair $(S_{2n},B_n)$, where $B_n$ is the hyperoctahedral subgroup of $S_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial universal algebra which projects on the Hecke algebra of $(S_{2n},B_n)$ for every $n$. To build it, we introduce new objects called partial bijections.


INTRODUCTION
For a positive integer n, let S n denote the symmetric group of permutations on the set [n] := {1, 2, · · · , n}, and let C[S n ] denote the group-algebra of S n over C, the field of complex numbers. The center of C[S n ], denoted by Z(C[S n ]) is a classical object in combinatorics. It is linearly generated by elements Z λ , indexed by partitions of n, where Z λ is the sum of permutations of [n] with cycle-type λ. The structure coefficients c ρ λδ describe the product in this algebra, they are defined by the equation: In other words, c ρ λδ counts the number of pairs of permutations (x, y) with cycle-type λ and δ such that x · y = z for a fixed permutation z with cycle-type ρ. It is known that these coefficients count the number of embeddings of certain graphs into orientable surfaces (see [Cor75]). One of the tools used to calculate these coefficients is the representation theory of the symmetric group, see [JV90,Lemma 3.3]. In [GS98, Theorem 2.1], Goupil and Schaeffer have a formula for c ρ λδ if one of the partitions λ, δ and ρ is equal to (n). There are no formulas for c ρ λδ in general. In 1958, Farahat and Higman proved the polynomiality of the coefficients c ρ λδ in n when λ, δ and ρ are fixed partitions, completed with parts equal to 1 to get partitions of n, [FH59, Theorem 2.2]. More recently, in [IK99], by using objects called partial permutations, the same result is obtained by Ivanov and Kerov. This more recent proof provides a combinatorial description of the coefficients of the relevant polynomials.
Here, we consider the Hecke algebra of the pair (S 2n , B n ), denoted by C[B n \S 2n /B n ], where B n is the hyperoctahedral group. It was introduced by James in [Jam61] and it also has a basis indexed by partitions of n. The algebra C[B n \S 2n /B n ] is a natural analogue of Z(C[S n ]) for several reasons. Goulden and Jackson proved in [GJ96] that its structure coefficients count graphs drawn on non-oriented surfaces. To get formulas for these coefficients, zonal characters are used instead of irreducible characters of the symmetric group, see [Mac95,Section VII,2].
In this paper we give a polynomiality property of the structure coefficients of the Hecke algebra of (S 2n , B n ). Namely, we prove that these coefficients can be written as the product of the number 2 n n! with a polynomial in n. In some specific basis, this polynomial has non-negative coefficients that have a combinatorial interpretation. Moreover, we are able to give an upper bound for its degree . Our proof is based on the construction of an universal algebra which projects onto the Hecke algebra of (S 2n , B n ) for every n 1 . This method was already used by Ivanov and Kerov in [IK99]. What is original in our approach is that the product in our universal algebra is computed as an average of combinatorial objects called partial bijections of n. Recently, P.-L. Méliot has used this same idea in [Mél13] to give a polynomiality property for the structure coefficients of the center of the group-algebra C[GL(n, F q )], where GL(n, F q ) is the group of invertible n × n matrices with coefficients in F q . Because of the similarities between Meliot's construction and ours, we are convinced we should build a general framework in which such a result (polynomiality of the structure coefficients) always holds. This is the subject of future work.
A weaker version of our polynomiality result (without non-negativity of the coefficients) for the structure coefficients of Hecke algebra of (S 2n , B n ) has been established by an indirect approach using Jack polynomials in [DF12,Proposition 4.4]. There is no combinatorial description in that proof. By a different approach than ours, in [AC12], Aker and Can study the Hecke ring C[B n \S 2n /B n ], however, it seems that there is a minor issue in their proof of polynomiality of the structure coefficients ( [Can13]). A universal algebra also appears in this paper, but it does not have a combinatorial realization as ours.
As explained, our proof goes through the construction of an universal algebra which projects onto the Hecke algebra of (S 2n , B n ) for every n. We are able to give a link between this algebra, and the algebra of shifted symmetric functions. Shifted symmetric functions have been introduced and studied by A. Okounkov and G. Olshansky in 1996, see [OO97]. They are deformations of usual symmetric functions that display remarkable properties.
The paper is organized as follows. In Section 2, we review all necessary definitions to describe the Hecke algebra of (S 2n , B n ). Then, we state our main result about its structure coefficients. We start Section 3 by introducing partial bijections of n then we build our universal algebra. We use this algebra in Section 4 to prove our main result, Theorem 2.1. In Section 5, we show how the universal algebra is related with the algebra of shifted symmetric functions and in Section 6 we exhibit some filtrations on this universal algebra, which implies the above mentioned upper bounds for the degree of the polynomials.

DEFINITIONS AND STATEMENT OF THE MAIN RESULT
2.1. Partitions. Since partitions index bases of the algebras studied in this paper, we recall the main definitions. A partition λ is a list of integers (λ 1 , . . . , λ l ) where λ 1 ≥ λ 2 ≥ . . . λ l ≥ 1. The λ i are called the parts of λ; the size of λ, denoted by |λ|, is the sum of all of its parts. If |λ| = n, we say that λ is a partition of n and we will write λ ⊢ n. The number of parts of λ is denoted by l(λ). We will also use the exponential notation λ = (1 m 1 (λ) , 2 m 2 (λ) , 3 m 3 (λ) , . . .), where m i (λ) is the number of parts equal to i in the partition λ. If λ and δ are two partitions we define the union λ ∪ δ as the following partition: A partition is called proper if it does not have any part equal to 1. The proper partition associated to a partition λ is the partitionλ := λ \ (1 m 1 (λ) ) = (2 m 2 (λ) , 3 m 3 (λ) , . . .).

Permutations and Coset type.
For a permutation ω, we use the word notation ω 1 ω 2 · · · ω n , where ω i = ω(i). The set S n of all permutations of [n] is a group for the composition called the symmetric group of size n.
To each permutation ω of 2n we associate a graph Γ(ω) with 2n vertices located on a circle. Each vertex is labelled by two labels (exterior and interior). The exterior labels run through natural numbers from 1 to 2n around the circle. The interior label of the vertex with exterior label i is ω(i). We link the vertices with exterior (resp. interior) labels 2i − 1 and 2i by an exterior (resp. interior) edge. As every vertex has degree 2, the graph Γ(ω) is a disjoint union of cycles since exterior and interior edges alternate, all cycles have even lengths 2λ 1 ≥ 2λ 2 ≥ 2λ 3 ≥ · · · . The coset-type of ω denoted by ct(ω) is the partition (λ 1 , λ 2 , λ 3 , . . .) of n.
A B n -double coset of S 2n is the set B n xB n = {bxb ′ ; b, b ′ ∈ B n } for some x ∈ S 2n . It is known, see [Mac95,page 401], that two permutations of S 2n are in the same B n -double coset if and only if they have the same coset-type. Thus, if x ∈ S 2n has coset-type λ, we have: B n xB n = {y ∈ S 2n such that ct(y) = λ}.
2.3. The Hecke algebra of (S 2n , B n ). The symmetric group algebra of n, denoted by C[S n ], is the algebra over C linearly generated by all permutations of [n]. The group B n × B n acts on C([S 2n ]) by the following action: (b, b ′ ) · x = bxb ′−1 , called the B n × B n -action. The Hecke algebra of (S 2n , B n ), denoted by C[B n \ S 2n /B n ], is the sub-algebra of C[S 2n ] of elements invariant under the B n × B n -action. Recall that B n -double cosets are indexed by partitions of n. Here, we rather index the basis by proper partitions of size less or equal to n, which are trivially in bijection with partitions of n. Therefore, the set {K λ (n) : λ is a proper partition with |λ| ≤ n} forms a basis for C[B n \ S 2n /B n ], where K λ (n) is the sum of all permutations from S 2n with coset-type λ ∪ 1 n−|λ| . So, for any two proper partitions λ and δ with size at most n, there exist complex numbers α ρ λδ (n) such that: (1) K λ (n) · K δ (n) = ρ proper partition |ρ|≤n α ρ λδ (n)K ρ (n).

Main result.
In this paper, we obtain a polynomiality property for the structure coefficients α ρ λδ (n) of the Hecke algebra of (S 2n , B n ). More precisely, we prove the following theorem. We will use the standard notation (n) k := n! (n−k)! = n(n − 1) · · · (n − k + 1).
a j (n − |ρ|) j is a polynomial in n and the a j 's are non-negative rational numbers.
2.5. Major steps of the proof. The idea of the proof is to build a universal algebra A ∞ over C satisfying the following properties: (1) For every n ∈ N * , there exists a morphism of algebras θ n : (2) Every element x in A ∞ is written in a unique way as an infinite linear combination of elements T λ , indexed by partitions. This implies that, for any two partitions λ and δ, there exist nonnegative rational numbers b ρ λδ such that: (3) The morphism θ n sends T λ to a multiple of Kλ(n). To build A ∞ , we use combinatorial objects called partial bijections. For every n ∈ N * , we construct an algebra A n using the set of partial bijections of size n. The algebra A ∞ is defined as the projective limit of this sequence (A n ).
The projection p n : A ∞ → A n involves coefficients which are polynomials in n. By defining the extension of a partial bijection of n to the set [2n], we construct a morphism from A n to C[B n \ S 2n /B n ]. Its coefficients involve the number 2 n n!. It turns out that the morphism θ n is the composition of those two morphisms: A n x x q q q q q q q q q q q The final step consists of applying the chain of homomorphisms in the diagram above to equation (2).
Remark. This method is based on Ivanov and Kerov's one to get the polynomiality of the structure coefficients of the center of the symmetric group algebra (see [IK99] for more details). Nevertheless, our construction is more complicated, mainly because a partial bijection does not have a unique trivial extension to a given set, see Definition 3.2.

THE PARTIAL BIJECTION ALGEBRA
In this section we define the set of partial bijections of n. With this set, we build the algebras and homomorphisms that appear in the diagram above.
3.1. Definition. We start by defining partial bijections of n and the partial bijection algebra. Then, we introduce the notion of trivial extension of a partial bijection of n and we use it to build a homomorphism between the partial bijection algebra of n and the symmetric group algebra of 2n.
Let N * denotes the set of positive integers. For n ∈ N * , we define P n to be the following set: The set d is the domain of (σ, d, d ′ ) while d ′ is its codomain. We denote by Q n the set of all partial bijections of n.
For any positive integer n, let R n be the set of all one-to-one maps f : . The set R n with the composition of maps is a monoid -that is the composition is associative and R n has an identity element -called the symmetric inverse semigroup. With this composition, the set of partial bijections Q n forms a submonoid of R 2n . It is known, see [Sol02], that R 2n is in bijection with the hook monoid R 2n . It is important to notice that this obvious structure on Q n does not enter the picture in here. The useful product in this work will be defined later in this section.
It should be clear that A permutation σ of 2n can be written as (σ, [2n], [2n]), so the set S 2n can be considered as a subset of Q n .
Notation. For any partial bijection α, we will use the convention that σ (resp. d, d ′ ) is the first (resp. second, third) element of the triple defining α. The same convention holds for α, α i ,α . . .
Observation 3.2. In the same way as in Section 2.2, we can associate to each partial bijection α of n a graph Γ(α) with |d| vertices placed on a circle. The exterior (resp. interior) labels are the elements of the set d (resp. d ′ ). Since the sets d and d ′ are in P n , we can link 2i with 2i − 1 as in the case So, the definition of coset-type extends naturally to partial bijections. We denote by ct(α) or ct(σ) the coset-type of a partial bijection α.
We denote by P α (n) the set of all trivial extensions of α in Q n . is also a trivial extension of α.
Lemma 3.1. Let α be a partial bijection of n and X an element of P n such that d ⊆ X. The number of trivial extensions α of α such that d = X is We have the same formula for the number of trivial extensions α such that d ′ = X.

Proof. Straightforward by induction.
Consider D n = C[Q n ] the vector space with basis Q n . We want to endow it with an algebra structure. Let α 1 and α 2 be two partial bijections. If d 1 = d ′ 2 , we can compose α 1 and α 2 and we define α 1 * α 2 = α 1 • α 2 = (σ 1 • σ 2 , d 2 , d ′ 1 ). Otherwise, we need to extend α 1 and α 2 to partial bijections α 1 and α 2 such that d 1 = d ′ 2 . Since there exist several trivial extensions of α 1 and α 2 , a natural choice is to take the average of the composition of all possible trivial extensions. Let E α 2 α 1 (n) be the following set: Elements of E α 2 α 1 (n) are schematically represented on Figure 3.
Note on a convention for figure. Throughout the paper, we will use the following conventions on figures that represent elements of some sets.
-The data defining the set (so fixed when we go from an element of the set to another) is drawn using plain shapes. -The element of the set is drawn using dashed shapes.
We define the product of α 1 and α 2 as follows: By Lemma 3.1, we have: Proposition 3.2. The product * is associative. In other words, D n is a (non-unital) algebra.
Proof. Postponed to the next section.
We will illustrate the associativity by a simple example given below.
3.3. Proof of the associativity of * . Let α 1 , α 2 and α 3 be three elements of Q n . By definition of the product we have: In the same way, with the notation X 2 , the product α 1 * (α 2 * α 3 ) can be written in the following way: Schematically, the elements in X 2 are represented on Figure 5.
To prove the associativity of the product, we build a set X and two surjective functions φ 1 : X −→ X 1 and φ 2 : X −→ X 2 in order to write both sums (equations (8) and (9)) as sums over the same set X. Let X be the set of elements satisfying the following properties: . The elements of this set are schematically represented on Figure 6. Note that (iii)) is a minimality condition. We will see in the proof of Lemma 3.4 below why it is useful.
Then, the codomain of α 2 and the domain of , which is given by condition (iii)). Thus, φ 1 is well defined. Now, fix ( α 1 , α 2 ), ( α 1 • α 2 , α 3 ) ∈ X 1 . We will count the number of its pre-images by φ 1 . To construct an element of φ −1 1 ( α 1 , α 2 ), ( α 1 • α 2 , α 3 ) , we have only to build ǫ 1 , ǫ 2 and δ 1 since the other elements ǫ 3 , δ 0 , δ 2 and δ 3 are determined by α 1 , α 2 , α 3 and α 1 • α 2 . First, to build δ 1 , we must extend d 1 ∪ d ′ 2 by adding pairs of form ρ(k) to obtain a set which has the same cardinality as . Since our choice must respect the condition that the extended set is in P n , this number is: is extended, we should extend σ 1 to τ 1 (we have the definition domain δ 0 and the arrival domain δ 1 ) by sending the pairs of form ρ(k) to pairs with same form. The number of ways to do so is: After extending σ 1 to τ 1 , we get immediately τ 2 because τ 1 • τ 2 = σ 1 • σ 2 is given. Thus, the cardinality of the set φ −1 In the same way, we can prove the following lemma.
3.4. Action of B n × B n on D n . In this section, we build the algebra A n as the algebra of invariant elements by an action of B n × B n on D n .
Definition 3.3. The group B n × B n acts on Q n by: Observation 3.5. Two partial bijections are in the same orbit if and only if they have the same coset-type.
We can extend this action by linearity to get an action of B n × B n on D n . Lemma 3.6. For any three permutations a, b and c of B n and for any partial bijections α 1 , α 2 of n, the set Proof. We can check easily that the two following functions: are well defined. Besides, they are inverse from each other: and, similarly, Thus Θ defines a bijection between E α 2 α 1 (n) and E It follows from this lemma that the action • is compatible with the product of Q n . Namely, we can prove the following corollary.
Corollary 3.7 . For any (a, b, c) ∈ B 3 n and for any partial bijections α 1 , α 2 of n, we have: Proof. If ( α 1 , α 2 ) ∈ E α 2 α 1 (n), we have: Then, we can write: We consider the set A n of invariant elements by the action of B n × B n on D n : For every partition λ such that |λ| ≤ n, we define the set A λ,n to be the set of all partial bijections α of n such that ct(α) = λ. The sum of all elements in A λ,n is denoted by S λ,n .
Proposition 3.8. The set A n is an algebra with basis the elements (S λ,n ) |λ|≤n .
Proof. For every (a, b) ∈ B n × B n , and for every x, y ∈ A n , we have by linearity: So A n is an algebra.
Any element x ∈ D n writes x = Thus, for any (a, b) ∈ B n × B n , we have . This means that if x ∈ A n , all partial permutations in the same orbit -that is with the same coset-type -have the same coefficients. Therefore, the elements (S λ,n ) |λ|≤n form a basis of A n .
Proof. We only have to prove the inequalities on the size of ρ. Let α 1 and α 2 be two partial bijections of n with coset-type λ and δ. By definition (see Figure 3), every partial bijection of n that appears in the sum of the product α 1 * α 2 has some coset-type ρ with |ρ| = Lemma 3.10. Let λ be a partition such that |λ| = r ≤ n, we have: Proof. We first prove the following equation: Fix a permutation ω ∈ Kλ(n), that is ω ∈ S 2n and ct(ω) =λ ∪ 1 n−|λ| . We are looking for the number of partial bijections α ∈ A λ,n such that ω is one of its trivial extensions. There is a unique set S such that ct(ω | S ) =λ. We call this set the support of ω and denote it supp(ω). The following condition is necessary so that ω is a trivial extension of α: supp(ω) ⊆ d and α | supp(ω) must be equal to ω | supp(ω) . Thus the partial bijections α we are looking for are the restrictions of ω to sets supp(ω) ⊔ x, with | supp(ω) ⊔ x| = 2|λ|. Since | supp(ω)| = 2|λ|, one has the necessarily |x| = 2(|λ| − |λ|). So the . This ends the proof of equation (13).
By applying ψ n to S λ,n , we get: This lemma implies that ψ n (A n ) ⊆ C[B n /S 2n \ B n ]. The homomorphism A n → C[B n /S 2n \ B n ] mentioned in Section 2.5 is the restriction ψ n | An .
3.6. homomorphism from A n+1 to A n . This paragraph is dedicated to the proof of the following proposition: Proposition 3.11. The function ϕ n defined as follows: is a homomorphism of algebras.
Note that the size of all partitions ρ in the sum index of this equation is n + 1. By applying ϕ n , we get: c ρ λδ (n + 1)ϕ n (S ρ,n+1 ) = 0.
Thus, ϕ n is a homomorphism if we have the following equality for any partition ρ with size at most n: . Let ρ be a partition with size at most n and α an element of A ρ,n . We define H ρ λδ (n) to be the following set: { α 1 , α 2 ∈ A λ,n × A δ,n such that there exists ( α 1 , α 2 ) ∈ E α 2 α 1 (n) with α = α 1 • α 2 }. This set depends on α by definition. However, α does not appear in our notation. This should not be an issue, since α is fixed in the whole proof.
This gives us after simplification: We will now evaluate the quotient |H ρ λδ (n+1)| |H ρ λδ (n)| . Let u = (u 1 , u ′ 1 , u 2 , u ′ 2 ) be an element of P 4 n such that: We introduce Figure 7. The set H ρ λδ (n) is the disjoint union of all N u with u satisfying the above conditions.

Its elements are represented on
be an element of P 4 n satisfying conditions above. If v ′ 1 = u ′ 1 and v 2 = u 2 , then there exists a bijection between N u and N v .
Thus, we have: . This proves that ϕ n is a homomorphism of algebras.
3.7. Projective limits. In this paragraph, we consider the projective limit A ∞ of the sequence (A n ). We prove in Proposition 3.15 that every element of A ∞ is written in a unique way as infinite linear combination of elements indexed by partitions.
For every n ≥ |ρ|, we have: Proof. We proceed by induction on n. For n = |ρ|, we have the equality. Assume we have the equality for some n ≥ |ρ| and let us prove it for n + 1. By equation (19), we have: . This gives us the following equality, using the induction hypothesis : Let A ∞ be the projective limit of (A n , ϕ n ): A ∞ = {(a n ) n≥1 | for every n ≥ 1, a n ∈ A n and ϕ n (a n+1 ) = a n }.
Lemma 3.14. An element a = (a n ) n≥1 is in A ∞ if and only if there exists a family (x a λ ) λ partition of elements of C such that for every n ≥ 1, a n = λ partition |λ|≤n x a λ n |λ| S λ,n .
Proof. Let a = (a n ) n≥1 be a sequence in A ∞ , a n ∈ A n for every n ≥ 1. By Proposition 3.8, the elements (S λ,n ) λ⊢r≤n form a basis of A n , thus for every n ≥ 1 and every partition λ such as |λ| ≤ n, there exists a scalar a λ (n) ∈ C such that a n = λ partition |λ|≤n a λ (n)S λ,n .
By considering the coefficients of S λ,n we get that for every partition λ such that |λ| ≤ n, we have: After an immediate induction, we get : Set x a λ = a λ (|λ|), this proves the "only if" statement. Converse is obvious. For every partition λ, we define the sequence T λ as follows: From Lemma 3.14, we obtain directly the following proposition:

Proposition 3.15. Every element a ∈ A ∞ is written in a unique way as infinite linear combination of elements T λ .
This proposition shows that the algebra A ∞ satisfies the second property required in Section 2.5. In particular, T λ * T δ writes as linear combination of elements T ρ . We can be more precise.
Proof. By Proposition 3.15, T λ * T δ writes as linear combination of elements T ρ .
It remains to prove how we get the conditions about the size of partitions ρ that appear in the sum index and the formula for b ρ λδ . If n < max (|λ|, |δ|), we have: (T λ * T δ ) n = 0. Let n ≥ max (|λ|, |δ|), we use Corollary 3.9 and Corollary 3.13 to get: Comparing both expressions for T λ * T δ , this proves our proposition.
Using the formulas for c ρ λδ (|ρ|) and b ρ λδ given in Corollary 3.13 and 3.16, we obtain: Corollary 3.17. The set of all finite linear combinations of (T λ ), denoted by A ∞ , forms a sub-algebra of A ∞ . The family (T λ ) λ partition is a basis of A ∞ .
Proof. This comes from the fact that the partitions ρ indexing the sum in the product T λ * T δ verify: The algebra A ∞ will be of interest in Section 5.

PROOF OF THEOREM 2.1
In the previous section, we built all algebras and homomorphisms that we need in order to prove Theorem 2.1.
This ends the proof of Theorem 2.1.
The polynomial of some structure coefficients is constant, especially we have the following corollary.

A LINK WITH SHIFTED SYMMETRIC FUNCTIONS
In [IK99, Section 9], Ivanov and Kerov have given an isomorphism between the algebra of 1-shifted symmetric functions and the algebra A ∞ , which is the universal algebra that projects on the center of the symmetric group algebra Z(C[S n ]), for each n. In this section, using the zonal spherical functions of the Gelfand pair (S 2n , B n ), we prove that there is an isomorphism between the algebra of 2-shifted symmetric functions and the algebra A ∞ .
We start with the definition of the algebra of shifted symmetric functions with coefficients in C(α), denoted by Λ * (α). An α-shifted symmetric function f in infinitely many variables (x 1 , x 2 , · · · ) is a family (f i ) i≥1 with the two following properties: ( The set of all shifted symmetric functions is an algebra denoted Λ * (α). In [Las08], Lassalle gives an isomorphism between the algebra of symmetric functions with coefficients in C[α], denoted by Λ(α), and Λ * (α). We will denote this isomorphism by sh α instead of (#), as used by Lassalle. We prefer this notation as it makes the dependence in the parameter α explicit.
Let f be an element of Λ * (α). For any partition λ = (λ 1 , λ 2 , · · · , λ l ), we denote by f (λ) the value f l (λ 1 , λ 2 , · · · , λ l ). The shifted symmetric function f is determined by its values on partitions, see [OO97, Section 2]. 5.1. Gelfand pairs and zonal spherical functions. Let G be a finite group and K a subgroup of G. We denote by C(G, K) the set of all complex-valued functions on G that are constant on each K-double coset in G. Namely, The set C(G, K) is an algebra with product defined as follows (usually called convolution product): The pair (G, K) is said to be a Gelfand pair if the algebra C(G, K) is commutative. More details about Gelfand pairs are given in [Mac95, Chapter VII, 1]. In particular, when (G, K) is a Gelfand pair, the algebra C(G, K) admit a relevant canonical basis (ω i ). The ω i are called zonal spherical functions. The pair (S 2n , B n ) is a Gelfand pair (see [Mac95,Chapter VII,2]) and its zonal spherical functions are indexed by partitions of n. They are denoted by ω ρ and defined by: for x ∈ S 2n , where χ 2ρ is the character of the irreducible S 2n -module corresponding to 2ρ := (2ρ 1 , 2ρ 2 , · · · ). Two permutations x and y in the same B n -double coset K λ (n) have the same image by ω ρ denoted by ω ρ λ .

Isomorphism between
A ∞ and Λ * (2). When α = 2, Jack polynomials are related to zonal spherical functions of (S 2n , B n ) by the following equation (cf. [Mac95, page 408]): for every partition ρ of n. This formula can be viewed as an analogue for α = 2 of the following formula known as Frobenius formula, see [Mac95, page 114]: where s ρ is the Schur function. Equations (22) and (24) give us the following equality when α = 2: Theorem 5.1. The linear mapping F : A ∞ −→ Λ * (2) defined by: is an isomorphism of algebras.
Proof. Let λ be a partition and T λ the corresponding element in A ∞ . Let n be an integer, n ≥ |λ|. By definition, T λ is a sequence and its n-th term (T λ ) n = 1 n |λ| S λ,n lies in A n . We project onto C[B n \ S 2n /B n ] by appying ψ n . By Lemma 3.10, we get: For any partition ρ with size equal to n, by applying ω ρ to ψ n ((T λ ) n ), we obtain: We denote by |Kλ(n)| the number of permutations of 2n with coset-typeλ ∪ (1 n−|λ| ). This number is equal to |Bn| 2 z 2(λ∪(1 n−|λ| ) (see [Mac95,page 402]). Thus, after simplification, we get: which together with (23) gives us the following equation: This formula is valid for any partition ρ such that |ρ| ≥ |λ|. We can check that it is also valid for any partition ρ with size less than |λ|, since in this case (T λ ) |ρ| and sh 2 (p λ )(ρ) are both equal to zero. Finally, for any partition ρ, its image by F (T λ ) as defined in the statement of Theorem 5.1, can also be written as follows: Let δ be a partition, for any partition ρ, we have: The last equality comes from the fact that ω ρ defines a homomorphism of C[B n \ S 2n /B n ] to C * (Proposition 5.1). Hence, for any two partitions λ and δ. That means that F is a homomorphism of algebras from A ∞ to Λ * (2). Since (T λ ) λ partition and (sh 2 (p λ )) λ partition are respectively bases of A ∞ and Λ * (2), F is actually an isomorphism of algebras.
Remark 5.2. The reader should remark while reading the proof that F (T λ ) could be defined by F (T λ )(ρ) = ω ρ (ψ |ρ| ((T λ ) |ρ| )), which would show directly that F (T λ ) is a homomorphism since it is the composition of homomorphisms. However, we prefer to use the definition given by the equation (25) because it gives us explicitly the action of F on the elements of basis of A ∞ .
5.4. Structure constants. As said in the beginning of this paragraph, sh α is an isomorphism between Λ(α) and Λ * (α). Thus, the family (sh α (p λ )) λ partition forms a linear basis of Λ * (α). This allows us to write the following equation: It is proven in [DF12] that the coefficients g ρ λ,δ (α) are polynomial in α−1 √ α . This structure constants are also related to the Matching-Jack conjecture of Goulden and ackson, see [DF12,Section 4.5].
We are interested in the case α = 2. We proved in 3.16 that the coefficients b ρ λδ that appear in the product T λ * T δ are non-negative rational numbers. By applying the isomorphism F given in 5.1 to the product T λ * T δ , we get directly the following proposition.

FILTRATIONS OF THE ALGEBRA A ∞
We gave in Theorem 2.1 a polynomiality property of the structure coefficients of the Hecke algebra of the pair (S 2n , B n ). In order to bound the degree of these polynomials, we study in this section some filtrations of the algebra A ∞ .
From the formula of the product of basis elements in A ∞ , given in Corollary 3.16, we can see that the function deg 1 (T λ ) = |λ| defines a filtration on A ∞ .
Observation 6.1. Let α, β and γ be three partial bijections such that γ ∈ α * β. If α ∈ x and β ∈ y where x and y are two elements of D n with non-negative coefficients, then γ ∈ x * y.
Therefore it is enough to prove the formula (31). Let θ be a partial bijection with coset-type δ. If c 2 : c 3 , c 4 ) is a cycle of length 1, we have two cases: (1) If {c 3 , c 4 } is in the domain of θ: In this case the partial bijections that appear in the expansion of the product θ * C have the same coset-type as θ and we have deg i (θ * C) ≤ deg i (θ) + deg i (C), for i = 2, 3.
(2) If not, then all the partial bijections that appear in the expansion of the product θ * C have the coset-type δ ∪ (1). Then, we can check easily that deg i (θ * C) ≤ deg i (θ) + deg i (C), for i = 2, 3. Now, if C = (c 1 , c 2 : c 3 , c 4 : c 5 , c 6 : c 7 , c 8 ) is a cycle of length 2, the figure of C is represented on Figure  8. We have 4 cases. We give for each case the general result without the details of the proofs. They simply consist in computing compositions of permutations.
(1) {c 3 , c 4 } and {c 7 , c 8 } do not appear in the domain of θ: In this case, the partial bijections that appear in the expansion of the product θ * C have cosettype δ ∪ (2).
(2) one of the sets {c 3 , c 4 } and {c 7 , c 8 } (for example {c 3 , c 4 }) appears in the domain of a cycle ω of θ and the other does not. Suppose that ω is as represented on Figure 9. Then, a cycle with the form drawn on Figure 10 appears in the expansion of the product θ * C. Note that some exterior labels are missing in this figure. To explain this, let us recall that the product α 1 * α 2 of two partial bijections α 1 and α 2 is defined using an average of some partial bijections α. When the extremity on an edge have no exterior labels, that means that the elements of any pair ρ(k) different from {c 1 , c 2 } and {c 5 , c 6 } can be used as labels and that we shall average over all possibilities. We will also use this convention in the last two cases. Thus, in this case, the coset-type of each partial bijection that appears in the expansion of the product θ * C has the same number of parts as δ and its size is equal to |δ| + 1. Consider the exterior labels of this cycle ω. Among them there are c 3 , c 4 , c 7 and c 8 and we know that c 3 and c 4 (resp. c 7 and c 8 ) appear consecutively. Then there are two cases that shall be considered separately. Either labels appear in cyclic order c 3 , c 4 , · · · , c 8 , c 7 or c 3 , c 4 , · · · , c 8 , c 7 . These two cases are represented on Figure 11 and Figure 12.  Note that, on Figure 12 , labels c 7 and c 8 are switched. The form of cycles that appear in the expansion of the product θ * C in each case is given on Figure 13 and Figure 14.  Thus, in the first case the cycle is cut into two cycles, then the coset-type of each partial bijection that appears in the expansion of the product θ * C has the same size as δ and l(δ) + 1 parts. While in the second nothing changes and the coset-type of each partial bijection that appears in the expansion of the product θ * C has the same size as δ and the same number of parts.
(4) the two sets {c 3 , c 4 } and {c 7 , c 8 } appear in the domain of two different cycles of θ.
For example we take the two cycles represented on Figure 15. Then a cycle with the form drawn on Figure 16 appears in the expansion of the product θ * C.
In this case the two cycles are joined to form a cycle, thus the coset-type of each partial bijection that appears in the expansion of the product θ * C has the same size as δ and l(δ) − 1 parts.
In these 4 cases, we can check that we have deg i (θ * C) ≤ deg i (θ) + deg i (C), for i = 2, 3 and this ends the proof of Proposition 6.3.
Proof. From the proof of Theorem 2.1, the degree of f ρ λδ (n) is as follows: On the other hand, since deg i is a filtration for i = 1, 2, 3, we obtain the following inequality: