C Édric

Nicolas Jacon
email: nicolas.jacon@univ-fcomte.fr

CELLULAR STRUCTURES ON HECKE ALGEBRAS OF TYPE B

Keywords:

The aim of this paper is to gather and (try to) unify several approaches for the modular representation theory of Hecke algebras of type B. We attempt to explain the connections between Geck's cellular structures (coming from Kazhdan-Lusztig theory with unequal parameters) and Ariki's Theorem on the canonical basis of the Fock spaces.

Crystal isomorphisms References

Introduction

The modular representation theory of Hecke algebras of type B was first studied by Dipper-James-Murphy [START_REF] Dipper | Hecke algebras of type B n at roots of unity[END_REF]: one of their essential tools was to construct a family of modules (called Specht modules) playing the same role as Specht modules in type A. Each of these new Specht modules have a canonical quotient which is zero or simple: one of the main problem raised by this construction is to determine which one are non-zero. Later, Graham and Lehrer [START_REF] Graham | Cellular algebras[END_REF] developed the theory of cellular algebras, which contains, as a particular case, the construction of Dipper-James-Murphy. The problem of parametrizing the simple modules and computing the decomposition matrix of Specht modules were then solved by Ariki [START_REF] Ariki | On the decomposition numbers of the Hecke algebra of G(m, 1, n)[END_REF] using the canonical basis of Fock spaces of higher level. In fact, Ariki's Theorem provides different parametrizations of the simple modules of the Hecke algebra: only one of them (asymptotic case) has an interpretation in the framework of Dipper-James-Murphy and Graham-Lehrer. Recently, Geck showed that the Kazhdan-Lusztig theory with unequal parameters should provide a cell datum for each choice of a weight function on the Weyl group (if Lusztig's conjectures (P1)-(P15) hold [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF]Conjecture 14.2]).

Our main aim in this paper is to present an overview of all these results, focusing particularly on conjectural connections between Uglov's point of view on the Fock space theory and Geck cellular structures. This should (if Lusztig's conjectures (P1)-(P15) hold in type B) lead to a unified approach for a better understanding of the representation theory of Hecke algebras. As a by-product, we should get an interpretation of all Ariki's parametrizations of simple modules.

More precisely, if Q and q are two indeterminates, if H n denotes the Hecke A-algebra with parameters Q and q (here, A = Z[Q, Q -1 , q, q -1 ]), if ξ is a positive irrational number (!) and if r denotes the unique natural number such that r ξ < r + 1, then Kazhdan-Lusztig theory should provide a cell datum C ξ = ((Bip(n), r ), SBT , C ξ , * ) where

• Bip(n) is the set of bipartitions of n and r is a partial order on Bip(n) depending on r (see §3.2); • If λ ∈ Bip(n), SBT (λ) denotes the set of standard bitableaux of (bi-)shape λ (filled with 1,. . . , n); • If S and T are two standard bitableaux of size n and of the same shape, C ξ S,T is an element of H n coming from a Kazhdan-Lusztig basis of H n (it heavily depends on ξ); • * : H n → H n is the A-linear anti-involution of H n sending the element T w of the standard basis to T w -1 ;

(see [START_REF] Bonnafé | On domino insertion and Kazhdan-Lusztig cells in type B n , to appear in Progress in Mathematics[END_REF]Conjecture C]). If this conjecture holds then, by the general theory of cellular algebras, we can associate to each bipartition λ of n a Specht module S ξ λ endowed with a bilinear form φ ξ λ . If K is the field of fractions of A, then KS ξ λ = K ⊗ A S ξ λ is the simple KH n -module associated to λ [START_REF] Geck | Characters of finite Coxeter groups and Iwahori-Hecke algebras[END_REF]Th. 10.1.5]. Now, if Q 0 and q 0 are two elements of C × then, through the specialization Q → Q 0 , q → q 0 , we can construct the CH n -module

D ξ λ = CS ξ λ / Rad(Cφ ξ λ )
. By the general theory of cellular algebras, it is known that the non-zero D ξ λ give a set of representatives of simple CH n -modules. At this stage, it must be noticed that, even if

KS ξ λ ≃ KS ξ ′ λ (where ξ ′ is another positive irrational number), it might happen that S ξ λ ≃ S ξ ′ λ and D ξ λ ≃ D ξ ′ λ (it is probable that (S ξ λ , φ ξ λ ) ≃ (S ξ ′ λ , φ ξ ′ λ
) for all λ ∈ Bip(n) if and only if we have also r ξ ′ < r + 1).

On the other hand, if we assume further that q 2 0 is a primitive e-th root of unity, if Q 2 0 = -q 2d 0 for some d ∈ Z (which is only well-defined modulo e), and if s = (s 0 , s 1 ) ∈ Z 2 is such that s 0s 1 ≡ d mod e, then Ariki's Theorem provides a bijection between the set of Uglov's bipartitions Bip s e (n) and the set of simple CH nmodules. Moreover, the decomposition matrix is given by d s λµ (1) λ∈Bip(n),µ∈Bip s e (n) , where d s λµ (q) λ,µ∈Bip(n) is the transition matrix between the standard basis and Uglov-Kashiwara-Lusztig's canonical basis of the Fock space (see §2.2). The first result of this paper insures that [6, Conjecture C] is "compatible" with Ariki's Theorem in the following sense:

Theorem. Assume that [6, Conjecture C] holds and assume that s 0 -s 1 ≡ d mod e and s 0 -s 1 r < s 0 -s 1 +e. Then D ξ λ = 0 if and only λ ∈ Bip s e (n) and, if λ ∈ Bip(n) and µ ∈ Bip s e (n), then

[CS ξ λ : D ξ µ ] = d s λµ (1)
. In particular, we have

[CS ξ λ : D ξ µ ] = 0 =⇒ λ r µ
Note that, in the asymptotic case (in other words, if ξ > n-1), then [6, Conjecture C] holds (see [START_REF] Geck | Lusztig's a-function in type B n in the asymptotic case[END_REF]) and the cellular datum C ξ is more or less equivalent to the one constructed by Dipper, James and Mathas (see the work of Geck, Iancu and Pallikaros [START_REF] Geck | Specht modules and Kazhdan-Lusztig cells in type B n , to appear in[END_REF]).

One of the problems raised by the previous theorem (in fact, essentially by Ariki's Theorem) is the following

: if s ′ = (s ′ 0 , s ′ 1 ) ∈ Z 2 is such that s ′ 0 -s ′ 1 ≡ s 0 -s 1 mod
e, then the sets Bip s e (n) and Bip s ′ e (n) are in bijection. Our second result (see Theorem 5.3) is to construct this bijection by means of an isomorphism between the crystals associated to the simple sub-modules M[s] and M[s ′ ] of the Fock spaces (see §2.1 for the definition of these modules). This shows that the "abstract crystal" of an irreducible highest weight module is canonically associated to the representation theory of Hecke algebras of type B. By contrast, each realization of this crystal is associated with a natural parametrization of the simple modules.

Once we accept [6, Conjecture C], it is natural to ask whether the matrix d s λµ (q) can be interpreted as a q-decomposition matrix using Jantzen's filtration (see our Conjecture C in §4.5). It would also be interesting to see if it should be possible to construct different Schur algebras of type B directly from Kazhdan-Lusztig's theory (for the asymptotic case, this construction should give rise to an algebra which is Morita equivalent to the two Schur algebras of type B constructed by Du and Scott [START_REF] Du | The q-Schur 2 algebra[END_REF] and Dipper, James and Mathas [START_REF] Dipper | Cyclotomic q-Schur algebras[END_REF]). If so, it is then natural to ask for a generalization of Varagnolo-Vasserot Theorem for Hecke algebras of type A as well as a generalization of Yvonne's Conjecture [START_REF] Yvonne | A conjecture for q-decomposition matrices of cyclotomic v-Schur algebras[END_REF]Conj. 2.13]. Note that a construction of Schur algebras of type B is provided by the theory of Cherednik algebras [START_REF] Ginzburg | On the category O for rational Cherednik algebras[END_REF], but this does not provide a generic Schur algebra.

If the reader wants more arguments for [6, Conjecture C], he or she is encouraged to read the original source of this conjecture [START_REF] Bonnafé | On domino insertion and Kazhdan-Lusztig cells in type B n , to appear in Progress in Mathematics[END_REF]. Note also that recent works by Gordon and Martino (see [START_REF] Gordon | Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras[END_REF], [START_REF] Gordon | Calogero-Moser space, reduced rational Cherednik algebras, and two-sided cells[END_REF]) on Cherednik algebras show other compatibilities between this conjecture and the geometry of the Calogero-Moser spaces, as well as with Baby Verma modules of Cherednik algebras at t = 0.

Finally, it is natural to ask whether there exist different cellular structures for Ariki-Koike algebras associated to the complex reflection groups G(d, 1, n), these cellular structures being indexed by d-cores (or by d-uples of elements of Z): however, in this case, no Kazhdan-Lusztig's theory is available at that time (will there be one in the future?) so we have no candidate for the different cellular bases.

This paper is organized as follows. In the first section, we present the setting of our problem. Then, we recall Ariki's Theorem which allows to compute the decomposition matrices for Hecke algebras of type B n using objects coming from quantum group theory. In the third section, we introduce the combinatorial notions that we need to describe the Kazhdan-Lusztig theory in type B n . The last two sections are devoted to the main results of our paper. Under some conjectures, we show the existence of several deep connections between Kazhdan-Lusztig theory and the canonical basis theory for U v ( sl e ).

Notation

Let (W n , S n ) be a Weyl group of type B n and assume that the elements of S n are denoted by t, s 1 ,. . . , s n-1 in such a way that the Dynkin diagram is given by

i i i • • • i t s 1 s 2 s n-1
The length function with respect to S n will be denoted by ℓ :

W n → N = {0, 1, 2 . . . }.
Let Γ be a torsion-free finitely generated abelian group and let A denote the group ring Z[Γ]. The group law on Γ will be denoted additively and we shall use an exponential notation for A: more precisely, A = ⊕ γ∈Γ Ze γ , where e γ e γ ′ = e γ+γ ′ for all γ, γ ′ ∈ Γ. We also fix two elements a and b in Γ and we denote by H n the Hecke algebra of W n over A associated to the choice of parameters t → b and s i → a for 1 i n -1 symbolized by the following diagram:

i i i • • • i b a a a t s 1 s 2 s n-1
More precisely, there exists a basis (T w ) w∈Wn of the A-module H n such that the multiplication on H n is A-bilinear and completely determined by the following properties:

     T w T w ′ = T ww ′ , if ℓ(ww ′ ) = ℓ(w) + ℓ(w ′ ), (T t -e b )(T t + e -b ) = 0, (T s i -e a )(T s i + e -a ) = 0, if 1 i n -1.
All along this paper, we also fix a field k of characteristic 0 and a morphism of groups θ : Γ → k × : the morphism θ extends uniquely to a morphism of rings Z[Γ] → k that we still denote by θ. We then set

kH n = k ⊗ A H n
and we still denote by θ the specialization morphism H n → kH n . Moreover, we assume that the following holds: there exist natural numbers d and e such that e 1 and    θ(a) 2 is a primitive e-th root of unity;

θ(b) 2 = -θ(a) 2d .
If M is an H n -module, we denote by kM the kH n -module k⊗ A M. The Grothendieck group of kH n will be denoted by R(kH n ) and, if L is a kH n -module, its class in R(kH n ) will be denoted by [L]. Note that the algebra kH n is split. Since Γ is torsion-free, the ring Z[Γ] is integral and we denote by K its field of fractions. Then the algebra KH n = K ⊗ A H n is split semisimple. Its simple modules are parametrized by the set Bip(n) of bipartitions of n: we shall denote this bijection by Bip(n) -→ Irr KH n λ -→ V λ . This bijection is chosen as in [19, 10.1.2]. We denote by Bip the set of all bipartitions (i.e. Bip = n 0 Bip(n)), the empty partition will be denoted by ∅ and the empty bipartition (∅, ∅) (which is the unique bipartition in Bip(0)) will be denoted by ∅. Since A is integrally closed (it is a Laurent polynomial ring in several algebraically independent indeterminates), there is a well-defined decomposition map [START_REF] Geck | Characters of finite Coxeter groups and Iwahori-Hecke algebras[END_REF]Theorem 7.4.3] d n : R(KH n ) -→ R(kH n ).

Fock space, canonical basis and Ariki's Theorem

The aim of this section is to recall Ariki's Theorem relating the canonical basis of the Fock space and the decomposition matrix d n . The main references are [START_REF] Ariki | Representations of Quantum algebras and combinatorics of Young tableaux[END_REF], [START_REF] Geck | Modular representations of Hecke algebras[END_REF] and [START_REF] Yvonne | Base canonique d'espaces de Fock de niveau supérieur[END_REF].

2.1. The Fock space. Let v be an indeterminate. Let h be a free Z-module with basis (h 0 , . . . , h e-1 , d) and let (Λ 0 , . . . , Λ e-1 , δ) be its dual basis in h * = Hom Z (h, Z). The quantum group U v ( sl e ) is defined as the unital associative C(v)-algebra generated by elements {e i , f i | 0 i e -1} and {k h | h ∈ h} subject to the relations given for example in [31, chapter 6]. We denote by U v ( sl e ) ′ the subalgebra of U v ( sl e ) generated by e i , f i , k h i , k -1 h i for i ∈ {0, 1, ..., e -1}. We fix a pair s = (s 0 , s 1 ) ∈ Z 2 . To s is associated a Fock space (of level 2) F s : this is an U v ( sl e )-module defined as follows. As a C(v)-vector space, it has a basis given by the symbols |λ, s where λ runs over the set of bipartitions:

F s = ⊕ λ∈Bip C(v) |λ, s .
The action of the generators U v ( sl e ) is given for instance in [START_REF] Uglov | Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials[END_REF]. By [START_REF] Jimbo | Combinatorics of representations of U q ( ŝl(n)) at q = 0[END_REF], F s is an integrable U v ( sl e )-module.

If m ∈ Z, we denote by m the unique element of {0, 1, . . . , e-1} such that m ≡ m mod e. We then set ∆(s) = 1 2

′ = (s ′ 0 , s ′ 1 ) ∈ Z 2 , then M[s] ′ ≃ M[s ′ ] ′ if and only if (s 0 , s 1 ) ≡ (s ′ 0 , s ′ 1 ) mod eZ 2 or (s 0 , s 1 ) ≡ (s ′ 1 , s ′ 0 ) mod eZ 2 .
2.2. Uglov's canonical basis. We shall recall here Uglov's construction of a canonical basis of the Fock space, which contains as a particular case the Kashiwara-Lusztig canonical basis of the simple U v ( sl e )-module M[s] (which is the same as the one of M[s] ′ ). The reader may refer to Uglov's original paper [START_REF] Uglov | Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials[END_REF] for further details. First, recall that there is a unique C-linear involutive automorphism of algebra : U v ( sl e ) -→ U v ( sl e ) such that, for all i ∈ {0, 1, ..., e -1} and h ∈ h, we have

v = v -1 , k h = k -h , e i = e i , f i = f i .
It is C(v)-antilinear (with respect to the restriction of ? to C(v)).

One of the main ingredients in Uglov's construction is a C(v)-antilinear involution : F s -→ F s which is defined using the wedge realization of the Fock space. This involution is U v ( sl e )-antilinear, that is, for all u ∈ U v ( sl e ) and f ∈ F s , we have

(2.2) u • f = u • f .
Moreover, recall that

(2.3) |∅, s = |∅, s . Now, if λ ∈ Bip, there exists a unique G(λ, s) ∈ F s satisfying    G(λ, s) = G(λ, s), G(λ, s) ≡ |λ, s mod vC[v].
The proof of this result is constructive and gives an algorithm to compute this canonical basis. This algorithm has been improved by Yvonne [START_REF] Yvonne | An algorithm for computing the canonical bases of higher-level q-deformed Fock spaces[END_REF]. Let us write,

for µ ∈ Bip, G(µ, s) = λ∈Bip d s λ,µ (v) |λ, s , where d s µ,µ (v) = 1 and d s λ,µ (v) ∈ vC[v] if λ = µ.

We set

Bip s e (n) = Bip s e ∩ Bip(n).

2.3. Ariki's Theorem. We shall recall here the statement of Ariki's Theorem, using the later construction of Uglov:

Ariki's Theorem. Assume that d ≡ s 0s 1 mod e. Then there exists a unique bijection

Bip s e (n) -→ Irr kH n µ -→ D s µ such that d n [V λ ] = µ∈Bip σ e (n) d s λ,µ (1) [D s µ ]
for all λ ∈ Bip(n).

This theorem, together with the algorithm for computing the Kashiwara-Lusztig canonical bases [START_REF] Jacon | An algorithm for the computation of the decomposition matrices for Ariki-Koike algebras[END_REF], provides an efficient tool for computing the decomposition map d n .

Domino insertion

We shall review here some combinatorial results about the domino insertion algorithm: the reader may refer to [START_REF] Lam | Growth diagrams, domino insertion, and sign-imbalance[END_REF], [START_REF] Vanleeuwen | The Robinson-Schensted and Schutzenberger algorithms, an elementary approach, The Foata Festschrift[END_REF] and [START_REF] Shimozono | Color-to-spin ribbon Schensted algorithms[END_REF] for further details.

3.1. Domino tableaux, bitableaux. Let r ∈ N. We denote by δ r the 2-core (r, r -1, . . . , 1), with the convention that δ 0 = ∅. Let P denote the set of all partitions and let P r denote the set of partitions with 2-core δ r . We denote by P r (n) the set of partitions of 2-weight n and 2-core δ r , so that

P r = n 0 P r (n) and P = r 0 P r .
Note that partitions in P r (n) are partitions of 2n + r(r -1) 2 . Let q : P → Bip the map sending a partition to its 2-quotient. By composition with the bijection

Bip

-→ Bip (λ 0 , λ 1 ) → (λ 0 , λ 1 ) if r is even (λ 1 , λ 0 ) if r is odd the restriction of q to P r induces a bijection

q r : P r ∼ -→ Bip .
The restriction of q r to P r (n) induces a bijection

P r (n) ∼ -→ Bip(n)
that we still denote by q r if no confusion may arise. Let SDT r (n) denote the set of standard domino tableaux whose shape lies in P r (n) (and filled with dominoes with entries 1, 2,. . . , n) and let SBT (n) denote the set of standard bitableaux of total size n (filled again with boxes with entries 1, 2,. . . , n). We denote by qr : SDT r (n) → SBT (n) the bijection obtained as a particular case of [START_REF] Carré | Splitting the square of a Schur function into its symmetric and antisymmetric parts[END_REF]Theorem 7.3]. If λ : SBT (n) → Bip(n) (respectively ∆) sends a bitableau (respectively a domino tableau) to its shape, then the diagram

(3.1) SDT r (n) qr ∼ / / ∆ SBT (n) λ P r (n) q r ∼ / / Bip(n) is commutative.
3.2. Orders between bipartitions. The bijection q r allows us to define several partial orders on the set of bipartitions. First, let denote the dominance order on P. We then define the partial order r on Bip as follows: if λ, µ ∈ Bip, then In the diagram, we put an arrow between two bipartitions λ -→ µ if µ i λ and if there is no other bipartition ν such that µ i ν i λ.

λ r µ ⇐⇒ q -1 r (λ) q -1 r (µ).
0 1 ∞ (3; ∅) (3; ∅) (3; ∅) (∅; 3) (21; ∅) (21; ∅) y y t t t t t t t t t % % L L L L L L L L L L
(2; 1)

% % K K K K K K K K K K
y y s s s s s s s s s s (2; 1) (2; 1)

% % J J J J J J J J J
(111; ∅) y y r r r r r r r r r r 3.3. Domino insertion algorithm. Recall that r ∈ N is fixed. If w ∈ W n , then the domino insertion algorithm, as described for instance in [START_REF] Lam | Growth diagrams, domino insertion, and sign-imbalance[END_REF], associates to w a standard domino tableau P r (w) ∈ SDT r (n). We set S r (w) = qr (P r (w)).

(1; 2) * * U U U U U U U U U U U U U U U U U U U U U (21; ∅) t t i i i i i i i i i i i i i i i i i i i i (∅; 3) (11; 1) (∅; 21) % % K K K K K K K K K K (11;
Then S r (w) is a standard bitableau of total size n. We also set Q r (w) = P r (w -1 ) and T r (w) = S r (w -1 ) = qr (Q r (w)).

It turns out that P r (w) and Q r (w) have the same shape, as well as S r (w) and T r (w). Also, if we set

SBT (2) (n) = {(S, T ) ∈ SBT (n) × SBT (n) | λ(S) = λ(T )},
then the map

(3.5) W n -→ SBT (2) (n) w -→ (S r (w), T r (w)) is bijective. If w ∈ W n , then we set λ r (w) = λ(S r (w)) ∈ Bip(n).
Note that

(3.6) λ r (w) = λ(T r (w)) = λ r (w -1 ). Remark 3.7 -If r, r ′ n -1, then the bijection W n → SBT (2) (n), w → (S r ′ (w), T r ′ (w)
) coincides with the bijection 3.5. We shall denote it by W n → SBT (2) (n), w → (S ∞ (w), T ∞ (w)). Similarly, we shall set λ ∞ (w) = λ(S ∞ (w)).

Kazhdan-Lusztig theory

The aim of this section is to show that Kazhdan-Lusztig theory with unequal parameters should provide cellular data on H n which are compatible with Ariki's Theorem. The reader may refer to [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF] for the foundations of this theory.

We denote by H n → H n , h → h the unique automorphism of ring such that e γ = e -γ and T w = T -1 w -1 for all γ ∈ Γ and w ∈ W n . It is an A-antilinear involutive automorphism of the ring H n . 4.1. Kazhdan-Lusztig basis. From now on, we fix a total order on Γ which endows Γ with a structure of ordered group.

A 0 = ⊕ γ 0 Ze γ and A <0 = ⊕ γ<0 Ze γ .
Then A 0 is a subring of A and A <0 is an ideal of A 0 . We also set

H <0 n = ⊕ w∈Wn A <0 T w .
It is a sub-A 0 -module of H n .

If w ∈ W n , then [30, Theorem 5.2] there exists a unique element

C w ∈ H n such that    C w = C w ;
C w ≡ T w mod H <0 n . Also, (C w ) w∈Wn is an A-basis of H n and this basis depends on the choice of the order on Γ. We define the preorders L, , R, and LR, on W n as in [30, §8.1] and we denote by ∼ L, , ∼ R, and ∼ LR, the equivalence relations respectively associated to these preorders. Here, the exponents or the indices stands for emphasizing the fact that the objects really depends on the order on Γ. Note that the flexibility on the choice of r whenever b > (n -1)a will not change the statements below (see Remarks 3.2 and 3.7). Note also that, once a and b are fixed elements of Γ, then the number r depends only on the choice of the order on Γ (again with some flexibility if b > (n -1)a).

Conjecture A. With the above definition of r, we have, for all w, w ′ ∈ W n : Note also that the statements (a), (b) and (c) have been proved whenever b > (n -1)a (asymptotic case: see [START_REF] Bonnafé | Left cells in type B n with unequal parameters[END_REF]Theorem 7.7] and [5, 3.9]): the statement (c + ) has been proved only if w and w ′ have the same t-length (i.e. if the numbers of occurrences of t in a reduced expression of w and w ′ are equal), see [5, 3.8].

(a) w ∼ L, w ′ if and only if T r (w) = T r (w ′ ). (b) w ∼ R, w ′ if and only if S r (w) = S r (w ′ ). (c) w ∼ LR, w ′ if and only if λ r (w) = λ r (w ′ ). (c + ) w LR, w ′ if and only if λ r (w) r λ r (w ′ ).
Statements (a), (b) and (c + ) have also been proved whenever a = 2b or 3a = 2b (see [START_REF] Bonnafé | On domino insertion and Kazhdan-Lusztig cells in type B n , to appear in Progress in Mathematics[END_REF]Theorem 3.14]).

Finally, we must add that Conjecture A is "highly probable": it has been checked for n 6 and is compatible with many other properties of Kazhdan-Lusztig cells and other conjectures of Lusztig. For a detailed discussion about this, the reader may refer to [START_REF] Bonnafé | On domino insertion and Kazhdan-Lusztig cells in type B n , to appear in Progress in Mathematics[END_REF].

From now on, and until the end of this paper, we assume that Conjecture A holds. We shall now define a basis of H n which depends on a, b and as follows. If (S, T ) ∈ SBT (2) (n), let w be the unique element of W n such that S = S r (w) and T = T r (w): we then set

C S,T = (C w ) † ,
where † :

H n → H n is the unique A-algebra involution such that T † s = -T -1 s
for all s ∈ S n . Now, if λ ∈ Bip(n), we denote by SBT (λ) the set of standard bitableaux of shape λ (i.e. SBT (λ) = λ -1 (λ)). Finally, we denote by H n → H n , h → h * the unique A-linear map such that T * w = T w -1 for all w ∈ W n . It is an involutive anti-automorphism of the A-algebra H n . Then we have constructed a quadruple ((Bip(n), r ), SBT , C , * ) where

• (Bip(n), r ) is a poset; • For each λ ∈ Bip(n), SBT (λ) is a finite set; • Let SBT (2) (n) = λ∈Bip(n) SBT (λ) × SBT (λ) then the map C : SBT (2) (n) -→ H n (S, T ) -→ C S,T
is injective and its image is an A-basis (C S,T ) (S,T )∈SBT (2) (n) of H n ; • The map * : H n → H n is an A-linear involutive anti-automorphism of the ring H n such that (C S,T ) * = C T,S for all (S, T ) ∈ SBT (2) (n).

The next conjecture is taken from [6, Conjecture C].

Conjecture B. The quadruple ((Bip(n), r ), SBT , C , * ) is a cell datum on H n , in the sense of Graham-Lehrer [START_REF] Graham | Cellular algebras[END_REF].

Remark 4.2 -For a discussion of some evidences for this conjecture, see again [START_REF] Bonnafé | On domino insertion and Kazhdan-Lusztig cells in type B n , to appear in Progress in Mathematics[END_REF]. Note that if Lusztig's Conjectures P1-P15 in [6, Chapter 14] hold and if Conjecture A above holds, then Conjecture B holds by a recent work of Geck [START_REF] Geck | Hecke algebras of finite type are cellular[END_REF].

In the rest of this paper, we shall give some more evidences for Conjectures A and B, towards their compatibilities with Ariki's Theorem.

Hypothesis: From now on, and until the end of this paper, we assume that Conjectures A and B hold. We shall denote by C the cell datum ((Bip(n), r ), SBT , C , * ).

4.3.

Interpretation of Ariki's Theorem. Since we assume that Conjectures A and B hold, the general theory of cellular algebras [START_REF] Graham | Cellular algebras[END_REF] shows that the cell datum C provides us with a family of Specht modules S λ associated to each bipartition λ of n. Moreover, S λ is endowed with a bilinear form φ λ : S λ × S λ → A such that, for all h ∈ H n and x, y ∈ S λ , we have

(4.3) φ λ (h • x, y) = φ λ (x, h * • y).
We denote by kφ λ the specialization of φ λ to kS λ (through θ : Γ → k × ). We then set D λ = kS λ / rad(kφ λ ).

By 4.3, rad(kφ λ ) is a kH n -submodule of S λ , so that D λ is a kH n -module. We set

Bip (k) (n) = {λ ∈ Bip(n) | D λ = 0}.
Then, by the theory of cellular algebras [START_REF] Graham | Cellular algebras[END_REF]Thm. 3.4], (D λ ) λ∈Bip (k) (n) is a family of representatives of isomorphism classes of irreducible kH n -modules.

Remark 4.4 -Note that the kH n -modules D λ , as well as the H n -modules S λ , depend heavily on the choice of the order on Γ (they depend on r): several choices for will lead to non-isomorphic kH n -modules. In particular, the set Bip (k) (n) does depend on (see §5).

Once we accept Conjectures A and B, it is highly probable that the Specht modules satisfy the following property.

Conjecture B'. For all n ∈ N and for all λ ∈ Bip(n), we have

KS λ ≃ V λ
Recall that θ is the specialization morphism H n → kH n such that there exist natural numbers d and e such that e 1 and    θ(a) 2 is a primitive e-th root of unity;

θ(b) 2 = -θ(a) 2d .
The following result is a generalization of [START_REF] Ariki | On the decomposition numbers of the Hecke algebra of G(m, 1, n)[END_REF] and [START_REF] Geck | Canonical basic sets in type B n[END_REF] (see also [START_REF] Geck | Hecke algebras of finite type are cellular[END_REF]).

then Bip (k) (n) = Bip s e (n) and [kS λ : D µ ] = d s λ,µ (1
) for all µ ∈ Bip (k) (n). In other words, D µ ≃ D s µ and

d n [V λ ] = µ Bip (k) (n) d s λ,µ (1)[D µ ].
Proof. Assume that p is the unique integer such that

d + pe r < d + (p + 1)e.
and put

s = (d + pe, 0) Let µ ∈ Bip s e (n)
. Then, because of the above condition on p, we can use the same strategy as in the proof of [START_REF] Geck | Canonical basic sets in type B n[END_REF]Theorem 5.4] to show that:

G(µ, s) = |µ, s + λ r µ, λ =µ d s λ,µ (v) |λ, s , Now, assume that ν ∈ Bip (k) (n).
Then by Ariki's Theorem there exists µ ∈ Bip s e (n) such that for all λ ∈ Bip(n): e (n) are known as Uglov bipartitions. They are constructed by using the crystal graph of the associated Fock space representation (see for example [START_REF] Ariki | Representations of Quantum algebras and combinatorics of Young tableaux[END_REF] for details on crystal graphs). In general, we only have a recursive definition for them. However, there exist non-recursive characterizations of such bipartitions in particular cases:

d λ,ν = d s λ,µ (1)
• in the case where -e ≤ d + pe < 0, see [START_REF] Foda | Branching functions of A (1) sn -1 and Jantzen-Seitz problem for Ariki-Koike algebras[END_REF] and [24, Proposition 3.1] (the Uglov bipartitions are then called the FLOTW bipartitions);

• in the case where d+pe > n-1-e, see [START_REF] Ariki | Dipper James Murphy's conjecture for Hecke algebras of type B n , to appear in Progress in Mathematics[END_REF], [START_REF] Ariki | On the tensor product of two basic representations of U v ( sl e )[END_REF] (the Uglov bipartitions are then called the Kleshchev bipartitions). Note that this includes the "asymptotic case", that is, the case where b > (n -1)a.

4.5. Jantzen filtration. Once we believe in Conjectures A and B, it is natural to try to develop the theory as in the type A case. For instance, one can define a Jantzen filtration on kS λ as follows. First, there exists a discrete valuation ring O ⊂ K containing A such that, if we denote by p the maximal ideal of O, then p ∩ A = Ker θ. Since K is the field of fractions of A, the map θ : A → k extends to a map θ : O → k with kernel p and we have

kH n = k ⊗ O OH n
where OH n = O ⊗ A H n . Similarly, if λ ∈ Bip(n), then OS λ is an OH n -module and the extension of scalars defines a bilinear form Oφ λ on OS λ . We then set, for all i 0,

OS λ (i) = {x ∈ OS λ | ∀ y ∈ OS λ , Oφ λ (x, y) ∈ p m }.
We then set kS λ (i) = OS λ (i) + pS λ )/pS λ .

Then there exists m 0 such that kS λ (m 0 ) = 0 and the S λ (i)'s are kH n -submodules of kS λ . Moreover, kS λ (0) = kS λ and kS λ (1) = rad kφ λ . The Jantzen filtration of kS λ is the filtration

0 = kS λ (m 0 ) ⊆ kS λ (m 0 -1) ⊆ • • • ⊆ kS λ (1) ⊆ kS λ (0) = kS λ .
The next conjecture proposes an interpretation of the polynomials d s λ,µ as a vdecomposition matrix. This is a generalization of a conjecture by Lascoux, Leclerc and Thibon [29, §9].

Conjecture C. Let λ ∈ Bip(n) and µ ∈ Bip (k) (n). Let p be the unique integer such that

(d + pe)a < b < (d + (p + 1)e)a. Put s = (d + pe, 0), then d s λ,µ (v) = i 0 [kS λ (i)/kS λ (i + 1) : D µ ] v i .
It would also be very interesting to find an analogue of Jantzen's sum formula. This formula could be obtained using the matrix described by Yvonne in [35, §7.4]. 4.6. An example: the asymptotic case. Assume here, and only in this subsection, that b > (n -1)a (in other words, if r n -1). Then Theorem 4.5 holds without assuming that Conjectures A and B are true in this case. However, note that, at the time this paper is written, the Conjecture A is not fully proved. As it is explained in Remark 4.1, only (a), (b) and (c) are proved and part of (c + ). However, we can define an order ′ ∞ on Bip(n) as follows: if λ, µ ∈ Bip(n), we write λ ′ ∞ µ if w LR, w ′ for some (or all) w and w ′ in W n such that λ ∞ (w) = λ and λ ∞ (w ′ ) = µ. Then, by the work of Geck-Iancu [START_REF] Geck | Lusztig's a-function in type B n in the asymptotic case[END_REF] and Geck [START_REF] Geck | Hecke algebras of finite type are cellular[END_REF], we have that ((Bip(n), ′ ∞ ), SBT , C , * ) is a cell datum on H n and it is easily checked that the proof of Theorem 4.5 remains valid if we replace everywhere r by ′ ∞ .

Note also that the cell datum ((Bip(n), ′ ∞ ), SBT , C , * ) is roughly speaking equivalent to the cell data constructed by Graham and Lehrer [20, Theorem 5.5] or Dipper, James and Mathas [9, Theorem 3.26].

Generic Hecke algebra

In the last section, we have shown that, if we assume that conjectures A and B hold, the choice of a pair (a, b) ∈ Γ 2 and the choice of a total order on Γ lead to an "appropriate" representation theory for the associated Hecke algebra of type B n . In this section, we show that these results can be applied to find several "Specht modules" theories for the same algebra: we shall work with the group Γ = Z 2 , with a = (1, 0) and b = (0, 1) and the main theme of this section will be to make the order vary.

Hypotheses and notation: From now on, we assume that Γ = Z 2 , a = (1, 0) and b = (0, 1). We set Q = e b and q = e a so that A

= Z[Q, Q -1 , q, q -1 ].
In this case, H n is called the generic Hecke algebra of type B n . Recall that

(T t -Q)(T t + Q -1 ) = (T s i -q)(T s i + q -1 ) = 0 for all i ∈ {1, 2, . . . , n -1}. Now, any choice of elements Q 0 and q 0 in k × defines a unique morphism θ : Γ → k × , a → v 0 , b → V 0 .
5.1. Orders on Z 2 . Our main theme in this section is to use the possibility of endowing Γ = Z 2 with several orders. For instance, if ξ ∈ R >0 \ Q (a positive irrational number), then we can define

(m, n) ξ (m ′ , n ′ ) if and only if m + ξn m ′ + ξn ′ .
This defines a total order on Z 2 . Moreover, 0 < ξ a and 0 < ξ b, so that all the results of this paper can be applied. If r denotes the entire part [ξ] of ξ, then ra < b < (r + 1)a.

For simplification, we shall denote by S ξ λ the H n -module S ξ λ and D ξ λ the kH nmodule D ξ λ . We also set Bip ξ (k) (n) = Bip ξ (k) (n). We also assume that the following holds: there exist natural numbers d and e such that e 1 and    θ(a) 2 is a primitive e-th root of unity;

θ(b) 2 = -θ(a) 2d .
We can now restate the Theorem 4.5 in this case: This Theorem shows that, if Conjectures A and B hold, then the different choices of ξ (or of other orders on Z 2 ) lead to:

• Different cellular structures on H n ; • Different families of Specht modules; • Different parametrizations of the simple kH n -modules, • Different "v-decomposition matrices" as defined in 4.5 (despite the fact that the decomposition matrices must be equal up to permutations of the rows and the columns).

For instance, if ξ and ξ ′ are two positive irrational numbers, then the KH n -modules KS ξ λ and KS ξ ′ λ are isomorphic, but the H n -modules S ξ λ and S ξ ′ λ might be nonisomorphic.

Remark 5.2 -If ξ > n -1, then Geck, Iancu and Pallikaros [START_REF] Geck | Specht modules and Kazhdan-Lusztig cells in type B n , to appear in[END_REF] have shown that the Specht modules S ξ λ are isomorphic to the ones constructed by Dipper, James and Murphy [START_REF] Dipper | Hecke algebras of type B n at roots of unity[END_REF]. 5.2. Crystal isomorphisms. Let ξ 1 and ξ 2 be irrational numbers. Let r 1 and r 2 be the natural numbers such that r 1 < ξ 1 < (r 1 + 1) and r 2 < ξ 2 < (r 2 + 1).

Then if we use the order ξ 1 , we obtain a complete set of non-isomorphic simple modules for the specialized algebra kH n :

{D ξ 1 λ | λ ∈ Bip ξ 1 (k) }.
On the other hand, if we use the order ξ 2 , we obtain a complete set of nonisomorphic simple modules for the same algebra:

{D ξ 2 λ | λ ∈ Bip ξ 2
(k) }. By Theorem 5.1, there exist s 1 ∈ Z 2 and s 2 ∈ Z 2 such that: For all µ ∈ Bip s 1 e (n), D ξ 1 µ ≃ D ξ 2 c(µ) . In this section, we want to explicitly determine the bijection c. To do this, we first note that this map induces a bijection between the vertices of the crystal graphs of M[s 1 ] and M[s 2 ].

Bip ξ 1 (k) (n) = Bip
The modules M[s 1 ] ′ and M[s 2 ] ′ are isomorphic as U v ( sl e ) ′ modules. As a consequence, the crystal graphs of M[s 1 ] and M[s 2 ] are also isomorphic. This bijection may be obtained by following a sequence of arrows back to the empty bipartition in the crystal graph of M[s 1 ] and then applying the reversed sequence from the empty bipartition of M[s 2 ]. One can also define it as follows. Let

Finally, recall that

  there exists a unique subset Bip s e of Bip such that (2.4) G(λ, s) λ∈Bip s e coincides with Kashiwara-Lusztig canonical basis of M[s].

Remark 3 . 2 -Example 3 . 3 -Example 3 . 4 -

 323334 If r ≥ n -1 and s ≥ n -1, then the partial orders r and s coincide on Bip(n). In fact, they coincide with the classical dominance order on bipartitions that we shall denote by ∞ . Here are the orders 0 and1 = 2 = • • • = ∞ on Bip(2):(∅; 11) 0 (11; ∅) 0 (1; 1) 0 (∅; 2) 0 (2; ∅) (∅; 11) ∞ (∅; 2) ∞ (1; 1) ∞ (11; ∅) ∞ (2; ∅) Here are the orders 0 , 1 and 2 = 3 = • • • = ∞ on Bip(3).

4. 2 .

 2 Bonnafé-Geck-Iancu-Lam conjectures. From now on, we shall assume that the parameters a, b in Γ are strictly positive and that b ∈ {a, 2a, . . . , (n -1)a}.If b < (n -1)a, then we denote by r the unique natural number such that ra < b < (r + 1)a. If b > (n -1)a, then we may choose for r any value in {n -1, n, n + 1, . . . } ∪ {∞}.

Remark 4 . 1 -

 41 First, note that the statements (a) and (b) in Conjecture A are equivalent. Note also that, if Lusztig's Conjectures P1-P15 in [30, Chapter 14] hold, then (a) and (b) imply (c).

Theorem 4 . 5 .

 45 Assume that Conjectures A, B and B' hold. Let λ ∈ Bip(n). Let p be the unique integer such that d + pe r < d + (p + 1)e. Put s = (d + pe, 0)

  , where d λ,ν = [kS λ : D ν ]. We have : d µ,ν = 1 and d λ,ν = 0 if µ r λ and µ = λ By the property of cellular algebras [20, Proposition 3.6], this implies that ν = µ. As we have a bijection between Bip (k) (n) and Bip s e (n), this implies that these two sets are equal. The rest of the theorem is now obvious. Remark 4.6 -The above theorem gives a conjectural interpretation of Ariki's parametrization of simple kH n -modules in terms of a new cellular structure on H n (coming from Kazhdan-Lusztig's theory).

4. 4 .

 4 Uglov bipartitions. The bipartitions in Bip s

Theorem 5 . 1 .

 51 Assume that Conjectures A, B and B' hold. Let λ ∈ Bip(n). Let p be the unique integer such that d + pe < ξ < d + (p + 1)e. Put s = (d + pe, 0) then Bip ξ (k) (n) = Bip s e (n) and [kS ξ λ : D ξ µ ] = d s λ,µ (1) for all µ ∈ Bip ξ (k) (n). In other words, D ξ µ ≃ D s µ and d n [V λ ] = µ∈Bip ξ (k) (n) d s λ,µ (1)[D ξ µ ].

s 1 e

 1 (n) and Bip ξ 2 (k) (n) = Bip s 2e (n) and we have a bijection c : Bip s 1 e (n) → Bip s 2 e (n) which is uniquely determined by the condition that:

B s 1

 1 := {G(µ, s 1 ) | µ ∈ Bip s 1 e } be the elements of the canonical basis of M[s 1 ] and letB s 2 := {G(µ, s 2 ) | µ ∈ Bip s 2e } be the elements of the canonical basis of M[s 1 ]. Then if we specialize the element of B s 1 or B s 2 to v = 1, we obtain a basis of the same irreducible highest weight U( sl e )-modules. Hence, for all µ ∈ Bip s 1 e (n), there exists γ(µ) ∈ Bip s 2 e (n) such that for all λ ∈ Bip(n), d s 1 λ,µ (1) = d s 2 λ,γ(µ)[START_REF] Ariki | On the decomposition numbers of the Hecke algebra of G(m, 1, n)[END_REF].

  We denote it by M[s] and it is isomorphic to the irreducible U v ( sl e )module with highest weight -∆(s)δ + Λ s0 + Λ s1 . In addition, the submodule U v ( sl e ) ′ |∅, s generated by |∅, s is also an irreducible highest weight module with weight -∆(s)δ + Λ s0 + Λ s1 . We denote it by M[s] ′ . If M and N are simple U v ( sl e ) ′ -modules with highest weight Λ and Λ ′ respectively, then M ≃ N if and only if Λ ≡ Λ ′ mod Zδ. Therefore, if s

	1 j=0 (s Remark 2.1 -s j -sj e

j + sje).

Since |∅, s is a highest weight vector of F s and as F s is an integrable module, it follows that the submodule U v ( sl e )|∅, s generated by |∅, s is an irreducible module.

  [START_REF] Ariki | On the decomposition numbers of the Hecke algebra of G(m, 1, n)[END_REF]

		(1; 2)	(1; 2)
	y y s s s s s s s s s s		y y t t t t t t t t t	% % L L L L L L L L L L
	(1; 11)	(11; 1)	(1; 11)	(∅; 3)
			% % J J J J J J J J J	y y r r r r r r r r r r
	(111; ∅)	(111; ∅)	(∅; 21)
	(∅; 111)	(1; 11)	(∅; 111)
		(∅; 21)
		(∅; 111)

Acknowledgements. We would like to thank Bernard Leclerc and Meinolf Geck for many useful discussions on the topic of this paper. Both authors are supported by the "Agence national de la recherche" No. JCO7 192339.

Then we have a bijection γ : Bip s 1 e (n) → Bip s 2 e (n).

This bijection has been explicitly described, in a non-recursive way in [START_REF] Jacon | Crystal graphs of irreducible highest weight U v ( sl e )-modules of level two and Uglov bipartitions[END_REF] (see also the generalization in [START_REF] Jacon | Lecouvey Crystal isomorphisms for irreducible highest weight U v ( sl e )modules of higher level[END_REF]).

Theorem 5.3. Assume that Conjectures A and B hold. Then for all µ ∈ Bip s 1 e (n), we have

Proof. Let r 1 (respectively r 2 ) denote the entire part of ξ 1 (respectively ξ 2 ). Let µ ∈ Bip s 1 e (n). By Ariki's Theorem and the theory of cellular algebras, c(µ) is the maximal element with respect to r 2 such that |c(µ), s 1 appears with non-zero coefficient in G(µ, s 1 ).

We have [CS ξ λ :

is the maximal element with respect to r 2 such that

This concludes the proof.

Remark 5.4 -Note that using the non-recursive parametrization of Bip s e (n) known in particular cases (see §4.4), it is possible to obtain nice characterizations of Bip s e (n) in all cases by applying the above bijection.

Example 5.5 -Assume that n = 4 and that θ(a) 2 = θ(b) 2 = -1. Then θ(a) is a primitive 2-root of unity and we have θ(b) 2 = -θ(a) 0 . If 0 < ξ 1 < 2 < ξ 2 < 4 (i.e. r 1 ∈ {0, 1} and r 2 ∈ {2, 3}), then we have

To compute Bip (0,0) 2 (4) (respectively Bip

(2,0) 2 (4)), we compute the crystal graph associated to the action of the quantum group U v ( sl 2 ) on the Fock space F (0,0) (respectively F (2,0) ). The submodule generated by the empty bipartition gives an irreducible highest weight module M[0, 0] (respectively M[2, 0]) with highest weight 2Λ 0 . We only give the part of both crystals containing the bipartitions up to rank 4. They are computed as explained in [24, 2.1, 2.2]

By Theorem 4.5, we then have


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Promoting cooperation between lactic acid bacteria within a community: towards new food applications

Fanny Canon

To cite this version:

Fanny••

  Canon. Promoting cooperation between lactic acid bacteria within a community: towards new food applications. STLOpendays, Institut National de Recherche Agronomique (INRA). UMR UMR INRA / AgroCampus Rennes : Science et Technologie du Lait et de l'?uf (1253)., Mar 2019, Rennes, France. Association of LAB based on their metabolic complementary induces cooperative behaviour within a community • Exchanges between LAB are partially linked with their nitrogen metabolism • Increase in aromatic/flavour compound production • Reduction of allergens I. What parameters favour positive interactions between LAB within a community ? II. What are the cooperation mechanisms ? What are the molecules exchanged during these interactions ? III. Have these interactions beneficial effects on the nutritional and organoleptic properties of a fermented milk-lupin mix ? Communities are studied in several fields : Bioprocesses : Development of synthetic communities to increase yields Natural ecosystems (intestinal, soil, oceans, lakes microbiota) : Studies of 1) microbial composition and its evolution, 2) interactions with host/environment • Knowledge transfer to multiple raw materials → Development of new plant-based fermented products → Development of innovative functionalized fermented products


Inflammation, DNA damage and Staphylococcus aureus infection

Berkova Nadia Nadejda, INRA, Agrocampus Ouest, Rennes, France, Oral presentation at conference BactInflam, Belo Horizonte, Mina Gerais, Brazil Staphylococcus aureus causes serious medical problems in human and animals. This oral presentation will demonstrate a newly identified mechanism used by S. aureus for fostering an infection.

We demonstrated that S. aureus can compromise host genomic integrity as indicated by bacteria-induced histone H2AX phosphorylation, a marker of DNA double strand breaks (DSBs) in the absence of apoptosis, in human cervix cancer HeLa and osteoblast-like MG-63 cells. This DNA damage is mediated by alpha phenol-soluble modulins (PSMα1-4), while a specific class of lipoproteins (Lpls), encoded on a pathogenicity island in S. aureus, dampens the H2AX phosphorylation thus counteracting the DNA damage. This DNA damage is mediated by reactive oxygen species (ROS), which promotes oxidation of guanine forming 7,8dihydro-8-oxoguanine (8-oxoG). DNA damage is followed by the induction of DNA repair that involves a canonical DNA damage response (DDR) comprising the triggering of the ATM kinase-signaling pathway.

An examination of S. aureus strains, isolated from the same patient during acute initial and recurrent bone and joint infections (BJI), shows that recurrent strains produce lower amounts of Lpls and induce stronger DNA-damage.

DNA damage triggers cellular responses such as transient cell cycle arrest to allow time for DNA repair. Consequently, we investigated the ability of S. aureus strains from human and animal origins to affect the host cell cycle in human and animal cells. Flow cytometry analysis revealed a S. aureus-induced delay in the G2/M phase transition. The results of Western blot experiments showed that the G2/M transition delay was associated with the accumulation of inactive cyclin-dependent kinase Cdk1, a key inducer of mitosis entry, and with the accumulation of unphosphorylated histone H3, which was correlated with a reduction of the mitotic cell number.

Using size exclusion chromatography of S. aureus (MW2, USA400) supernatant, followed by mass spectroscopy analysis of corresponding peaks, we identified phenol-soluble modulin alpha (PSMα) peptides as the likely candidates for this effect. Indeed, synthetic PSMα1 and PSMα3 caused a G2/M phase transition delay. The implication of PSMα in cell cycle alteration was confirmed by comparison of S. aureus USA300 (LAC wt) with the isogenic mutant LAC∆psmα, which lacks the psmα operon encoding PSMα1 to 4; only the supernatant of LAC wt and not that of the mutant alters cell cycle progression. By testing the supernatant of S. aureus isolates from patients with enterocolitis, we found that the degree of G2/M phase transition delay correlated with PSMα1 production. Analysis of S. aureus proliferation in asynchronous, G1-and G2-phase-enriched HeLa cells showed that the G2 phase was preferential for bacterial infective efficiency, suggesting that the G2 phase delay may be used by S. aureus for propagation within the host. We show that PSMs secreted by S. aureus induce DNA damage and alter the host cell cycle that promotes S. aureus-associated infection. Our findings redefine our understanding of mechanisms of S. aureus-host interaction and suggest that the balance between the levels of PSMα and Lpls expression impacts the persistence of the infection.


Assessing impacts of cropping systems on biodiversity: comparison of indicators for four systems along an intensity gradient

Assessing impacts of cropping systems on biodiversity: comparison of indicators for four systems along an intensity gradient

Christian Bockstaller



  Any implementation of indicator  Relevance of their outputs?  Different approaches (Bockstaller et al., 2008). Aucune note nulle attribuée  La récompense a moins de poids que l'attraction visuelle et l'accessibilité à la fleur Système experiment at Arvalis-Institut du Végétal station (South of Paris)  4 cropping systems : rotations  similar (in ORG 2 alfafa ley)

Material and methods .010 Montevideo FSD 2019 Results: Some causal indicators relevant Results

		Indicator	MACH	CONV	INT	ORG
		Average N rate
		% cropped area with
		> 150 kg N/ha
		Number of crops in
		the rotation
		Diversity of crop
		families
		% of semi natural
		habitats in 1 km
		buffer
		Diversity of semi
	Tillage	No natural habitats in 1 Optimized		Reduced	Plough
	N	High km buffer	Optimized		Reduced	No
	Pesticides	High Inconsistent	Optimized intermediate	Reduced consistent	No not concerned
				Intensification gradient
						Montevideo
						.011

FSD 2019 DEXiPM V1 & V2 performing for fauna Results

	Criterion	Indicator	Auximore	MASC 2.0	DEXiPM	DEXiPM
			DEXi		v1	v2
	Global			Inconsistent	Consistent with the
	biodiversity			with gradient	gradient
	Flying	Flying		CONV = INT
	insects	insects
		Flying	(MACH,		Consistent with the
		natural	CONV) and		gradient
		enemies	(INT, ORG).
	Soil fauna	Soil macro		Only MACH
		fauna		is ≠
		Soil natural			Consistent with the
		enemies			gradient
		Crawling
		natural
		enemies
	Inconsistent	intermediate	consistent	not concerned Montevideo
						.012

FSD 2019 But for flora (weeds) DEXiPM V2 needs enhancement Results

	Discussion & conclusion
	Criterion	Indicator	Auximore	MASC 2.0	DEXiPM	DEXiPM
			DEXi		v1	v2
	Flora				Consisten	Inconsist
		Weeds			t with the	ent with
					gradient	gradient
		Semi		Only ORG	Only	Only
		natural flora	is >	ORG is >	ORG is >
	Micro	Soil micro	Only ORG is
	organisms	organisms	>
	Inconsistent	intermediate	consistent	not concerned
					Montevideo FSD 2019
						.013

 Some causal indicators (N management) consistent with gradient data  Need to repeat such study  with datasets of several locations  taxon diversity/abundance measurements  DEXiPM V1 and V2 : able to provide consistent results with intensity gradient  Exception for weeds: this indicator should be enhanced Ex ante assessment: simulation Very qualitative Easy to implement Low integration of processes .014

Thank you for your attention

Aknowledgement: authors are thankful to the GIS "Grande Culture à Hautes Performances Economiques et Environnementales" (GC HP2E) for its support to the master thesis of Thomas Denoirjean


Yu Xiaoxi

Nativif

Keywords:

Conception of an infant formula by filtration: nutritional, physiological and sensory impacts

  30 billion $ in sales (Nielsen, 2015) -France: 1.2 billion € in sales, 1st market in Europe (Deloitte, 2013) Infant formulas  41% of the market  60% of 1-month-old infants, 80% of 3-month-old infants (Wagner, 2015) -Sector with robust competition, which requires innovation -Consumers searching for naturalness • Infant formula : Numerous heat treatments for sanitary reasons  Denaturation/aggregation of proteins (Joyce et al., 2017 ; Buggy et al., 2017)  Formation of Maillard reaction products • Membrane Filtration: Alternative Non-Denaturing Process -Bacteriological stabilization of milk -Fractionation of milk components Project objectives :  Develop new technological routes to produce the most native and microbiologically safe infant formula  Evaluate the nutritional, physiological and sensory interest of such a formula ingredients using membrane filtration -Production of three powdered infant formulas, using vacuum evaporation and spray-drying and with different levels of heat treatment : -without heat treatment (T-) -with minimum heat treatment to ensure the sanitary quality (T+) -with multiple heat treatments (T +++) Characteristics of the infant formulas -Denaturation/aggregation levels of proteins -Maillard reaction products -Microbiological characteristics (e.g. absence of Cronobacter Sakazakii)


HeteroProtein Complex coacervates: mechanisms and potential applications

Saïd Bouhallab a* , Thomas Croguennec a a. UMR1253, INRA, Agrocampus-Ouest, STLO, 65 rue de Saint-Brieuc, F-35042 Rennes, France * said.bouhallab@inra.fr

The application of fundamental physicochemical concepts for rational design of functional assemblies from food proteins constitute a response to the growing trend toward the development of new and innovative food products and also an opportunity to generate new protein-based supramolecular structures with new applications. Because of their omnipresence in food systems and their biodegradability, proteins are the focus of many attempts for their use as building blocks for such supramolecular structures. Controlled self-co-assembly of proteins can generate a variety of supramolecular structures that vary in shape, size and density (fibrils, spherulites, nanotubes, etc). For instance, well-defined microspheres called heteroprotein complex coacervates (HPCC) can be formed by control mixing of oppositely charged proteins. [START_REF] Croguennec | Heteroprotein complex coacervation: A generic process[END_REF] The objective of our research is to understand the mechanisms behind HPCC process from initial spontaneous molecular interaction to micro-scale characterization. In this communication, we will summarize our results on several binary protein systems and will show that co-assembly of proteins into complex coacervates (Figure ) is a generic process that is, de facto, independent of the protein amino-acid sequence. We will report on the requirements that drive such spontaneous coassembly: protein conformational state and flexibility, molar stoichiometry, total protein concentration, charge anisotropy, etc. The Research challenges and the promising uses of HPCC in food and non-food sectors (encapsulation of bioactives, design of edible films) will be discussed.


Mathieu Buoro

Julien Papaix

Cyril Piou

Etienne Prévost

Investigating the demogenetic responses of exploited Atlantic salmon populations to climate change

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

  , Univ.Pau & Pays Adour Aquapôle -Quartier Ibarron n°173, RD 918 -route de St Jean de Luz 64310 St Pée sur Nivelle, France www.ecobiop.com A demo-genetic individual-based model for Atlantic salmon populations (IBASAM, Piou & Prévost 2012) → mimics a small population typical of french coastal streams → connect demo-genetic dynamics with biotic (e.g. fisheries) & abiotic factors (e.g. Water temperature, Flow) . salmon population size and composition (age structure) could be severely affected under CC and no selective exploitation strategies mitigate the effects of CC But current selective exploitation of Multiple Sea Winter fish could worsens the effects of CCPerspectives: Explore the potential of management strategies that would be unselective and that would increase intra-population biodiversity, favoring the "portfolio effect" and resilience to CC.

  W a t e r t e m p e r a t u r e : + 3 °C A m p l i t u d e o f F l o w : + 2 5 % G r o w t h c o n d it io n s a t s e a : -2 5 % Mathieu Buoro, Julien Papaïx, Cyril Piou and Etienne Prévost