Coline Emprin

Dana Hunter

Muriel Livernet

Christine Vespa

Inna A Zakharevich

Struc- Ture Prop

Partitions On

A PROP STRUCTURE ON PARTITIONS

Keywords: prop, partitions, Yoneda product, Karoubi envelope. 2024 Mathematics Subject Classification. 18M85, 18G15, 05A17. 1

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

Introduction

The notions of props and operads arose in the work of Mac Lane [START_REF] Lane | Categorical algebra[END_REF], in the aim of encoding algebraic structures. While operads encode products with a single output, props allow for working with algebraic structures involving operations with multiple outputs. Such structures include Hopf algebras, Frobenius algebras, or Lie bialgebras, which arouse interest after the discovery of quantum groups for instance, see [START_REF] Drinfeld | Hamiltonian structures on lie groups, lie bialgebras and the geometric meaning of classical yang-baxter equations[END_REF][START_REF] Drinfeld | Quantum groups[END_REF]. A prop is a symmetric monoidal category with objects the natural numbers and whose symmetric monoidal structure ⊗ is given by the sum of integers on objects. The notions of operads and props are intrinsically related: one can consider the prop freely generated by a given operad and to any prop, one can associate its underlying operad. Working with set operads, it is well known that the prop freely generated by the terminal operad Com is the category of surjections Surj, see e.g. [START_REF] Hartl | Polynomial Functors from Algebras over a Set-Operad and Nonlinear Mackey Functors[END_REF].

The present paper focuses on the graded linear prop E freely generated by the operadic suspension of the commutative operad. In this context, the suspension gives rise to signs. Let K denote the underlying ground field of characteristic zero. As a graded K-vector space, the space E • (m, n) is concentrated in degree m -n, where we have an isomorphism

E m-n (m, n) ≃ K[Surj(m, n)] .
In [START_REF] Kawazumi | On the wheeled PROP of stable cohomology of Aut(Fn) with bivariant coefficients[END_REF], the authors exhibit a system of generators having the advantage that the monoidal structure simply corresponds to the disjoint union of sets. However, the composition involves signs. In Section 1, we provide another system of generators which has the advantage that the left and right actions of the symmetric groups are by sign, and the composition agrees with the set composition of surjections. Considering idempotents of the symmetric groups, one can construct various categories out of the prop E, defined as subcategories of its Karoubi envelope. The general construction is recalled in Section 2. In particular, for any prop C, one can consider a certain subcategory ΛC of the Karoubi envelope of C. It is equivalent to a category C Λ where the vector space of morphisms is obtained from C by taking the quotient by the symmetric group actions. Its composition is described in Theorem 2.7. In the case of the prop E, the space E • Λ (m, n) is concentrated in degree m -n and is spanned by partitions of m into n parts. The category structure, coming from that of E, is described in Section 3. Nonetheless, the monoidal product of E, does not induce any prop structure on E Λ . In Theorem 3.10, we introduce a different monoidal product which turns the category E Λ into a prop and thus leads to a particular prop structure on partitions.

Our interest in these props comes from their link with extensions between functors from free groups which is explored in Section 4. In fact, functor homology turned out to be a useful tool for computing stable homology with twisted coefficients of various families of groups, and in particular of automorphism groups of free groups Aut(Z * n ), for n ∈ N. Djament proved in [START_REF] Djament | Décomposition de hodge pour l'homologie stable des groupes d'automorphismes des groupes libres[END_REF] that stable cohomology of Aut(Z * n ) with coefficients given by a reduced polynomial covariant functor is governed by Ext-groups in the category F(gr) of functors from finitely generated free groups to K-vector spaces. From this perspective, [START_REF] Vespa | Extensions between functors from free groups[END_REF] gives an explicit computation of the graded K-vector spaces

Ext • F (gr) ((T n • a) ⊗ K, (T m • a) ⊗ K) ,
where a is the abelianization functor and T n is the n-th tensor power functor. Together with the Yoneda product and external product of extension, this family forms a prop which is shown to be isomorphic to the prop E. This prop structure was leveraged to give explicit computations of stable homology of Aut(Z * n ) with coefficients given by particular contravariant functors, see [START_REF] Vespa | Extensions between functors from free groups[END_REF]Theorem 4]. The aformentionned results were extended in [START_REF] Kawazumi | On the wheeled PROP of stable cohomology of Aut(Fn) with bivariant coefficients[END_REF][START_REF] Djament | Décomposition de hodge pour l'homologie stable des groupes d'automorphismes des groupes libres[END_REF] in order to deal with bivariant coefficients. One can study extensions between other functors, by replacing T n by exterior power functors Λ n for example. By [START_REF] Vespa | Extensions between functors from free groups[END_REF], the extensions between these functors are concentrated in one degree and are spanned by partitions. A prop structure on these groups can thus be derived from that of E Λ . We conclude this paper by exploiting the construction of Section 2 in order to give some other explicit computations of extension groups between simple functors.

Notation.

(1) Let K be a characteristic zero field and let grVect K be the symmetric monoidal category of graded vector spaces over K. We use the cohomological grading convention V • , and the degree of an element x in V is denoted d(x). For every set S, we denote by K[S] the K-vector space spanned by S.

(2) We denote by |S| the cardinal of a finite set S.

(3) The set of surjections from {1, . . . , m} to {1, . . . , n} is denoted Surj(m, n). Composition of surjections is denoted •. Given f ∈ Surj(m, n) and g ∈ Surj(m ′ , n ′ ) we denote by f × g the element in Surj(m + m ′ , n + n ′ ) defined by

(f × g)(i) = f (i), for 1 ⩽ i ⩽ m, g(i -m) + n, for m + 1 ⩽ i ⩽ m + m ′ .
(4) The symmetric group on n letters is denoted S n .

(5) τ i,i+1 denotes the transposition of S n that permutes i and i + 1. ϵ(σ) denotes the sign of the permutation σ. (6) A (p, q)-unshuffle is the inverse of a (p, q)-shuffle permutation, that is, a permutation σ such that σ -1 (1) < . . . < σ -1 (p) and σ -1 (p + 1) < . . . < σ -1 (p + q). Similarly we define a (p 1 , . . . , p n )-unshuffle. For a surjection f ∈ Surj(m, n), we denote by Sh f the set of (p 1 , . . . , p n )-unshuffles with

p i = |f -1 (i)|. (7) We denote by Surj or (m, n) the set of order-preserving surjections. For f ∈ Surj(m, n),
there is a unique decomposition f = s • α with s ∈ Surj or (m, n) and α ∈ Sh f . (8) A partition λ of m into n parts is a sequence of positive integers λ 1 ⩾ . . . ⩾ λ n such that i λ i = m. We denote by Part(m, n) the set of partitions of m into n parts. To a surjection f ∈ Surj(m, n), we can associate a partition of m into n parts given by ordering the cardinals of its fibers in the decreasing order. We denote by

proj : Surj(m, n) → Part(m, n)
this map. By linear extension we have a surjective morphism

proj : K[Surj(m, n)] → K[Part(m, n)] .

A graded linear prop spanned by surjections

The aim of this section is to give an explicit description of the prop freely generated by the suspension of the commutative operad. We emphasize a choice of generators for which the symmetric group actions are given by the sign and which behave well with respect to the composition of maps.

1.1. Recollections on props. This section recalls the definition of a prop and the freely generated prop associated to an operad. We refer the reader to [START_REF] Markl | Operads and props, handbook of algebra[END_REF] for more details on props and to [START_REF] Loday | Algebraic operads[END_REF] for more details on operads. Definition 1.1 (Graded linear prop). A graded linear prop, or simply prop, is a symmetric monoidal category (C, ⊗, 1), enriched over the category of graded vector spaces grVect K , with objects the natural numbers and whose symmetric monoidal structure ⊗ is given by the sum of integers on objects. In other words, a graded linear prop is the data of a collection {C(m, n)} m,n∈N of graded K-vector spaces together with compatible morphisms: a vertical composition given by the categorical composition,

⋄ : C(n, l) ⊗ C(m, n) -→ C(m, l)
an horizontal composition coming from the monoidal product,

⊗ : C(m, n) ⊗ C(m ′ , n ′ ) -→ C(m + m ′ , n + n ′ ) isomorphisms s m,m ′ ∈ C(m + m ′ , m + m ′ ) such that we have (-1) d(f )d(g) (g ⊗ f ) ⋄ s m,m ′ = s n,n ′ ⋄ (f ⊗ g) ,
for all f ∈ C(m, n) and all g ∈ C(m ′ , n ′ ). Throughout the paper, we will use the terminology prop to refer to a graded linear prop.

Remark 1.2. The isomorphisms s m,m ′ induce a morphism of K-algebras φ : K[S n ] → P(n, n). This gives a K[S m ]-right action and K[S n ]-left action on P(m, n) defined for σ ∈ S n , τ ∈ S m , and f ∈ P(m, n) by

(1) σ • f • τ = φ(σ) ⋄ f ⋄ φ(τ ).
In particular, one can find an equivalent definition in the literature of prop using only the action of the symmetric groups and its compatibility with vertical and horizontal compositions, see e.g. [START_REF] Hackney | On the category of props[END_REF].

Remark 1.3. For every prop (C, ⊗, 1), the collection {C(n, 1)} n∈N forms an operad in grVect K where the composition maps are given for all k, n 1 , . . . , n k ⩾ 0 by

C(k, 1) ⊗ C(n 1 , 1) ⊗ . . . ⊗ C(n k , 1) → C(k, 1) ⊗ C(n 1 + . . . + n k , k) → C(n 1 + . . . + n k , 1) ,
where the first map is induced by the monoidal product ⊗ and the second map by the composition in the category C. We will refer to it as the underlying operad of the prop (C, ⊗, 1). This leads to a forgetful functor from the category of props to the one of operads. Its restriction to reduced props and operads admits a left ajoint Ω which associates to any reduced operad P, the prop ΩP freely generated by P.

Definition 1.4 (Freely generated prop). Let P be a reduced operad in grVect K (i.e. P(0) = 0). The graded linear prop ΩP freely generated by P is given by the following data.

The collection {ΩP(m, n)} m,n∈N : It is defined by

ΩP(m, n) = p 1 +...+pn=m P(p 1 ) ⊗ . . . ⊗ P(p n ) ⊗ Sp 1 ×...×Sp n K[S n ] = f ∈Surj(m,n) P(|f -1 (1)|) ⊗ . . . ⊗ P(|f -1 (n)|) .
We denote by ΩP f the summand corresponding to a surjection f ∈ Surj(m, n) in the previous decomposition.

The monoidal product is obtained by concatenation, i.e. we consider

x ⊗ y ∈ ΩP f ×g ,
for all x ∈ ΩP f and all y ∈ ΩP g . The S n -left action is defined for all σ ∈ S n , all f ∈ Surj(m, n) and all x = x 1 ⊗. . .⊗x n ∈ ΩP f , where

x i ∈ P(|f -1 (i)|) by the formula (2) σ • x = ±x σ -1 (1) ⊗ . . . ⊗ x σ -1 (n) ∈ ΩP σ•f ,
where ± is the Koszul sign rule induced by the degrees of each element

x i . The S m -right action is defined for τ ∈ S m , f ∈ Surj(m, n) and x = x 1 ⊗ . . . ⊗ x n ∈ ΩP f as follows. Decompose f as s • α with s ∈ Surj or (m, n) and α ∈ Sh f , and decompose α • τ as α • τ = (σ 1 × . . . × σ n ) • u with σ i ∈ S |f -1 (i)| and u ∈ Sh f . Then, we have (3) (x 1 ⊗ . . . ⊗ x n ) • τ = (x 1 • σ 1 ) ⊗ . . . ⊗ (x n • σ n ) ∈ ΩP f •τ .
The composition morphisms

ΩP(m, n) ⊗ ΩP(l, m) ⋄ -→ ΩP(l, n)
are described as follows. Let s ∈ Surj or (m, n) be an order-preserving surjection and let g ∈ Surj(l, m). Let us consider x = x 1 ⊗ . . . ⊗ x n ∈ ΩP s and y = y 1 ⊗ . . . ⊗ y n ∈ ΩP g , with

y i = y 1 i ⊗ . . . ⊗ y p i i where p i = |s -1 (i)|. Then, we have (4) x ⋄ y = (-1) ω γ(x 1 ; y 1 ) ⊗ . . . ⊗ γ(x n ; y n ) ∈ ΩP s•g ,
where γ denotes the composition in the operad P and ω is given by the Koszul sign rule, namely

ω := n i=1 d(y i-1 ) (d(x i ) + • • • + d(x n )) . For f ∈ Surj(m, n), let us consider the decomposition f = s • α with s ∈ Surj or (m, n) and α ∈ Sh f . Let us denote by x s ∈ ΩP s the unique element satisfying x s • α = x ∈ ΩP f . Then, we have x ⋄ y = (x s • α) ⋄ y = x s ⋄ (α • y)
, which can be computed through the composition with an order-preserving surjection and the left action of the symmetric group already defined.

The isomorphisms s n,m ∈ ΩP(n + m, n + m) are defined as s n,m = 1 ⊗n+m ∈ ΩP σ where σ is the unshuffle defined as

σ(i) = m + i for 1 ≤ i ≤ n and σ(i) = i -n for n + 1 ≤ i ≤ n + m.
The map φ : K[S n ] → ΩP(n, n) sends σ to 1 ⊗n ∈ ΩP σ where 1 ∈ P(1) denotes the unit of the operad P.

1.2. The prop freely generated by the operadic suspension of Com. Let us consider sCom the operadic suspension of the commutative operad, see [LV12, Section 7.2.2] for more details. The graded operad sCom is generated by a single operation µ 2 , of degree 1 and arity 2, subject to the relation

µ 2 • 1 µ 2 = -µ 2 • 2 µ 2
, and the action of S 2 on µ 2 is by sign. By using the conventions of [KV23, Section 9], we have that sCom(n) = Kµ n ⊗ sgn n is concentrated in degree n -1, and

µ 3 = µ 2 • 1 µ 2 = -µ 2 • 2 µ 2 , so that µ n • i µ k = (-1) (i-1)(k-1) µ n+k-1 ,
where µ 1 = 1 is the unit of the operad sCom. The operadic composition in sCom is then given by µ n (µ p 1 , . . . , µ pn ) = (-1) κ(p 1 ,...,pn) µ p 1 +...+pn where (5)

κ(p 1 , . . . , p n ) = n j=1 (p j -1)(p 1 + . . . + p j-1 ) = n k=1 p k (p k+1 -1 + . . . + p n -1) .
Notation 1.5. Let us denote by E the graded linear prop ΩsCom freely generated by the operad sCom.

By Definition 1.4, for f ∈ Surj(m, n), we have

E • f = sCom(|f -1 (1)|) ⊗ . . . ⊗ sCom(|f -1 (n)|) ≃ K if • = m -n and m ⩾ n, 0 otherwise
as a graded vector space. In order to give an explicit description of this prop using Definition 1.4 we need to choose, for any f ∈ Surj(m, n), a generator in the vector space E m-n f

. In [START_REF] Kawazumi | On the wheeled PROP of stable cohomology of Aut(Fn) with bivariant coefficients[END_REF], the authors consider the generators

µ f := µ p 1 ⊗ . . . ⊗ µ pn of E m-n f for f ∈ Surj(m, n) and p i = |f -1 (i)|.
For these generators the composition ⋄ and the action of the symmetric groups give rise to complicated signs whereas the monoidal structure is simply given by concatenation without signs. More precisely, the left action by τ i,i+1 is given by ( 6)

τ i,i+1 • µ f = (-1) (p i -1)(p i+1 -1) µ τ •f .
For the right action, following Definition 1.4, we decompose

f as f = s • α, with s ∈ Surj or (m, n) and α ∈ Sh f . Given τ ∈ S m , there exist σ = (σ 1 × . . . × σ n ), σ i ∈ S p i and u ∈ Sh f such that α • τ = σ • u.
The right action on ΩsCom is given by sign, so that (7)

µ f • τ = ϵ(σ)µ f •τ .
We give below an alternative choice of generators in E m-n f having the advantage that the description of the composition ⋄ is easy and that the action of the symmetric groups on these generators on both sides is given by sign.

Notation 1.6. Let f ∈ Surj(m, n) be a surjection such that p i = |f -1 (i)| and let κ(f ) be the integer κ(p 1 , . . . , p n ) defined in (5). Let us denote

ν f := ϵ(α)(-1) κ(f ) µ p 1 ⊗ • • • ⊗ µ pn = ϵ(α)(-1) κ(f ) µ f ,
where s • α is the decomposition of f with s ∈ Surj or (m, n) and α ∈ Sh f . Let us note that ν σ = ϵ(σ)µ σ for all σ ∈ S m , or equivalently that φ(σ) = ϵ(σ)ν σ . The degree is given by d

(ν f ) = m -n and E m-n f ≃ Kν f .
The following theorem describes the prop structure on E • in terms of the generators ν f .

Theorem 1.7. Let E be the graded linear prop ΩsCom.

(a) The S n -left action and the S m -right action are given by

σ • ν f • τ = ϵ(σ)ϵ(τ )ν σ•f •τ , for all f ∈ Surj(m, n), all σ ∈ S n and all τ ∈ S m . (b) Let s ∈ Surj or (m, n) and t ∈ Surj or (m ′ , n ′ )
be order-preserving surjections. The monoidal product is given by the concatenation up to a sign, i.e.

ν s ⊗ ν t = (-1) d(νt)m ν s×t .
If f and g are not order-preserving surjections, the formula for ν f ⊗ ν g can be derived by decomposing f and g into order-preserving surjections and unshuffles. (c) The composition is defined for f ∈ Surj(m, n) and h ∈ Surj(l, m) by

ν f ⋄ ν h = ν f •h . Proof. Let f ∈ Surj(m, n) be a surjection which decomposes uniquely as s • α with s ∈ Surj or (m, n) and α ∈ Sh f . Let us denote p i = |f -1 (i)| = |s -1 (i)|. We have κ(f ) = κ(s).
Let us prove Point (a). The S m -right action is given for τ ∈ S m as follows. Let us write α • τ = σ • u with σ = (σ 1 × . . . × σ n ), σ i ∈ S p i and u ∈ Sh f . By definition of µ f and Relation (7), we have

ν f • τ = (-1) κ(f ) ϵ(α)ϵ(σ)µ f •τ . Since s • σ = s, we have f • τ = s • u and µ f •τ = ϵ(u)(-1) κ(f ) ν f •τ . This leads to ν f • τ = ϵ(τ )ν f •τ .
The S n -left action is given for σ ∈ S n as follows. Let s ∈ Surj or (m, n) be an order preserving surjection such that

p i = |s -1 (i)| and let us write σ • s = t • β with t ∈ Surj or (m, n) and β ∈ Sh σ•s . First, we claim that ϵ(σ)σ • µ s = ϵ(β)(-1) n i=1 p i (σ(i)-i) µ σ•s
Since this formula is clearly multiplicative, it is enough to prove it for σ = τ i,i+1 . By Formula (6), we have

τ i,i+1 • µ s = (-1) (p i -1)(p i+1 -1) µ τ i,i+1
•s , and ϵ(β) = (-1) p i p i+1 , since β exchanges the fibers i and i + 1, which gives the desired formula. It leads to

σ • ν s = (-1) κ(s) σ • µ s = (-1) κ(s) ϵ(σ)ϵ(β)(-1) n i=1 p i (σ(i)-i) µ σ•s . One can prove that (-1) κ(σ•s) = (-1) κ(s) (-1) n i=1 p i (σ(i)-i) , which gives σ • ν s = (-1) κ(σ•s) ϵ(σ)ϵ(β)µ σ•s = ϵ(σ)ν σ•s .
We get the result for any surjection f ∈ Surj(m, n) using its decomposition f = s • α and the formula of the right action. Let us prove Point (b). Let s ∈ Surj or (m, n) and t ∈ Surj or (m ′ , n ′ ) be order-preserving surjections. It is sufficient to prove that

κ(s × t) = κ(s) + κ(t) + (m ′ -n ′ )m .
We have κ(s × t) = κ(s) + κ(t) + n ′ i=1 (q i -1)m, for q i = |t -1 (i)|, which gives the desired result. Let us prove Point (c). For any h ∈ Surj(l, m), the surjection α • h decomposes as t • β with t an order-preserving surjection and β an unshuffle. We have

ν f ⋄ ν h = ϵ(α)(ν s • α) ⋄ ν h = ν s ⋄ ν α•h = ϵ(β)(ν s ⋄ ν t ) • β.
Hence, it is sufficient to prove the result for order-preserving surjections s ∈ Surj or (m, n) and t ∈ Surj or (l, m). Let us denote

m 1 = |s -1 (1)|, m 2 = m -m 1 , l 1 = m 1 i=1 |t -1 (i)| and l 2 = l -l 1 .
We proceed by induction on n. For n = 1, it is a consequence of the operadic composition in sCom and the definition of κ. We decompose s = s 1 × s 2 with s 1 ∈ Surj or (m 1 , 1), s 2 ∈ Surj or (m 2 , n -1) and t = t 1 × t 2 with t 1 ∈ Surj or (l 1 , m 1 ) and t 2 ∈ Surj or (l 2 , m 2 ). We have

ν s ⋄ ν t = ν s 1 ×s 2 ⋄ ν t 1 ×t 2 = (-1) d(νs 2 )m 1 +d(νt 2 )l 1 (ν s 1 ⊗ ν s 2 ) ⋄ (ν t 1 ⊗ ν t 2 ) = (-1) (d(νs 2 )+d(νt 2 ))l 1 ν s 1 •t 1 ⊗ ν s 2 •t 2 = ν (s 1 •t 1 )×(s 2 •t 2 ) = ν s•t . □ 2.
On subcategories of the Karoubi envelope of a prop

In this section, we recall the construction of the Karoubi envelope of a category (also called idempotent completion or pseudo-abelian hull), see [Kar68, Section 1.2]. This construction aims to add objects and morphisms to a category so that every idempotent, i.e. every endomorphism e A : A → A satisfying e A ⋄ e A = e A , is split.

Definition 2.1 (Karoubi envelope). The Karoubi envelope of a given category C is the category Kar(C) defined by the following data.

The objects are the pairs (A, e A ) where A is an object in C and e A : A → A is an idempotent. A morphism f : (A, e A ) → (B, e B ) between two objects in Kar(C) is a morphism

f : A → B in C such that f = f ⋄ e A = e B ⋄ f = e B ⋄ f ⋄ e A .
The composition is the same as the one in C and the identity morphism on (A, e A ) is e A .

There is a fully faithful functor η C : C → Kar(C) defined as follows. For an object A in C, we have that η C (A) = (A, 1 A ) and for a morphism f in C, we have that η

C (f ) = f . Remark 2.2. A morphism f : A → B in a category C is a morphism in Kar(C) from (A, e A ) to (B, e B ) if and only if f = e B ⋄ f ⋄ e A .
Proposition 2.3. Suppose that (C, ⊗, I) is a symmetric monoidal structure on C. Then (Kar(C), ⊗, (I, 1 I )) is a symmetric monoidal category with

(A, e A ) ⊗ (B, e B ) = (A ⊗ B, e A ⊗ e B ) f ⊗ g = f ⊗ g .
Proof. We observe that e A ⊗ e B is an idempotent. For f : (A, e A ) → (A ′ , e A ′ ) and g : (B, e B ) → (B ′ , e B ′ ) two morphisms in Kar(C), we have that f ⊗ g is a morphism from (A, e A ) ⊗ (B, e B ) to (A ′ , e A ′ ) ⊗ (B ′ , e B ′ ) since

(e A ′ ⊗ e B ′ ) ⋄ (f ⊗ g) ⋄ (e A ⊗ e B ) = (e A ′ ⋄ f ⋄ e A ) ⊗ (e B ′ ⋄ g ⋄ e B ) = f ⊗ g .
If ρ is the left unitor in C, that is, a natural isomorphism defined by ρ A : A ⊗ I → A, then by naturality one has e A ⋄ ρ A = ρ A ⋄ (e A ⊗ 1 I ). Hence ρ(A,e A ) := e A ⋄ ρ A ⋄ (e A ⊗ 1 I ) is a well defined morphism in Kar(C) from (A, e A ) ⊗(I, 1 I ) to (A, e A ). It is natural and it is an isomorphism. The associator, the right unitor and the swap map are defined similarly, and the desired diagrams between the natural transformations (the commutative pentagon, hexagon, and triangle) commute. □

Let C and D be two categories. A semifunctor F : C → D maps objects (resp. arrows) in C to objects (resp. arrows) in D, preserving domain, codomain and composition, see [START_REF] Mitchell | The dominion of Isbell[END_REF]Section 4]. The difference between semifunctors and functors is that semifunctors need not preserve identities. Definition 2.4. There is a forgetful semifunctor ϵ C : Kar(C) → C defined as follows. For (A, e A ) an object of Kar(C), we have that ϵ C (A, e A ) = A and any morphism is sent to itself. In particular, the identity e A of (A, e A ) is sent to e A which is not equal to 1 A in general.

Let (C, ⊗, 1) be a prop. By Remark 1.2, the morphism φ : K[S n ] → C(n, n), maps any idempotent e in K[S n ] to an idempotent in C(n, n). Via an abuse of notation, we will denote the idempotents φ(e) by e, so that by relation (1), the action • of the symmetric group on C corresponds to the composition ⋄ in the prop C. There is a well-known construction of a set of primitive orthogonal idempotents of K[S n ] indexed by the partitions of n (see [FH91, Section 4.1]). For λ a partition of n, we denote by e λ the associated idempotent. For example, we have:

e (1 n ) := 1 n! σ∈Sn ϵ(σ)σ and e (n) := 1 n! σ∈Sn σ
In this section, we are interested in particular subcategories of Kar(C) built from the idempotents e (1 n ) .

Definition 2.5. Let (C, ⊗, 1) be a prop. We denote by ΛC the full subcategory of Kar(C) whose objects are given by (n, e (1 n ) ) for all n ∈ N.

By Definition 2.1 and Remark 2.2, an element of ΛC((m, e (1 m ) ), (n, e (1 n ) )) is necessarily of the form

e (1 n ) ⋄ f ⋄ e (1 m ) = 1 n!m! σ∈Sn,τ ∈Sm ϵ(σ)ϵ(τ )σ • f • τ , with f ∈ C(m, n).
In what follows, we denote objects in ΛC simply n instead of (n, e (1 n ) ).

In order to give in Section 3, an explicit description of the composition of partitions we introduce the category C Λ which is equivalent to ΛC. More generally, if the graded vector spaces C(m, n) are endowed with a basis such that composition of elements of the basis is up to a sign an element of the basis, then the combinatorics of the category C Λ is more easily understood than that of ΛC. Definition 2.6. Let (C, ⊗, 1) be a prop. We denote by C Λ (m, n) the quotient of C(m, n) by the relation

f ∼ ϵ(σ)ϵ(τ )τ • f • σ , for τ ∈ S n , σ ∈ S m . The normalization map f → e (1 n ) ⋄ f ⋄ e (1 m ) induces an isomorphism C Λ (m, n) ∼ = → ΛC(m, n) ,
for all m, n ∈ N. Via these isomorphisms, one can define a category C Λ isomorphic to the category ΛC. Next Theorem gives the explicit composition in C Λ induced by that in ΛC.

Theorem 2.7. Let C be a prop and φ : K[S n ] → C(n, n) be the map induced by the action of the symmetric group. For

[f ] ∈ C Λ (m, n) and [g] ∈ C Λ (l, m), the composition [f ] * [g] in the category C Λ has the following form [f ] * [g] = 1 m! σ∈Sm ϵ(σ)[f ⋄ φ(σ) ⋄ g] .
Proof. For all n ∈ N, let us denote e n := e (1 n ) . Let f ∈ C(m, n) and g ∈ C(l, m). The normalization map sends any

[f ] in C Λ (m, n) to e n ⋄ f ⋄ e m in ΛC(m, n).
The composition of e n ⋄ f ⋄ e m with e m ⋄ g ⋄ e l in the category ΛC, or equivalently in the category C(m, n) is :

e n ⋄ f ⋄ e m ⋄ e m ⋄ g ⋄ e l = e n ⋄ f ⋄ e m ⋄ g ⋄ e l
which is the image of [f ⋄ e m ⋄ g] ∈ C Λ (l, n) by the normalization map. This leads to

[f ] * [g] = [f ⋄ e m ⋄ g].
In addition, we have [

1 m ] * [g] = [e m ⋄ g] = [g] and [f ] * [1 m ] = [f ⋄ e m ] = [f ]. □
Remark 2.8. Note that the family of quotient maps C(m, n) → C Λ (m, n) does not provide a functor from C to C Λ . The equivalence of categories → C Λ is, on morphisms, the composition of the restriction of the semi-functor

ϵ C : Kar(C)((m, e (1 m ) ), (n, e (1 n ) )) → C(m, n)
to the category ΛC((m, e (1 m ) ), (n, e (1 n ) )) with the quotient map.

A graded linear prop spanned by partitions

In Section 2, we associate to every prop C a category C Λ , which is equivalent to a subcategory ΛC of the Karoubi envelope of C. This section focuses on the particular case of the prop E = ΩsCom (see Section 1) and its associated category E Λ (see Section 2). The Karoubi envelope Kar(E) inherits a prop structure from that of E, see Proposition 2.3. However, it does not induce a prop structure at the level of a given subcategory of Kar(E) in general. For ΛE, a necessary condition would be that

e (1 m ) ⊗ e (1 n ) = e (1 m+n ) ,
in E which is not the case. Nonetheless, in Section 3.2, we extend the category E Λ into a graded linear prop by introducing another monoidal product ⊙.

3.1.

The composition in the category E Λ . In this section, we describe the category structure E Λ given by Theorem 2.7 in the case of C = E. We start by proving that this category is spanned by partitions.

Proposition 3.1. For all n, m ∈ N, there is an isomorphism Proof. The result follows directly from Lemma 3.2. □ Lemma 3.2. For all f, g ∈ Surj(m, n), the following propositions are equivalent :

E • Λ (m, n) ∼ = K [Part (m, n)] if • = m -n,
(i) [ν f ] = [ν g ] ∈ E m-n Λ (m, n) ; (ii) there exist τ ∈ S m and σ ∈ S n such that g = σ • f • τ ; (iii) proj(f ) = proj(g),
where proj : Surj(m, n) → Part(m, n) is the map defined in Notation (8).

Proof. We have [ν f ] = [ν g ] if and only if there exist τ ∈ S m and σ ∈ S n such that

ν g = ϵ(σ)ϵ(τ )σ • ν f • τ = ν σ•f •τ .
This proves the equivalence (i) ⇐⇒ (ii). The implication (ii) =⇒ (iii) is immediate. Conversely, we assume that proj(f ) = proj(g) and let us denote by λ 1 ⩾ . . . ⩾ λ n this partition.

There is a unique s ∈ Surj or (m, n) such that |s -1 (i)| = λ i , for all i and there exists a permutation β ∈ S n such that |f -1 (i)| = λ β(i) . In particular, the surjection β • f writes uniquely as

β • f = s • u with u ∈ Sh β•f , so that f decomposes as β -1 • s • u. Similarly, there exist γ ∈ S n , v ∈ S m such that g = γ -1 • s • v. By construction, σ = γ -1 • β and τ = u -1 • v are such that g = σ • f • τ , which proves (iii) =⇒ (ii).
□ Notation 3.3. For a given partition λ ∈ Part(m, n), we denote

ρ λ := [ν f ] ∈ E m-n Λ (m, n)
the class of ν f , where f is any surjection such that proj(f ) = λ.

Theorem 3.4. The composition of two basis elements in the category E Λ is a weighted average of basis elements, i.e. for λ ∈ Part(m, n) and µ ∈ Part(l, m), we have

ρ λ * ρ µ = α∈Part(l,n) c λ,µ α ρ α
where m!c λ,µ α ∈ N and α c λ,µ α = 1. The identity morphisms are given by ρ (1 m ) ∈ E Λ (m, m).

Proof. Let f ∈ Surj(m, n) and g ∈ Surj(l, m) such that proj(f ) = λ and proj(g) = µ. By Theorem 2.7 we have

[ν f ] * [ν g ] = 1 m! σ∈Sm ϵ(σ)[ν f ⋄ φ(σ) ⋄ ν g ]
and φ(σ) = ϵ(σ)ν σ , see Notation 1.6. This leads to

□ (8) [ν f ] * [ν g ] = 1 m! σ∈Sm [ν f •σ•g ] .
Corollary 3.5. For any partition λ = λ 1 ⩾ . . . ⩾ λ n ∈ Part(m, n), we have

ρ λ * ρ (2,1 m-1 ) = 1 m n i=1 λ i [ν s(λ 1 ,...,λ i-1 ,λ i +1,λ i+1 ,...,λn) ] ,
where s(p 1 , . . . , p n ) denotes the unique order-preserving surjection s ∈ Surj or (m, n) such that

|s -1 (i)| = p i and m = p 1 + • • • + p n .
Proof. Let f = s(λ 1 , . . . , λ n ) ∈ Surj or (m, n) and g = s(2, 1, . . . , 1) ∈ Surj or (m + 1, m). By Formula (8), we have

ρ λ * ρ (2,1 m-1 ) = 1 m! σ∈Sm [ν f •σ•g ] . For σ ∈ S m , there is a unique decomposition σ • g = t • u with t ∈ Surj or (m + 1, m) and u ∈ Sh σ•g . For all 1 ⩽ j ⩽ m, we have that |t -1 (j)| = 2 ⇐⇒ σ(1) = j .
There are (m-1)! permutations σ such that σ(1) = j. In that case, we have t = s(1 j-1 , 2, 1 m-j ). By Lemma 3.2, we have

[ν f •t•u ] = [ν f •t ] , so that ρ λ * ρ (2,1 m-1 ) = 1 m m j=1 [ν f •s(1 j-1 ,2,1 m-j ) ].
Finally, we decompose the sum as

m j=1 [ν f •s(1 j-1 ,2,1 m-j ) ] = n i=1 λ 1 +•••+λ i-1 +λ i j=λ 1 +•••+λ i-1 +1 [ν f •s(1 j-1 ,2,1 m-j ) ] .
We conclude by noting that for λ 1 + . . .

+ λ i-1 + 1 ⩽ j ⩽ λ 1 + . . . + λ i-1 + λ i , we have [ν f •s(1 j-1 ,2,1 m-j ) ] = [ν s(λ 1 ,...,λ i-1 ,λ i +1,λ i+1 ,...,λn) ] . □
Example 3.6. The formula of Corollary 3.5 gives

ρ (3,3,1) * ρ (2,1 6 ) = 1 7 (3[ν s(4,3,1) ] + 3[ν s(3,4,1) ] + [ν s(3,3,2) ]) = 1 7 (6ρ (4,3,1) + ρ (3,3,2) ) .

3.2.

A particular prop structure. In this section, we introduce a particular monoidal product ⊙ which turns the category E Λ into a prop.

Notation 3.7. Let us define a family of elements

P m,n ∈ E m-n Λ (m, n) for m ⩾ n as follows: (1) P m,m = ρ (1 m ) ∈ E 0 Λ (m, m), (2) P m,m-1 = ρ (2,1 m-2 ) ∈ E 1 Λ (m, m -1), (3) P m,n = P n+1,n * . . . * P m,m-1 ∈ E m-n Λ (m, n). Lemma 3.8. The element P m,n ∈ E m-n Λ (m, n) satisfies P m,n = 1 |Surj or (m, n)| s∈Surj or (m,n) [ν s ] = 1 |Surj or (m, n)| s∈Surj or (m,n) ρ proj(s) .
Proof. Let us prove the result by induction on m ⩾ n. For m = n, we have by definition P n,n = ρ (1 n ) and there is only one s ∈ Surj or (n, n) such that proj(s) = (1 n ). Similarly, Part(n + 1, n) has only one element, so that the formula boils down to

P n+1,n = ρ (2,1 n-1 ) .
Assume the formula is true for P m,n . We have P m+1,n = P m,n * P m+1,m , that is,

P m+1,n = 1 |Surj or (m, n)| s∈Surj or (m,n) [ν s ] * ρ (2,1 m-1 ) .
By Theorem 3.4, we know that P m+1,n is a weighted average of the elements ρ λ , where λ runs in Part(m+1, n). For s = s(p 1 , . . . , p n ) the order-preserving surjection such that |s -1 (i)| = p i , we have

[ν s ] * ρ (2,1 m-1 ) = 1 m n i=1 p i [ν s(p 1 ,...,p i-1 ,p i +1,p i+1 ,...,pn) ] ,
by Proposition 3.5. Let s(q 1 , . . . , q n ) ∈ Surj or (m + 1, n). In the composition P m,n * ρ (2,1 m-1 ) , the element [ν s(q 1 ,...,qn )] appears with weight

1 m|Surj or (m, n)| n i=1 (q i -1) = m + 1 -n m|Surj or (m, n)|
which is independant of the chosen order-preserving surjection in Surj or (m + 1, n). This gives the desired result. We also obtain that for m + 1 > n,

|Surj or (m + 1, n)| = |Surj or (m, n)| m m + 1 -n . □
Remark 3.9. One can also express P m,n in the basis ρ λ with λ ∈ Part(m, n), by counting the number of order-preserving surjections s such that proj(s) = λ. As an example, we have P 6,3 = 1 10 (3ρ (4,1,1) + 6ρ (3,2,1) + ρ (2,2,2) ) .

Theorem 3.10. For α ∈ Part(m, n) and β ∈ Part(m ′ , n ′ ), the monoidal product defined as

ρ α ⊙ ρ β = (-1) d(ρα)n ′ P m+m ′ ,n+n ′ ,
endows the category E Λ with a structure of symmetric monoidal category. On the one hand, we have

(ρ λ ⊙ ρ λ 2 ) * (ρ β ⊙ ρ β 2 ) = (-1) d(ρ λ )n 2 +d(ρ β )m 2 P l+l 2 ,n+n 2 ,
and on the other hand, we have

(ρ λ * ρ β ) ⊙ (ρ λ 2 * ρ β 2 ) = α∈Part(l,n) α 2 ∈Part(l 2 ,n 2 ) c λ,β α c λ 2 ,β 2 α 2 ρ α ⊙ ρ α 2 = (-1) (l-n)n 2 ( α∈Part(l,n) α 2 ∈Part(l 2 ,n 2 ) c λ,β α c λ 2 ,β 2 α 2 )P l+l 2 ,n+n 2 .
By Theorem 3.4, we obtain that

(ρ λ ⊙ ρ λ 2 ) * (ρ β ⊙ ρ β 2 ) = (-1) d(ρ β )d(ρ λ 2 ) (ρ λ * ρ β ) ⊙ (ρ λ 2 * ρ β 2 ) .

Finally, let us set

s m,m ′ := (-1) mm ′ ρ (1 m+m ′ ) ∈ E Λ (m + m ′ , m + m ′ ) . We have (-1) d(ρα)d(ρ β ) (ρ β ⊙ ρ α ) * s m,m ′ = s n,n ′ * (ρ α ⊙ ρ β ) = (-1) mn ′ P m+m ′ ,n+n ′ . □
Remark 3.11. The underlying operad associated to the prop E Λ is isomorphic to the operad sCom via the isomorphism given by

ρ (m) ∈ E Λ (m, 1) -→ (-1) m(m-1) 2 µ m .
For dimension reasons, the prop E Λ is nevertheless not isomorphic to the prop freely generated by the operad sCom, and thus to E. Let us note that the prop E Λ is also not finitely generated since any family of generators would necessary contain a generator in each E 1 Λ (m, m -1).

Additional computations suggest the following conjecture.

Conjecture 3.12. The prop structure given by Theorem 3.10 is the unique prop structure on the category E • Λ which is the sum of integers on objects and such that for all m ⩾ 0,

ρ (1) ⊙ ρ (1 m ) = ρ (1 m+1 ) and ρ (1) ⊙ ρ (2,1 m ) = ρ (2,1 m+1 ) .

Relation with functor homology on free groups

This work is motivated by the relation between the prop freely generated by the operadic suspension of Com and the extension groups between the tensor powers of the abelianisation functor obtained in [START_REF] Vespa | Extensions between functors from free groups[END_REF]. More precisely, let gr be the category of finitely generated free groups, ab the category of finitely generated free abelian groups and Vect K the category of K-vector spaces. We denote by F (gr) the category of functors from gr to Vect K . Let a : gr -→ ab be the abelianization functor and let T n : Vect K -→ Vect K be the n th tensor product functor. Let us consider the functor

a K := a ⊗ Z K : gr -→ Vect K .
We consider the category E T enriched in graded K-vector spaces whose objects are the integers N and whose morphisms are given by E

• T (m, n) := Ext • F (gr) (T n • a K , T m • a K )
with the composition given by the Yoneda product. It was shown in [START_REF] Vespa | Extensions between functors from free groups[END_REF] that, together with the external product of extensions, this category forms a prop which is freely generated by its underlying operad. This operad is identified in [KV23, Section 9] as being the suspension of Com. The authors introduce the generator

[π ⊗n ] ∈ E n-1 T (n, 1)
and they define an isomorphism of props ι :

E T → E given by [π ⊗p 1 ] ⊗ . . . ⊗ [π ⊗pn ] ∈ E T (m, n) -→ µ p 1 ⊗ . . . ⊗ µ pn .
By Section 1.2 we have an isomorphism E ∼ = E given for every surjection f ∈ Surj(m, n) by,

E m-n f -→ E m-n f µ p 1 ⊗ . . . ⊗ µ pn -→ ν f .
The composition of ι with the previous isomorphism leads to a natural equivalence:

Kar(E T ) ∼ -→ Kar(E) .
Let Λ n : Vect K -→ Vect K be the n th exterior power functor. We consider the category ΛE T whose objects are the integers and whose morphisms are given by

ΛE T (m, n) = Ext • F (gr) (Λ n • a K , Λ m • a K )
, with the composition given by the Yoneda product. The prop isomorphism

E T ≃ E induces isomorphisms ΛE T ∼ = ΛE ∼ = E Λ . Thus, we recover the following isomorphism established in [Ves18, Theorem 4.2] Ext • F (gr) (Λ n • a K , Λ m • a K ) ∼ = K[Part(m, n)] if • = m -n 0 otherwise ,
and ΛE T inherits a prop structure from that of E Λ . The Yoneda product of ΛE T coincides with the composition and Theorem 3.4 thus gives an explicit formula.

Remark 4.1. We could hope that the monoidal product of ΛE T corresponds to an external product defined on extensions of exterior powers functors, as it was the case for E. However, this is not the case: exploiting the Hopf structure of the functor Λ, one can actually define an external product on ΛE T but it is not compatible with the Yoneda product.

Recall that the regular representation of the symmetric group S d decomposes as Email address: livernet@math.univ-paris-diderot.fr Christine Vespa, Aix Marseille Univ, CNRS, I2M, Marseille, France Email address: christine.vespa@univ-amu.fr

Inna Zakharevich, Math. Dept., Cornell University, Ithaca, NY, USA Email address: zakh@math.cornell.edu

  and m ⩾ n, 0 otherwise where Part (m, n) denotes the partitions of m into n parts.

Proof.

  It is immediate to prove that the product ⊙ is associative. Let us prove the compatibility between the composition * and the monoidal product ⊙. Let us consider λ ∈ Part(m, n), λ 2 ∈ Part(m 2 , n 2 ), β ∈ Part(l, m), and β 2 ∈ Part(l 2 , m 2 ) .

K

  [S d ] ∼ = λ⊢d S ⊕dim(S λ ) λ, where S λ is the simple module indexed by the partition λ of d. Moreover, a simple K[S d ]module is isomorphic to S λ for a unique partition λ of d. We deduce from the previous decomposition thatT d • a K ∼ = λ⊢d S ⊕ dim(S λ ) λ where S λ = (T d • a K ) ⊗ S d S λis a simple functor. For λ a partition of m and µ a partition of n we have:Kar(E T )((m, e λ ), (n, e µ )) = e µ • Ext • F (gr) (T n • a K , T m • a K ) • e λ = Ext • F (gr) (S µ , S λ ) , and thus (9) Ext • F (gr) (S µ , S λ ) ≃ e µ • E(m, n) • e λ .Using Formula (9) and the generators ν f introduced in Notation 1.6, one can compute these extension groups. We conclude by giving some examples below.Example 4.2. For all n and all m, we haveΛE T (m, n) = Ext • F (gr) S (1 n ) , S (1 m ) . For example, we have Ext 1 F (gr) S (1 2 ) , S (1 3 ) ≃ e (1 2 ) • E(3, 2) • e (1 3 ) , with e (1 2 ) = 1 2 (1 -τ 1,2) and e (1 3 )By Theorem 1.7, any surjection f ∈ Surj(3, 2) satisfiese (1 2 ) • ν f • e (1 3 ) = 1 6 g∈Surj(3,2) ν g , confirming that Ext 1 F (gr) S (1 2 ) , S (1 3 ) is 1-dimensional.Example 4.3. We recover the following result of [Ves18, Theorem 4.2] :Ext • F (gr) (T n • a K , S m • a K ) = K if n = m and • = 0, 0 else.We already know that the graded vector space is concentrated in degree m -n. For m = n,Ext 0 F (gr) (T m • a K , S m • a K ) is one dimensional, generated by the image of the idempotent e (m) of S m via the isomorphism K[S m ] ≃ E(m, m). If m ̸ = n, we can assume m > n. Working in the Karoubi envelope of E T , we have Kar(E T )((m, e (m) ), (n, id)) = Ext • F (gr) (T n • a K , S m • a K ) ≃ E(m, n) • e (m) . Let f ∈ Surj(m, n) be a surjection which decomposes as s • α with s ∈ Surj or (m, n). Since ν f = ϵ(α)ν s • α, it is enough to prove that ν s • e (m) = 0. Let us denote p i = |s -1 (i)|. AnyDana Hunter, Kalamazoo College, 1200 Academy Street, Kalamazoo, Michigan, 49006-3295, USA Email address: Dana.Hunter@kzoo.edu Muriel Livernet, Univ. Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS, SU, DMA, ENS-PSL, Paris, France

Acknowledgements. We would like to thank the Hausdorff Research Institute for Mathematics for hosting the Women in Topology IV workshop and for financial support. We would also like to thank the Foundation Compositio Mathematica, the Foundation Nagoya Mathematical Journal and the K-theory Foundation for financial support for this event.

σ ∈ S m writes uniquely as σ = (σ 1 ×. . .×σ n )•u with σ i ∈ S i and u ∈ Sh s , so that s

If m > n, then there exists i such that p i > 1. In particular, in S i , there are as many odd permutations as even permutations, so that ν s • e (m) = 0.

Example 4.4. With the help of Formula (9), let us compute

and Ext 1 F (gr) S (2) , S (2,1) . Following [FH91, Section4], we have

where ( 132) is the cyclic permutation. Using Theorem 1.7, we obtain that Ext 1


Clément François

Daniele Schön

Clément Francois
email: clement.francois@incm.cnrs-mrs.fr

Learning of musical and linguistic structures: comparing eventrelated potentials and behavior

To learn a new language, it is necessary for the learner to succeed in segmenting the continuous stream of sounds into significant units. Previous behavioral studies have shown that it is possible to segment a language or musical stream based only on probabilities of occurrence between adjacent syllables/tones. Here we used a sung language and tested participants' learning of both linguistic and musical structures while recording electroencephalography. Although behavioral results showed learning of the linguistic structure only, event-related potential results for both dimensions showed a negative component sensitive to the degree of familiarity of items. We discuss this component as an index of lexical search, also pointing to the greater sensitivity of the event-related potentials compared to the behavioral responses.

INTRODUCTION

The very first steps in language acquisition require the ability to segment speech into separate words. Speech being a continuous stream of linguistic sounds, the challenge is to be able to segment it into separate (and possibly) meaningful units or words. Several studies have been conducted to better understand the cognitive mechanisms underlying language learning, and more precisely, its segmentation. These studies have shown the importance of regularities and statistical structure in word segmentation and confirm the observation that conditional probabilities between adjacent syllables, namely the probability of a syllable Y given the occurrence of a syllable X, are typically higher within words than between words [START_REF] Saffran | Word segmentation: the role of distributional cues[END_REF].

The importance of conditional probabilities in language acquisition in infants and adults has been shown by Saffran and colleagues [START_REF] Saffran | Word segmentation: the role of distributional cues[END_REF][START_REF] Saffran | Statistical learning by 8-month old infants[END_REF]. In these studies, the authors built an artificial language in such a way that the only cues to word segmentation were conditional probabilities between syllable pairs, which were high within words and lower across word boundaries. The design typically used in such studies comprises a passive listening session followed by a two-alternative forced choice procedure (or head turn procedure when studying infants) in which participants have to choose the most familiar of the two items. Thereafter, this type of design also proved effective in showing implicit learning in the visual modality [START_REF] Fiser | Encoding multielement scenes: statistical learning of visual feature hierarchies[END_REF] and in the auditory modality for nonlinguistic stimuli, such as tone sequences [START_REF] Saffran | Statistical learning of tones sequences by human infants and adults[END_REF] or timbre sequences [START_REF] Tillmann | Implicit learning of musical timbre sequences: statistical regularities confronted with acoustical (dis) similarities[END_REF]. Recently we showed that performance in the test with spoken items is facilitated when the learning material is a sung language compared with a spoken language [START_REF] Schön | Song as an aid for language acquisition[END_REF]. Overall, these results point to the fact that a similar learning mechanism might be at work independent of the modality (auditory or visual) or stimulus type (linguistic or musical) of the input.

In this study, we used a sung artificial learning language and we collected behavioral and electrophysiological responses during two separate tests on linguistic and musical dimensions. The aim was two-fold. First, we wanted to compare the sensitivity of the behavioral and electrophysiological measures and the extent to which they correlate. Second, we wanted to compare eventrelated potentials (ERPs) with linguistic and musical items with ERPs described in the literature on the familiarity effect.

Concerning the first aim, we hypothesized that ERP measures might be more sensitive than behavioral measures to the subtle changes that may undergo implicit learning [START_REF] Mc Laughlin | Neural correlates of second-language word learning: minimal instruction produces rapid change[END_REF]. Indeed, while behavior reflects the combined effects of several processing stages, ERPs, thanks to their continuous time resolution, can (sometimes) show differences in the brain activity even when this is not backed by an overt behavioral response [START_REF] Sergent | Timing of the brain events underlying access to consciousness during the attentional blink[END_REF][START_REF] Tremblay | The time course of auditory perceptual learning: neurophysiological changes during speech-sound training[END_REF].

Concerning the second aim, few earlier studies have attempted to investigate statistical learning using electroencephalography (EEG) [START_REF] Abla | On-line assessment of statistical learning by event-related potentials[END_REF][START_REF] Cunillera | The effects of stress and statistical cues on continuous speech segmentation: an event related brain potential study[END_REF][START_REF] Cunillera | Time course and functional neuroanatomy of speech segmentation in adults[END_REF][START_REF] De | Different neurophysiological mechanisms underlying word and rule extraction from speech[END_REF][START_REF] Sanders | Segmenting nonsense: an eventrelated-potential index of perceived onsets in continuous speech[END_REF], reporting learning-related modifications of two well-known ERP components, the N100, related to perceptual processing, and the N400 whose amplitude is sensitive to lexical/semantic memory processes [START_REF] Kutas | Reading senseless sentences: Brain potentials reflect semantic incongruity[END_REF]. With one exception [START_REF] De | Different neurophysiological mechanisms underlying word and rule extraction from speech[END_REF] the above cited studies recorded (or reported) ERPs during the learning phase only. Overall, while earlier studies investigated the emergence of 'lexical units', we investigated the contrast between more or less stable/familiar (lexicalized) units directly after the learning phase. Earlier research showed that unfamiliar words give rise to a larger N400 than familiar words possibly reflecting the greater difficulty in accessing semantic memory for unfamiliar words [START_REF] Young | Word frequency and multiple repetition as determinants of the modulation of event-related-potentials in a semantic classification task[END_REF]. In this study, as in general in artificial language learning studies, participants are familiarized to a language during passive listening. This seems to allow lexicalization of the learning output [START_REF] Fernandes | The metamorphosis of the statistical segmentation output: lexicalization during artificial language learning[END_REF]; also enabling participants to choose, during the test session, the more familiar out of the two presented items. Therefore, we hypothesized that unfamiliar items would yield greater N400-like responses than familiar items for both linguistic and musical dimensions.

METHODS

Participants

Fourteen native French participants (mean age 25 years, 13 right handed, 11 males, nonmusicians) were paid for their participation in the experiment (approved by the national ethical committee).

Material

The sung language was that used by [START_REF] Schön | Song as an aid for language acquisition[END_REF]. Eleven syllables were combined to give rise to six trisyllabic sung words. Each syllable was associated with a distinct tone (C3, D3, F3, G3, A3, B3, C4, Db4, D4, E4, and F4); therefore, each word was always sung on the same melodic contour (gimysy C3 D3 F3, mimosi E4 Db4 G3, pymiso B3 E4 F4, pogysi D4 C3 G3, sipygy G3 B3 C4, sysipi F3 G3 A3). Transitional probabilities ranged from 0.31 to 1.0 within words and from 0.1 to 0.2 across word boundaries. This was obtained by combining 216 repetitions of each of the six words in pseudorandom order (same word was never repeated twice in a row). The language stream was synthesized using the Mbrola speech synthesizer (http://tcts.fpms.ac.be/synthesis/mbrola.html).

Procedure

During the learning phase, participants were asked to listen carefully to the sound stream. During the linguistic test, participants had to press one of two buttons to indicate which of the two strings (first or second spoken item) resembled the most to what they just heard in the stream (Fig. 1). In each of the 36 trials, while one item was a 'word' from the nonsense language, the other was not (hereafter, part-word). The part-words consisted of either the last syllable of a word plus the first syllable pair of another word, or the last syllable pair of a word plus the first syllable of another word (gysimi, mosigi, pygymi, pysipi, sogimy, and sypogy). All items were pronounced using a flat contour, that is, without melodic information. During the musical test, the stimuli were 'midi' sequences of three notes generated with a piano sound. The six sequences, called tone-melodies, were built with the same pitches defining the melodic contour of the words heard during the learning phase (C5 D5 F5, E6 Eb6 G5, B5 E6 F6, D6 B5 G5, G5 B5 C5, F5 G5 A5). The sequences, called part-tone melodies, were built using the pitches of the part-words. The instructions were the same as in the linguistic test. Stimuli were presented using loudspeakers. Both the linguistic and musical tests lasted 5 min each and their order was counter balanced across participants (Fig. 1).

Data acquisition and analysis

The EEG was recorded from 32 scalp electrodes located at standard positions (international 10/20 system sites: Fz, Cz, Pz, Oz, Fp1, Fp2, Af3, Af4, F3, F4, C3, C4, P3, P4, P7, P8, Po3, Po4, O1, O2, F7, F8, T7, T8, Fc5, Fc1, Fc2, Fc6, Cp5, Cp1, Cp2, and Cp6). The data were then re-referenced offline to the algebraic average of the mastoids. Four participants were discarded because of major EEG artifacts. Artifacts were detected by eyeballing and using a 75 mV maximum amplitude criterion (in approximately 10% of trials). The EEG was amplified by Biosemi amplifiers (Amsterdam University) with a band pass of 0-102. [START_REF] Saffran | Statistical learning of tones sequences by human infants and adults[END_REF] Hz, digitized at 512 Hz and analyzed using Brain Vision Analyzer (Brain Products, Munich, Germany). ERPs were analyzed by computing the mean amplitude, starting at the onset of the stimulus presentation and ending after 1500ms, using latency windows of 50ms. Analyses were performed for correct trials only and for all trials (correct and incorrect trials). A repeated measures analysis of variance was used for statistical assessment. To test the distribution of the effects, six regions of interest (ROIs) were selected: frontal, central, and parietal (left and right). P values were adjusted using the Greenhouse-Geisser correction, when appropriate, and Sidak tests were used in post-hoc comparisons. Owing to the increased likelihood of type I errors associated with the large number of statistical tests, only effects that reached significance (P < .05) in at least two consecutive time windows were considered significant.

RESULTS

Behavioural data

The participants' level of performance in the linguistic test was significantly above chance (62.2% of correct responses, P = 0.007, Wilcoxon test). By contrast, performance in the musical test (54.5%) fell between performances with linguistic material (P = 0.06, Wilcoxon test) and chance level (P = 0.2, Wilcoxon test). We also ran nonparametric one-way analyses of variances using the items as a factor (six levels). Items analysis showed no effect of item type in the musical task (P = 0.1, Friedman test). By contrast, there was a significant item effect in the linguistic task: two items with low-transitional probabilities (P = 0.5 and 0.41) were not learned as well as the others (P = 0.002, Friedman test).

Electrophysiological data

Language

A negative ERP component starting 200 ms after stimulus onset and lasting until 1400 ms is visible at fronto-central sites (Fig. 2). The amplitude of this component is modulated by the degree of familiarity: part-words elicited a larger negativity between 450 and 800 ms than words. The main effect of familiarity was significant between 600 and 750 ms [F(1,9) = 7.5, P =.03; 2.4 µV of effect size: unfamiliar-familiar]. Interestingly, an even stronger difference was found in the same latency band when using all trials rather than correct trials only [F(1,9) = 9.4, P =.01; 2.6 µV of effect size]. The correlation between the effect size (main effect of Familiarity in the 600-750 ms latency range) and performance was significant (r = 0.77, P = .018). Thus, the higher the participants' performance, the greater is the difference between part-words and words.

Music

A negative ERP component starting 250 ms after stimulus onset and lasting until 1300 ms is visible at fronto-central sites (Fig. 2). The amplitude of this component was modulated by the degree of familiarity, with part-tone words eliciting a larger negativity between 600 and 1000 ms than tonewords. The interaction of familiarity and ROIs was significant between 750 and 900 ms [F(2,18) = 5.1, P = .03]. Indeed part-tone words elicited a larger negativity over fronto-central regions than tone-words. Interestingly, an even stronger difference was found in the same latency band when using all trials rather than correct trials only [F(2,18) = 6.3, P = .01]. This effect size did not significantly correlate with performance. Finally, a main effect of familiarity was also significant between 1250 and 1400 ms, with parttone words eliciting a larger positivity than tone-words [F(1,9) = 5.7, P = .04, 1.81 µV of effect size].

DISCUSSION

The aim of this study was two-fold. First, we wanted to compare the sensitivity of behavioral and electrophysiological responses. Second, we wanted to compare ERPs to linguistic and musical items to ERPs described in the literature on familiarity effect. Participants were asked to listen to a continuous stream of sung language, built using a pseudo-random concatenation of six sung words and were then tested on both the linguistic and musical dimensions using a two-alternative forced choice task. Concerning the sensitivity of the methods, while the performance in the musical test was not significantly different from chance at a group level, ERP data revealed a significant familiarity effect. Interestingly, this familiarity effect was even stronger when using all trials rather than correct trials only. This seems to point to the fact that participants, on average, learned the musical structure better than they actually performed in the behavioral test. It seems that, after learning has taken place, tone and parttone sequences do have distinct, although probably, poor quality representations. These poor quality representations might be enough to yield different ERP responses, but might be hidden by the noise surrounding the decisional and response-related processing required by the task (see Ref. [START_REF] Cleeremans | Conscious and unconscious cognition: a graded, dynamic, perspective[END_REF] for a graded perspective on conscious and unconscious cognition). Results in the linguistic test also point in the same direction. Although there was quite a robust positive correlation between participants' performance and the amplitude of the ERP effect (partwords-words) in the 600-750 ms time windows, ERPs seem to be more sensitive than behavior. In fact, all participants, but one had the expected ERP effect. Thus, even participants whose performance was barely above chance (e.g. two participants with 55% of correct responses) show the typical familiarity effect ( -2.2 and -3.2 µV). Concerning the comparison of ERPs with linguistic and musical items, electrophysiological data from both the sessions showed a larger negativity at fronto-central sites in the 400-900 ms latency band for part-words (or parttone sequences) than for words (or tone-sequences). To interpret this ERP difference it is important to clarify the respective 'linguistic' status of the two conditions. In fact, because each word/tone-sequence has been repeated 216 times, they have 'popped out' of the continuous sung stream, thanks to statistical learning or associative mechanisms. By contrast part-words/part-tone sequences are heard five times less frequently during learning. Therefore, the linguistic status of the items used in the test is that of more or less familiar item (words and part-words, respectively). Earlier ERP studies have already shown that, compared with high-frequency (familiar) words, lowfrequency (unfamiliar) words elicit a larger negativity peaking around 400 ms [START_REF] Young | Word frequency and multiple repetition as determinants of the modulation of event-related-potentials in a semantic classification task[END_REF][START_REF] Van Petten | Interactions between sentence context and word frequency in event-related brain potentials[END_REF]. This negative component is often referred to as N400 and its amplitude seems to be sensitive to the ease of retrieving long-term memory traces [START_REF] Federmeier | It's about time[END_REF]. In our study, only sufficiently reinforced (repeated) items may 'survive' concurrent interference and end up being stored in a long-lasting manner [START_REF] Perruchet | The self-organizing consciousness[END_REF]. The negative ERP components found in the linguistic and musical tests might be an index for the search of the memory traces that have been stored during implicit learning [START_REF] Fernandes | The metamorphosis of the statistical segmentation output: lexicalization during artificial language learning[END_REF]. It is interesting to note that the topographic distribution of this ERP component is similar in the linguistic and musical tests (only one significantly different ROI out of six). Although this ERP component peaks later in the musical test, such a difference might be because of the fact that participants needed more time in the musical test than in the linguistic test to 'search' whether an item was familiar or not.

CONCLUSION

In the framework of implicit language learning, the electrophysiological approach seems to be more revealing than the behavioral approach. Finally, our findings showed that superposition of two identical linguistic and musical structures in song facilitates segmentation of both linguistic and musical information.

Figure 1 :

 1 Figure1: The experimental design. A learning phase using a sung language was followed by two tests on linguistic and musical structures. Stimuli were presented in the auditory modality.

Figure 2 :

 2 Figure 2: Grand average across all participants for the linguistic and musical sessions. Solid lines illustrate ERPs to words/tone sequences; dashed lines illustrate ERPs to part-words/part-tone sequences. Each trace represents the mean of the three electrodes included in the ROI (LC, left central; LF, left frontal; LP, left parietal; RC, right central; RF, right frontal; RP, right parietal). Maps illustrate the topographical distribution of the peak of the familiarity effect (unfamiliar-familiar).

Acknowledgements

The authors thank Benedicte Poulin-Charronnat, Deirdre Bolger, and Mireille Besson for constructive comments on a previous version. This study has been supported by the French National Research Agency (ANR 05-BLAN-0214 'Music and Memory' to D. Schön). The authors have no competing financial interests to declare.