F Digne

A GARSIDE PRESENTATION FOR ARTIN-TITS GROUPS OF TYPE Cn

Keywords: 2010 Mathematics Subject Classification. 20F36, 20F05 Braids, Garside, Artin-Tits groups, affine Coxeter groups

We prove that an Artin-Tits group of type C is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the "generated group" method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type C, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof is to show that this group is a group of fixed points in an Artin-Tits group of type à under an involution. Another important point is the study of the Hurwitz action of the usual braid group on the decomposition of a Coxeter element into a product of reflections.

Introduction

The aim of this paper is to define a Garside structure on the Artin-Tits group of type Cn . A Garside structure on a group means that this group is the group of fractions of a Garside monoid. A Garside monoid is a monoid for which the two posets given by right or left divisibility have nice properties. In particular these two posets are lattices: there exist least common multiples and greatest common divisors, and moreover these two lattices have a common sublattice which generates the monoid and has a greatest element for both orders, the Garside element (see Definition 2.3 below). It is known that all Artin-Tits groups of spherical type have two nice Garside structures given respectively by the classical monoid, obtained by generating the Artin-Tits group by lifts of the simple reflections, and the dual monoid (see [START_REF]The dual braid monoid[END_REF]), obtained by generating the Artin-Tits group by elements lifting all reflections which divide (see below beginning of Section 3) a given Coxeter element. In the case of non-spherical Artin-Tits groups the classical Artin-Tits monoid exists but is only locally Garside (i.e., two elements have not always a common multiple, in particular there is no Garside element). An open question in general is the existence of a dual Garside structure for general Artin-Tits groups. Such a structure is known for type à and it has been conjectured by John Crisp and Jon McCammond that no Artin-Tits group of affine type other than type à and maybe C can have such a structure. A Garside structure provides normal forms for the elements of the group and is a tool for solving the word problem. It also allows to compute centralisers of periodic elements (roots of powers of the Garside element).

To get a dual Garside structure for an Artin-Tits group of type Cn , we shall view this group as the group of fixed points under an involution in an Artin-Tits group of type Ã2n-1 . In [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF] this last group has been shown to be the group of fractions of several monoids only one of which, up to automorphism, is Garside, but unfortunately this one is not stable by the involution. On the other hand only one of these non-Garside monoids, up to automorphism, is stable by the involution. We show that by taking fixed points in this last monoid one gets a Garside structure for Cn .

The paper is organised as follows. In Section 2 we introduce Garside monoids and give methods for getting a Garside monoid from a partially defined product on a subset called a germ. In Section 3 we recall and improve results from [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF] on presentations of Artin-Tits groups of type Ã. In Section 4 we get a Garside structure from the fixed points of an involution in a Garside group of type Ã. In Section 5 we show that an Artin-Tits group of type C can be seen as a group of fixed points in an Artin-Tits group of type Ã. In Section 6 we show that the Garside structure we have got in Section 4 can be obtained by the method of the "generated group" of [2, 0.4]. In Section 7 we prove that the group of fractions of our Garside monoid is the Artin-Tits group of type Cn and we give a a dual presentation of this group similar to what has been done in [START_REF]The dual braid monoid[END_REF], [START_REF]A dual braid monoid for the free group[END_REF] and [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF] for the other known dual monoids, where the generators are in oneto-one correspondence with a set of reflections in the Coxeter group. One of the intermediate results is that the Hurwitz action is transitive on the set of shortest decompositions of a Coxeter element of W ( Cn ) into a product of reflections. The analogous property is known for all finite Coxeter groups ([2, 2.1.4]), for all well-generated complex reflexion groups ([1, 7.5]), for Coxeter groups of type à ( [10, 3.4]) and is conjectured to be true for all Coxeter groups. In section 8 we deduce from the Garside structure the centralizer of a power of a lift of a Coxeter element in the Artin-Tits group.

I thank Eddy Godelle for having carefully read a preliminary version of this paper, allowing me to fix an error in one of the proofs.

Germs, Garside groups

In this section we recall some definition and results on Garside monoids and groups.

Definition 2.1 ( [START_REF] Digne | Garside and locally Garside categories[END_REF]).

(i) A germ of monoid is a set P endowed with a partially defined product (x, y) → xy, which has a unit, i.e., an element 1 such that 1p and p1 are defined and equal to p for any p ∈ P . (ii) A germ is associative if for any a, b, c in P such that one of the products a(bc) or (ab)c is defined then the other one is also defined and both products are equal. (iii) A germ is left (resp. right) cancellative if ab = ac (resp. ba = ca) implies b = c. It is cancellative if it is cancellative on both sides.

We say that an element a of a germ P left divides an element b ∈ P if there exists c ∈ P such that b = ac. Right divisibility is defined similarly. In an associative germ right and left divisibility are preorder relations. They are order relations if moreover the germ is cancellative and there is no invertible element different from 1. Definition 2.2. An associative germ is said to be left (resp. right) Noetherian if there is no strictly decreasing infinite sequence for left (resp. right) divisibility. It is called Noetherian if it is both left and right Noetherian.

Note that in a Noetherian germ there is no non-trivial invertible element. Note also that to be left (resp. right) Noetherian is equivalent to the fact that there is no strictly increasing bounded infinite sequence for right (resp. left) divisibility.

A morphism from a germ to a monoid is a map which sends the (partial) product on P to the product in the monoid, and the unit of P to the unit of the monoid. The monoid M(P ) (resp. group G(P )) defined by a germ P is the monoid (resp. group) which has the universal property that it factorizes any morphism from P to a monoid (resp. group). In other words it is the monoid (resp. group) generated by P with only relations the relations given by equalities of products in P . It is known ( [11, 3.5]) that P injects into M(P ) and is stable by left and right divisibility in M(P ). The following definition is the definition of Garsideness that we will use in the present paper (for small variations and generalizations of this definition see [START_REF] Dehornoy | Gaussian groups and Garside groups, two generalisations of Artin groups[END_REF], [START_REF] Dehornoy | Groupes de Garside[END_REF] and [START_REF] Digne | Garside and locally Garside categories[END_REF]). Definition 2.3.

• We say that a monoid is Garside if it is cancellative, Noetherian, if it is a lattice for both left and right divisibility and if there exists an element ∆ (called a Garside element) whose sets of right and left divisors coincide and generate the monoid.

• A group is Garside if it is generated by a submonoid which is a Garside monoid.

Note that here we do not assume the set of divisors of ∆ to be finite. When this set is infinite what we call here a Garside monoid (resp. group) is what is usually called a quasi-Garside monoid (resp. group). A general reference for Garside monoids can be [START_REF] Digne | Garside and locally Garside categories[END_REF]. The following result is a combination of [11, 3.31, 5.4 and 5.5].

Proposition 2.4. Let P be an associative Noetherian germ satisfying the following properties:

(i) two elements of P have a least right common multiple in P ;

(ii) for all m ∈ M(P ), and a, b in P , if am = bm or ma = mb, then a = b;

(iii) the elements of P have a both left and right common multiple ∆ ∈ P . then M(P ) is a Garside monoid with Garside element ∆.

Conversely, in a Garside monoid M the divisors of the Garside element form a germ P satisfying the above properties and such that the canonical map M(P ) → M is an isomorphism. Definition 2.5. A germ satisfying the assumptions of 2.4 is called a Garside germ.

Given a Garside germ P , elements in M(P ) have normal forms: any element can be written uniquely p 1 p 2 . . . p k with p i ∈ P such that p i is the greatest element for left divisibility dividing p i p i+1 . . . p k . In the (Garside) group of a Garside monoid M with Garside element ∆ any element can be written as ∆ k x with x ∈ M and k ∈ Z.

We will use proposition 2.4 through its following corollary. Before stating this corollary we need: Definition 2.6. An automorphism of a germ P is a bijection f : P → P mapping the unit to the unit and such that ab is defined if and only if

f (a)f (b) is defined, in which case f (ab) = f (a)f (b).
Corollary 2.7. Let P be an associative and Noetherian germ having a (unique) both left and right common multiple ∆; assume that M(P ) is cancellative and that P has an automorphism σ such that any two elements of P have a unique minimal σ-stable common right multiple; then P σ is a Garside germ.

Proof. It is clear that P σ is an associative and Noetherian germ. We get the result by proving that P σ satisfies the assumptions of Proposition 2.4. Since P is cancellative, if x, y ∈ P σ are such that x divides y in P then x divides y in P σ . Hence the assumption of the corollary implies that any two elements of P σ have an lcm in P σ , whence (i).

The inclusion P σ → M(P ) extends to a morphism M(P σ ) → M(P ). If am = bm or ma = mb as in (ii), taking the images in M(P ) we get a = b, since M(P ) is cancellative, whence (ii).

Unicity of ∆ as the maximal element of P implies ∆ ∈ P σ , hence ∆ is a common right and left multiple of P σ , whence (iii).

The monoids of type Ã2n-1

Before applying the previous results to Artin-Tits groups of type Cn , we need to recall the presentation of an Artin-Tits group of type à given in [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF] as group of fractions of the monoid M(P ) generated by an associative germ P .

We recall first a general method for constructing a germ. Given a group G generated as a monoid by a set S, i.e., any element of G is a product of elements of S, without inverses, we let l S be the length on G with respect to S: the length of g ∈ G is the length of a shortest expression of g as a product of elements of S. We say that a ∈ G left divides b ∈ G, denoted by a G b if l S (a) + l S (a -1 b) = l S (b), and similarly for right divisibility. Starting with a balanced element δ (an element which has the same set of right and left divisors), we call D the set of (left or right) divisors of δ. Then D is a germ, the product of a and b in D being defined and equal to ab if ab ∈ D and l S (ab) = l S (a) + l S (b). Associated to this germ we have a monoid M(D) and a group G(D). If D is a lattice then M(D) is a Garside monoid with Garside element δ (result due to J. Michel, see [2, Theorem 0.5.2]). We call this construction the method of the "generated group". This construction starting with any finite Coxeter group, its set of Coxeter generators and taking the longest element for δ, gives the associated Artin-Tits monoid (or group). Starting with a finite Coxeter group with set of generators all reflections it gives the dual monoid if we take for δ any Coxeter element. Starting with a Coxeter element and all reflections in a Coxeter group of type à it gives the monoids M(P ) that we describe in this section. For these results see [START_REF]The dual braid monoid[END_REF], [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF] and [START_REF] Digne | Garside and locally Garside categories[END_REF].

We see a Coxeter group of type à as a subgroup of the periodic permutations of Z. We need some notation. Definition 3.1.

(i) A permutation w of Z is said to be n-periodic if w(i + n) = w(i) + n for any i ∈ Z. (ii) A cycle is an n-periodic permutation which has precisely one orbit up to translation by n. We say that a cycle is finite if its orbits are finite. We call length of a cycle the cardinality of one of its orbits. We call support of a permutation w the union of its non-trivial orbits.

Any n-periodic permutation of Z can be written uniquely as a product of disjoint cycles. There are two types of cycles: either all the orbits are finite, or all the non-trivial orbits are infinite. In the former case we will represent the cycle by one of its non-trivial orbits; in the latter case we will represent a cycle as (a 1 , a 2 , . . . , a k ) [h] , with all a i distinct modulo n, meaning that the image of a i is a i+1 for 0 ≤ i < k and the image of a k is a 1 + nh (this cycle has |h| non trivial orbits). To each n-periodic permutation w of Z we can associate its total shift 1 n x=n x=1 (w(x) -x). We recall the following facts: Proposition 3.2. The Coxeter group W ( Ãn-1 ) of type Ãn-1 is isomorphic to the group of n-periodic permutations of Z with total shift equal to 0. The reflections of W ( Ãn-1 ) correspond to the permutations (a, b) with a and b distinct modulo n. The simple reflections are the permutations s i = (i, i + 1) for i = 1, 2, . . . , n. The reflections s i and s j commute unless i -j = ±1 (mod n) in which case their product has order 3.

In [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF] a germ generating the Artin-Tits group of type Ãn-1 is defined for each partition of Z into two non-empty subsets X and Ξ stable by translation by n; such a partition corresponds to the choice of a Coxeter element. To recall this construction we need the following definitions. A graphical representation of these definitions will be given after Definition 3.6 We make the convention that Latin letters denote elements of X and Greek letters elements of Ξ. Definition 3.3. We say that a cycle is positive and self non-crossing if it has one of the following forms: (a 1 , a 2 , . . . , a k , α 1 , α 2 , . . . , α l ) or (a 1 , a 2 , . . . , a k ) [START_REF] Bessis | Complex reflection arrangements are K(π, 1)[END_REF] or (α 1 , α 2 , . . . , α l ) [-1] with a i ∈ X, α j ∈ Ξ satisfying the conditions a 1 < a 2 < . . . < a k < a 1 + n and α 1 > α 2 > . . . > α l > α 1 -n. Definition 3.4. We say that two positive self non-crossing cycles are non-crossing if they satisfy one of the following:

(i) One of them is of the form (a 1 , a 2 , . . . , a k ) or (a 1 , a 2 , . . . , a k ) [START_REF] Bessis | Complex reflection arrangements are K(π, 1)[END_REF] and the other one

(α 1 , α 2 , . . . , α l ) or (α 1 , α 2 , . . . , α l ) [-1] . (ii)
One of them is of the form (a 1 , a 2 , . . . , a k , α 1 , α 2 , . . . , α l ), and the other (b 1 , . . . , b q , β 1 , . . . , β r ) with k, l, q, r > 0, a k < b i < a 1 + n for all i and α l < β i < α 1 + n for all i. (iii) One of them is of the form (a 1 , a 2 , . . . , a k , α 1 , α 2 , . . . , α l ) with k, l ≥ 0, and the other one (b 1 , . . . , b m ) (resp. (β 1 , . . . , β m )) with a j < b i < a j+1 for some j and all i (resp. α j+1 < β i < α j for some j and all i), where in this condition we put a k+1 = a 1 + n and α l+1 = α 1 -n, (there is no condition on b i if k = 0 and no condition on β i if l = 0). (iv) One of them is of the form (a 1 , a 2 , . . . , a k ) [START_REF] Bessis | Complex reflection arrangements are K(π, 1)[END_REF] or (α 1 , α 2 , . . . , α k ) [-1] with k > 0 and the other one (b 1 , . . . , b m ) (resp. (β 1 , . . . , β m )) with a j < b i < a j+1 for some j and all i (resp. α j+1 < β i < α j for some j and all i), where in this condition we put a k+1 = a 1 +n and α k+1 = α 1 -n.

Definition 3.5. We say that a periodic permutation of Z is positive and self non-crossing if it is the product of disjoint positive self non-crossing and pairwise non-crossing cycles.

Note that a finite cycle has total shift 0. Note also that a periodic positive self non-crossing permutation of Z has at most 2 infinite cycles, one in X and one in Ξ and that, if it has total shift 0, it has either 0 or 2 infinite cycles. Definition 3.6. We call pseudo-cycle a positive self non-crossing n-periodic permutation of Z which is the product of two infinite disjoint cycles.

A pseudo-cycle is a permutation (a 1 , . . . , a h ) [1] (α 1 , α 2 , . . . , α l ) [-1] with the a i in X, the α i in Ξ and a 1 < a 2 < . . . < a h < a 1 + n, α 1 > α 2 > . . . > α n > α 1 -n. A pseudo-cycle has total shift 0.
We now give a graphical representation of the above definitions (see Figure 1 and Figure 2; in these figures we have taken n = 9, and modulo n there are 5 elements in X and 4 in Ξ). Consider two parallel oriented lines D and ∆ in the plane, with same orientation. On D we put a discrete set of points in one-to-one ordered correspondence with X and on ∆ we put a discrete set of points in one-to-one ordered correspondence with Ξ. Let S be the strip delimited in the plane by D and ∆. We associate to a permutation w of Z a union of oriented paths (one for

• 17 • 16 • 15 • 14 • 9 • 8 • 7 • 6 • 5 • 13 • 12 • 11 • 10 • 4 • 3 • 2 • 1 / / / / / / / / / / / / / r r 1 1 / / / / / / / / / / / / / r r 1 1 ∆ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o D _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o Figure 1.
The 9-periodic cycle (5, 7, 8, 3, 2)

• 17 • 16 • 15 • 14 • 9 • 8 • 7 • 6 • 5 • 13 • 12 • 11 • 10 • 4 • 3 • 2 • 1 r r o o / / / / ∆ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o D _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o Figure 2. The 9-periodic pseudo-cycle (5, 7, 8) [1] (3, 2) [-1]
each orbit of w) contained in the strip S joining i to w(i) for all i, up to homotopy in the strip. A cycle is positive self non-crossing if it can be represented by a union of disjoint oriented paths such that each of them intersects X (resp. Ξ) along an increasing (resp. decreasing) or empty subsequence. Two positive self non-crossing cycles are non-crossing if they can be represented by two unions of paths which do not cross each other. Other figures (with n = 10) can be found in Section 6.

Following [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF] we now describe the germ P associated to the X and Ξ. The elements of P are the n-periodic positive self non-crossing permutations. The permutation c given by x i → x i+1 and ξ i → ξ i-1 where X = (x i ) i∈Z and Ξ = (ξ i ) i∈Z with x i < x i+1 and ξ i < ξ i+1 , is a Coxeter element of W ( Ãn-1 ) and the elements of P are precisely the left (or right) divisors of c (see [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF]Proposition 2.19]). The germ is obtained by the method of the generated group described at the beginning of this section, hence the product of two elements w and w ′ of P is defined in P if the product ww ′ of the permutations is in P and if l Ãn-1 (ww ′ ) = l Ãn-1 (w) + l Ãn-1 (w ′ ) where l Ãn-1 is the length in W ( Ãn-1 ) with respect to the generating set consisting of all reflections. We will give below an equivalent and more tractable condition.

By general results P is an associative, cancellative and Noetherian germ, the monoid M(P ) embeds in the group G(P ) and any element G(P ) can be written a -1 b with a and b in M(P ). Since the length l Ãn-1 is invariant by conjugation, right and left divisibility in P coincide.

The following is proved in [10, Theorem 4.1].

Proposition 3.7. The group G(P ) is isomorphic to the Artin-Tits group of type Ãn-1 .

Note that by [10, Proposition 5.5] P is a Garside germ if and only if either X or Ξ contains exactly one element modulo n.

We now give a more tractable definition of divisibility in P . To each element w ∈ P we associate the partition p w of Z whose parts are the finite orbits of w and the union of the two infinite orbits of w if they exist. Such a partition is invariant by translation by n. We say that it is periodic. Moreover such a partition has at most one infinite part and this infinite part must meet both X and Ξ. Also such a partition is non-crossing in the following sense: Definition 3.8.

• Two subsets A and B of X ∪ Ξ are non-crossing if for any a, a ′ ∈ A and any b, b ′ ∈ B there exist in the strip S a path γ from a to a ′ and a path δ from b to b ′ such that γ and δ have an empty intersection.

• We say that a partition is non-crossing if any two of its parts are non-crossing.

By [10, Corollary 2.22] w → p w is a bijection from P onto the set of non-crossing periodic partitions of Z such that any infinite part meets both X and Ξ. Such a partition can have at most one infinite part. We order P by divisibility and the set of partitions by refinement. Note that the largest element of P for the divisibility order is c and that p c is the partition with only one part. Proposition 3.9. The bijection w → p w is an isomorphism of ordered sets.

Proof. Let v, w ∈ P . We write v and w as products v = v 1 . . . v k and w = w 1 . . . w l where each v i and each w i is either a a positive self non-crossing finite cycle (see Definition 3.1(ii)) or is a pseudo-cycle (see Definition 3.6) and the v i (resp. the w i ) are pairwise non-crossing.

By [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF]Lemma 2.20] a reflection of P divides w in W ( Ãn ) if and only if its support is a subset of the support of w i for some i. Now the following lemma (see [10, Corollary 2.9]) shows that any v i can be written as a product r 1 r 2 . . . r h in P of reflections whose union of supports is the support of v i and such that the supports of r i and r i+1 have a non-empty intersection.

Lemma 3.10. The two following formulas give shortest decompositions of a finite cycle and of a pseudo-cycle respectively into products of reflections of Ãn-1 .

(a 1 , a 2 , . . . , a h ) = (a 1 , a 2 )(a 2 , a 3 ) . . . (a h-1 a h ) (a 1 , . . . , a h ) [1] (α 1 , α 2 , . . . , α l ) [-1] = (a 1 , a 2 )(a 2 , a 3 ) . . . (a h-1 , a h )(a h , α 1 )(a h , n + α 1 )(α 1 , α 2 )(α 2 , α 3 ) . . . (α l-1 , α l ) Proof. By [10, 2.8] we know that l Ãn-1 ((a 1 , . . . , a h ) [1] (α 1 , α 2 , . . . , α l ) [-1] ) = h + l and that l Ãn-1 ((a 1 , a 2 , . . . , a h )) = h -1.
Assume that v = v 1 . . . v k divides w in P ; fix i and let us we write v i = r 1 . . . r h as above. Then every r j divides w so its support must be a subset of the support of some w s . But since the supports of r j and r j+1 have a non-empty intersection, they have to be included in the support of the same w s , so that the whole support of v i is a subset of the support of some w s . This, being true for all i, means that p v is finer than p w .

Conversely, assume that p v is finer than p w . We prove by induction on l Ãn-1 (v) that v divides w in P . If v is trivial the result is true. If v is not trivial, there exists a reflection s dividing v, so that the support of s is included in the support of some v i . Hence the support of s is also included in the support of some w j , so that s divides w. Put v = sv ′ and w = sw ′ ; then v divides w if and only if v ′ divides w ′ . We will be done by induction if we prove that p v ′ is finer than p w ′ . We have seen that s divides a cycle or a pseudo-cycle v i of v and a cycle or pseudo-cycle w j of w. Since disjoint cycles and pseudo-cycles commute we may assume that s divides v 1 and w 1 . We have only to show that p v ′ 1 is finer than p w ′ 1 where v 1 = sv ′ 1 and w 1 = sw ′ 1 . This will be a consequence of the following lemma:

Lemma 3.11. Let u ∈ P be a finite cycle or a pseudo-cycle and let s be a reflection dividing u in P . We put u = su ′ .

• If u = (a 1 , a 2 , . . . , a h ) ∈ P is a finite cycle (with a i ∈ X ∪ Ξ) and s = (a 1 , a j ), then u ′ = (a 1 , a 2 , . . . , a j-1 )(a j , a j+1 , . . . , a h ).

• If u = (a 1 , . . . , a h ) [1] (α 1 , α 2 , . . . , α l ) [-1] with a i ∈ X and α i ∈ Ξ is a pseudo-cycle and s = (a 1 , a j ) then u ′ = (a 1 , a 2 , . . . , a j-1 )(a j , a j+1 , . . . , a h ) [1] (α 1 , α 2 , . . . , α l ) [-1] . • If u = (a 1 , . . . , a h ) [1] (α 1 , α 2 , . . . , α l ) [-1] with a i ∈ X and α i ∈ Ξ is a pseudo-cycle and s = (α 1 , α j ) then u ′ = (a 1 , a 2 , . . . , . . . , a h ) [1] (α 1 , α 2 , . . . , α j-1 )(α j , . . . , α l ) [-1] . • If u = (a 1 , . . . , a h ) [1] (α 1 , α 2 , . . . , α l ) [-1] with a i ∈ X and α i ∈ Ξ is a pseudo-cycle and s = (a 1 , α 1 ) then u ′ = (a 1 , a 2 , . . . , . . . , a h , α 1 + n, α 2 + n, . . . , α l + n).
This lemma is an easy computation (see also [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF]Lemma 2.5]).

Applying the lemma with u = v 1 and with u = w 1 shows that p v ′ 1 is finer than p w ′ 1 .

Fixed points in Ã2n-1

It is well known that the Coxeter group of type Cn is the group of fixed points under the involution of W ( Ã2n-1 ) which maps s i to s 2n-i , with the notation of Proposition 3.2, the subscript i being taken modulo 2n (see e.g., [START_REF] Hée | Système de racines sur un anneau commutatif totalement ordonné[END_REF]). This involution can be lifted to the Artin-Tits group using the same formula. We shall see in Section 5 that similarly the Artin-Tits group of type Cn is the group of fixed points under this lifted involution in the Artin-Tits group of type Ã2n-1 .

To get a Garside germ for an Artin-Tits group of type Cn we shall start with a particular choice of X and Ξ in the construction of the previous section, compatible with this involution. We take X to be the the set of odd integers and Ξ to be the set of even integers. This corresponds to choosing the Coxeter element c = (2, 3)(4, 5) . . . (2n, 2n + 1)(1, 2)(3, 4) . . . (2n -1, 2n) = s 2 s 4 . . . s 2n s 1 s 3 . . . s 2n-1 in the previous section. Then the germ P has an automorphism σ coming from the map i → 1 -i which interchanges X and Ξ. This involution lifts to the Artin-Tits group the involution s i → s 2n-i of the Coxeter group. We still denote by σ this involution of X ∪ Ξ.

Theorem 4.1. The germ P σ is Garside Proof. We show that P and σ satisfy the assumptions of Corollary 2.7.

First P has a unique both right and left multiple of all its elements, which is c. Note that with our choice of X and Ξ, the element c seen as a 2n-periodic permutation of Z is i → i + 2 for odd i, i -2 for even i.

We have to show that any two elements v and w in P have a unique minimal σ-stable common multiple z. By Proposition 3.9, this amounts to show the existence of a unique minimal σ-stable non-crossing periodic partition of the form p z coarser than p v and p w . Note that a σ-stable non-crossing partition has precisely 0, one or two infinite parts. Hence the condition for such a partition to be of the form p z , i.e., that any infinite part has a non empty intersection with both X and Ξ is equivalent to the condition for the partition to have at most one infinite part. A σ-stable partition coarser than p v and p w is also coarser than the partitions σ p v and σ p w . We claim that it is sufficient to show the existence of a unique minimal non-crossing partition coarser than these 4 partitions: such a partition will be periodic and σ-stable by unicity. Moreover since it is non-crossing and symmetric it will have precisely 0, one or two infinite parts. If it has zero or one infinite part it is of the form p z and we are done. If it has two infinite parts these parts are interchanged by σ, in which case there is exactly one minimal coarser partition with one infinite part meeting both X and Ξ, obtained by putting together the two infinite parts. This partition is σ-stable and periodic. By Proposition 3.9 it corresponds to a minimal σ-stable common multiple of v and w and we are done also in this case.

The theorem is thus a consequence of the following lemma which ensures by induction the existence of a unique non-crossing partition coarser than any (finite) given number of partitions of X ∪ Ξ. Lemma 4.2. There exists a unique minimal non-crossing partition coarser than any two given partitions of X ∪ Ξ.

Proof of the lemma. Consider a graph whose vertices are the parts of the given partitions and such that there is an edge between two vertices if the corresponding parts cross each other (this includes the case where two parts have an intersection). Then the desired partition is the partition whose parts are the union of all the parts lying in the same connected component of our graph.

The Artin-Tits group of type Cn

We denote by G( Ã2n-1 ) the Artin-Tits group of type Ã2n-1 and by G( Cn ) the Artin-Tits group of type Cn . They are the groups of fractions of the corresponding classical Artin-Tits monoids, respectively M( Ã2n-1 ) and M( Cn ). The monoid M( Ã2n-1 ) has a presentation with generators s 1 , . . . , s 2n and relations s i s j = s j s i if |i -j| = 1 mod 2n and s i s i+1 s i = s i+1 s i s i+1 for all i, where the indices are taken modulo 2n; the monoid M( Cn ) of type Cn has a presentation with generators σ 0 , . . . , σ n and relations the braid relations given by the Coxeter diagram

σ 0 σ 1 σ 2 • • • σ n-2 σ n-1 σn
Like any Artin-Tits monoid, by [START_REF] Paris | Artin monoids embed in their groups[END_REF], the monoids M( Ã2n-1 ) and M( Cn ) embed in their respective groups G( Ã2n-1 ) and G( Cn ).

By Proposition 3.7 the group G( Ã2n-1 ) is isomorphic to the group of fractions of the monoid M(P ) generated by the germ P defined in Section 4. This isomorphism maps the generator s i of M( Ã2n-1 ) to the element (i, i + 1) ∈ P , so that M( Ã2n-1 ) is a submonoid of M(P ). The involution σ considered in Section 4 restricts to M( Ã2n-1 ) in the diagram automorphism of Ã2n-1 which maps s i to s 2n-i where the indices are taken modulo 2n.

By [15, 4.4] the monoid of fixed points M( Ã2n-1 ) σ is isomorphic to M( Cn ). We will identify these two monoids. Under this identification we have

σ 0 = s 2n , σ 1 = s 1 s 2n-1 , . . . , σ n-1 = s n-1 s n+1 , σ n = s n .
We summarise all these facts in the commutative diagram:

(5.1)

M( Cn ) ∼ / / _ M( Ã2n-1 ) σ / / _ M( Ã2n-1 ) _ M(P ) σ / / _ M(P ) _ G( Cn ) / / G( Ã2n-1 ) σ / / G( Ã2n-1 ) = G(P )
The aim of this section is to prove the following theorem. I thank John Crisp and Dave Margalit for having pointed out that this theorem can be easily deduced from the results of Birman-Hilden and Maclachlan-Harvey.

Theorem 5.2. The morphism G( Cn ) → G( Ã2n-1 ) σ of Diagram 5.1, which maps σ 0 to s 2n , σ i to s i s 2n-i for i = 1, . . . , n -1 and σ n to s n is an isomorphism.

We need first to recall how the Artin-Tits groups of type à and C can be embedded into mapping class groups. Let E and F be two finite subsets of a 2-sphere S 2 . We denote by M(S 2 , E, F ) the subgroup of the mapping class group of the 2-sphere with punctures the points of E ∪ F , represented by the diffeomorphisms which fix each point of F and stabilize E.

The following result appears in [7, Section 2]: Proposition 5.3 (Charney-Crisp).

(i) Let E = {P 1 , . . . , P n } and F = {P ′ , P ′′ }) be two sets of points of S 2 ; the group G( Ãn-1 ) embeds into the group M(S 2 , E, F ). This embedding maps the standard generator s i of G( Ãn-1 ) to the positive braid twist exchanging P i and P i+1 , for 1 ≤ i ≤ n, where the indices are taken modulo n. (ii) Let E = {P 1 , . . . , P n } and F = {P 0 , Q ′ , Q ′′ } be subsets of S 2 then the group G( Cn ) is isomorphic to M(S 2 , E, F ). This isomorphism maps the standard generator (numbered as in the beginning of Section 5) σ i of G( Cn ) to the positive braid twist exchanging P i and P i+1 for 1 ≤ i ≤ n -1 and maps σ 0 (resp. σ n ) to the square of the positive braid twist exchanging P 0 and P 1 (resp. P n and Q ′′ ).

We need also the following result of Birman and Hilden (see [START_REF] Birman | On isotopies of homeomorphisms of Riemann surfaces[END_REF] and [START_REF] Maclachlan | On mapping class groups and Teichmüller spaces[END_REF]).

Theorem 5.4 (Birman-Hilden). Let E 1 and E 2 be two finite sets of points of a 2-sphere S 2 , let π : S 2 → S 2 /G be a ramified covering realizing the quotient of S 2 by a finite group G of diffeomorphisms stabilizing E 1 and E 2 and let F ⊂ S 2 /G be the set of ramification points. Then the projection induces an isomorphism from the normalizer of G in the mapping class group

M(S 2 , E 1 , E 2 ) to the mapping class group M(S 2 /G, π(E 1 ), π(E 2 ) ∪ F ).
Proof of Theorem 5.2. Consider a set E consisting of 2n points P 1 , . . . , P 2n regularly placed on the equator of a sphere S 2 and let P ′ and P ′′ be the north and south poles. We apply Proposition 5.3(i), for embedding the group G( Ã2n-1 ) into the mapping class group M(S 2 , E, F ) where F = {P ′ , P ′′ }. We then apply Theorem 5.4 with G the group of order 2 generated by the symmetry σ which exchanges P ′ and P ′′ and exchanges P i and P 2n-i for all i. We can view S 2 /G as a sphere so that π : S 2 → S 2 /G is a ramified covering of a 2-sphere with 2 ramification points A ′ , A ′′ ∈ S 2 /G which are antipodal on the equator. The points π(P i ) = π(P 2n-i ) are regularly placed on one half of the equator and the point P = π(P ′ ) = π(P ′′ ) is on the other half of this equator. We get an isomorphism M(S 2 , E, F ) σ ≃ M(S 2 /G, π(E), {P, A ′ , A ′′ }).

Since π(E) has cardinality n, we get by Proposition 5.3(ii) that M(S 2 /G, π(E), {P, A ′ , A ′′ }) is isomorphic to G( Cn ). Since σ maps the product of braid twists s i s n-i to the braid twist σ i for i = 1, . . . , n -1 and maps the braid twist s n (resp. s 2n ) to the square σ 0 (resp. σ n ) of the positive braid twist exchanging A ′ and π(P 1 ) (resp. π(P n ) and A ′′ ), we get that the isomorphism restricted to G( Ã2n-1 ) σ is onto which means that G( Ã2n-1 ) σ is in fact equal to M(S 2 , E, F ) σ . Moreover we also see that the above isomorphism coincides with the morphism of Theorem 5.2, whence the result.

A generated group

Our aim in this section is to show that the germ P σ is obtained from the Coxeter group of type Cn by the method of the generated group described at the beginning of Section 3. We denote the Coxeter group of type Cn by W ( Cn ) and see it as the group of fixed points under the involution σ : s i → s 2n-i in the Coxeter group W ( Ã2n-1 ) of type Ã2n-1 (see the beginning of Section 4). Lemma 6.1. The reflections of W ( Cn ) are the σ-stable reflections of W ( Ã2n-1 ) and the products r. σ r where r is a reflection of W ( Ã2n-1 ) which is not σ-stable.

Proof. This is well known. We recall a proof. By [START_REF] Hée | Système de racines sur un anneau commutatif totalement ordonné[END_REF], if σ is an automorphism of a Coxeter group permuting the simple reflections, the group of σ-fixed elements is a Coxeter group with simple reflections the longest elements of the parabolic subgroups generated by the σ-orbits of spherical type of simple reflections. The simple reflections of W ( Cn ) are thus s n , s 2n and the products s i . σ s i = s i s 2n-i for 1 ≤ i ≤ n -1. An arbitrary reflection of W ( Cn ) is a conjugate under W ( Cn ) to a simple reflection, whence the result since this conjugation commutes with σ.

We put c = s 2 s 4 . . . s 2n s 1 s 3 . . . s 2n-1 as in Section 4. We denote the simple reflections of W ( Cn ) by σ 0 = s 2n , σ 1 = s 1 s 2n-1 , σ 2 = s 2 s 2n-2 ,. . . , σ n-1 = s n-1 s n+1 and σ n = s n . We have c = σ 0 σ 2 σ 4 . . . σ 1 σ 3 σ 5 . . ., hence c is both a σ-fixed Coxeter element of W ( Ã2n-1 ) and a Coxeter element of W ( Cn ). We denote by W ( Ã2n-1 ) (resp. W ( Cn) ) the divisibility in W ( Ã2n-1 ) (resp. W ( Cn )) with respect to the reflection length. Beware that the reflection length in W ( Cn ) is not the restriction of the reflection length in W ( Ã2n-1 ), in particular the reflection length of c is n + 1 in W ( Cn ) and 2n in W ( Ã2n-1 ), since a Coxeter element in a Coxeter group of rank h has reflection length equal to h (see e.g., [START_REF] Dyer | On minimal lengths of expressions of Coxeter group elements as product of reflections[END_REF]). Recall that divisibility for the reflection length in a Coxeter group can be defined by saying that w divides w ′ if for one, or for any, shortest decomposition of w into a product of reflections, there exists a shortest decomposition of w ′ as a product of reflections which begins with that decomposition of w. We now show that divisibility in W ( Ã2n-1 ) restricts to divisibility in W ( Cn ) for the divisors of c. Recall that by the results of [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine A[END_REF] P is the set of divisors of c in W ( Ã2n-1 ), so that P σ is the set of elements of W ( Cn ) which divide c in W ( Ã2n-1 ). Proposition 6.2. Let w and w ′ be two elements of W ( Cn ); then w W ( Cn) w ′ W ( Cn) c if and only if w W ( Ã2n-1 ) w ′ W ( Ã2n-1 ) c. r r ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Proof. We first prove that divisibility in W ( Cn ) implies divisibility in W ( Ã2n-1 ) for divisors of c in W ( Cn ).

• 19 • 17 • 15 • 13 • 11 • 9 • 7 • 5 • 3 • 1 • 20 • 18 • 16 • 14 • 12 • 10 • 8 • 6 •
1 1 1 1 ∆ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o D _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o Figure 3. A σ-stable cycle in Ã9 • 19 • 17 • 15 • 13 • 11 • 9 • 7 • 5 • 3 • 1 • 20 • 18 • 16 • 14 • 12 • 10 • 8 • 6 • 4
∆ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o D _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o
We claim that in any decomposition of c into a product of reflections of W ( Cn ) there is at least one reflection conjugate to σ 0 and one reflection conjugate to σ n . Indeed the quotient of W ( Cn ) by its commutator subgroup is the direct product of 3 groups of order 2, generated respectively by the image of σ 0 , the image of σ n and the image of any one of σ 1 , . . . , σ n-1 which have all same image since they are conjugate. Hence the image of c in this quotient must involve the images of σ 0 and σ n . Whence the result since two conjugate reflections have same image in this quotient.

Let c = r 0 . . . r n be a decomposition of c into a product of n+1 reflections in W ( Cn ). Let k be the number of reflections in that decomposition which are conjugate to either σ 0 or σ n . These k reflections are also reflections in W ( Ã2n-1 ) and the other n + 1 -k reflections are products of 2 reflections in W ( Ã2n-1 ). So this product is the product of k +2(n+1-k) = 2n+2-k reflections in W ( Ã2n-1 ). But c cannot be written with less than 2n reflections in W ( Ã2n-1 ), whence k ≤ 2. Since we know that k ≥ 2, we have k = 2. We have proved that, replacing a reflection of W ( Cn ) by its image in the embedding W ( Cn ) ֒→ W ( Ã2n-1 ), any shortest decomposition of c as a product of reflection of W ( Cn ) becomes a shortest decomposition of c as a product of reflections of W ( Ã2n-1 ). This gives that if w W ( Cn) w ′ W ( Cn) c then w W ( Ã2n-1 ) w ′ W ( Ã2n-1 ) c. To prove the converse we first prove a formula for the reflection length l W ( Cn) in W ( Cn ) of the elements of P σ . An element of P σ can be written uniquely as a commuting product of elements of three types: either σ-stable non-crossing finite cycles, or products of a finite cycle and its image by σ, or σ-stable pseudo-cycles. There can be at most one pseudo-cycle and any two of these factors are non-crossing. In Figures 3, 4 and5 we have given examples in C5 , i.e., the partitions are 10-periodic and σ comes from i → 1 -i modulo 10.

• 19 • 17 • 15 • 13 • 11 • 9 • 7 • 5 • 3 • 1 • 20 • 18 • 16 • 14 • 12 • 10 • 8 • 6 • 4 • 2 r r 1 1 0 0 r r 1 1 0 0 ∆ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o D _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ o o Figure 5. A σ-stable pseudo-cycle in Ã9
Lemma 6.3. Let w = w 1 . . . w r be a decomposition as above of w ∈ P σ ; then l W ( Cn) (w) = i=r i=1 l W ( Cn) (w i ) and

l W ( Cn) (w i ) =      k if w i is a σ-stable cycle of length 2k k -1 if w i is
the product of a cycle of length k and its image by σ k + 1 if w i is a pseudo-cycle whose support modulo 2n has cardinality 2k

Proof. We denote by f (w) the formula of the lemma. We have to prove that l W ( Cn) (w) = f (w). First, it is clear that f (w) = 0 if and only if w = 1. We now prove: Lemma 6.4. For any w = 1 in P σ and any reflection ρ ∈ W ( Cn ) such that ρ W ( Ã2n-1 ) w we have f (ρw) = f (w) -1.

Proof. Recall that σ is induced by the bijection a → 1-a of Z so that by Lemma 6.1 a reflection in W ( Cn ) is either equal to (a, 2kn + 1 -a) for some a and k in Z or (a, b)(1 -a, 1 -b) for some a and b in Z with a + b ≡ 1 mod 2n. Decompose w into a product w = w 1 w 2 . . . w r as in lemma 6.3. Let ρ be a reflection of W ( Cn ) which divides w in W ( Ã2n-1 ). By Proposition 3.9, if ρ = (a, 2kn + 1 -a) for some a, then a and 2kn + 1 -a have to be in the support of the same w i and if ρ = (a, b)(1 -a, 1 -b), then a and b have to be in the support of the same w i and 1 -a and 1 -b are then in the support of σ w i . Lemma 6.4 is then a consequence of the following formulas, obtained by an immediate computation. In each of these formulas the left hand side has the form ρw i with w i as above and one checks easily that f has value f (w i ) -1 on the the right hand side.

(a 1 , 2kn + 1 -a 1 )(a 1 , a 2 , . . . , a h , 2kn + 1 -a 1 , 2kn + 1 -a 2 , . . . , 2kn + 1 -a h ) = (a 1 , a 2 , . . . , a h )(1 -a 1 , . . . , 1 -a h ) (a 1 , 2kn + 1 -a 1 )(a 1 , . . . , a h ) [1] (1 -a 1 , . . . , 1 -a h ) [-1] = (a 1 , . . . , a h , (2k + 1)n + 1 -a 1 , . . . , (2k + 1)n + 1 -a h ) (a 1 , b 1 )(1 -a 1 , 1 -b 1 )(a 1 , . . . , a h , b 1 , . . . , b k )(1 -a 1 , . . . , 1 -a h , 1 -b 1 , . . . , 1 -b k ) = (a 1 , . . . , a h )(b 1 , . . . , b k )(1 -a 1 , . . . , 1 -a h )(1 -b 1 , . . . , 1 -b k ) (a 1 , b 1 )(1 -a 1 , 1 -b 1 )(a 1 , . . . , a h , b 1 , . . . , b k ) [1] (1 -a 1 , . . . , 1 -a h , 1 -b 1 , . . . , 1 -b k ) [-1] = (a 1 , . . . , a h )(1 -a 1 , . . . , 1 -a h )(b 1 , . . . , b k ) [1] (1 -b 1 , . . . , 1 -b k ) [-1] (a 1 , b 1 )(1 -a 1 , 1 -b 1 )(a 1 , . . . , a h , 1 -b 1 , . . . , 1 -b k ) [1] (1 -a 1 , . . . , 1 -a h , b 1 , . . . , b k ) [-1] = (a 1 , . . . , a h , 1 -a 1 , . . . , 1 -a h )(b 1 , . . . , b k , 2n + 1 -b 1 , . . . , 2n + 1 -b k )
We can now finish the proof of Lemma 6.3 by induction on l W ( Cn) (w). If w = ρ 1 ρ 2 . . . ρ l is a decomposition of w as a product of reflections in W ( Cn ), with l = l W ( Cn) (w), then ρ 1 W ( Cn) w, so that ρ 1 W ( Ã2n-1 ) w and by Lemma 6.4 we have f (ρ 1 w) = f (w) -1. Since f (ρ 1 w) = l W ( Cn) (ρ 1 w) by induction and l W ( Cn) (ρ 1 w) = l W ( Cn) (w) -1, we get the result.

We now end the proof of Proposition 6.2. First, since l W ( Cn) = f on P σ , Lemma 6.4 shows that for any w ∈ P σ and any reflection ρ ∈ W ( Cn ) such that ρ W ( Ã2n-1 ) w we have ρ W ( Cn) w.

Assume now that w W ( Ã2n-1 ) w ′ W ( Ã2n-1 ) c with w = 1 and w, w ′ ∈ W ( Cn ), so that w, w ′ ∈ P σ ; choose a reflection ρ in W ( Cn ) such that ρ W ( Cn) w, hence ρ W ( Ã2n-1 ) w W ( Ã2n-1 ) w ′ and ρw W ( Ã2n-1 ) ρw ′ , so that in particular ρ W ( Cn) w ′ by what we have just proved. We can assume by induction on l W ( Ã2n-1 ) (w) that ρw W ( Cn) ρw ′ . Since l W ( Cn) (ρw) = l W ( Cn) (w) -

1 and l W ( Cn) (ρw ′ ) = l W ( Cn) (w ′ ) -1, we get l W ( Cn) (w) + l W ( Cn) (w -1 w ′ ) = 1 + l W ( Cn) (ρw) + l W ( Cn) ((ρw -1 )(ρw ′ )) = 1 + l W ( Cn) (ρw ′ ) = l W ( Cn) (w ′ ) so that w W ( Cn) w ′ .
Corollary 6.5.

(i) The germ P σ is in one-to-one correspondence with {w ∈ W ( Cn ) | w W ( Cn) c} and this correspondence is compatible with divisibility.

(ii) The monoid M(P σ ) has a presentation with set of generator R in one-to-one correspondence r → r with the set R of reflections r ∈ W ( Cn ) such that r W ( Cn) c and relations r 0 r 1 . . . r n = r ′ 0 r ′ 1 . . . r ′ n for any two n-tuples of reflections such that r 0 . . . r n = r ′ 0 . . . r ′ n = c. Assertion (i) says that the germ P σ is obtained from W ( Cn ) and the balanced element c by the method of the "generated group".

Proof. Assertion (i) is precisely the content of Proposition 6.2.

Assertion (ii) is always true for a monoid obtained by the method of the generated group. We give a proof for the sake of completeness. If w ∈ W ( Cn ) is such that w W ( Cn) c, we denote by w the corresponding element of the germ P σ . For such a w there exist a sequence of reflections r 0 , . . . , r n and a non-negative integer k ≤ n such that w = r 0 r 1 . . . r k and c = r 0 . . . r n , so that we have w = r 0 . . . r k . A defining relation of M(P σ ) such as w.w ′ = ww ′ can be written r 0 . . . r n = c if w = r 0 . . . r k , w ′ = r k+1 . . . r l and c = ww ′ r l+1 . . . r n , using the cancellability of the germ. This means that we can reduce the set of generators to R and have the presentation given in (ii). Definition 6.6. We call dual monoid of type Cn the monoid M(P σ ). Remark 6.7. Note that since all Coxeter elements are conjugate in W ( Cn ) changing the Coxeter element in the presentation of the dual monoid given in Corollary 6.5(ii) leads to isomorphic monoids.

Hurwitz action; presentations

In this section we show that the Artin-Tits group of type Cn is a Garside group: more precisely it is the group of fractions G(P σ ) of the Garside monoid M(P σ ). We also get simpler presentations for the monoid and the group than in Corollary 6.5(ii). For this we study the Hurwitz action on reduced decomposition of elements of P σ . Recall if g is an element of some group G, the braid group with n stands acts on the set of decompositions of g into a product of n elements of G in the following way: if (g 1 , . . . , g n ) is such that g = g 1 g 2 . . . g n , the action of the elementary braid s i maps (g 1 , . . . , g n ) onto (g 1 , . . . , g i-1 , g i g i+1 g -1 i , g i , g i+2 , . . . , g n ). This action is called the Hurwitz action. Our first aim is to prove the following result where we call "reduced decomposition" of an element w ∈ W ( Cn ) a decomposition of w into a product of l W ( Cn) (w) reflections.

Theorem 7.1. Let c be a Coxeter element of W ( Cn ) and w be a divisor of c; then the Hurwitz action is transitive on the set of reduced decompositions of w.

Proof. Note first that, since a conjugate of a reflection is a reflection, the Hurwitz action on reduced decompositions of an element w is well defined.

By induction on l W ( Cn) (w) it is sufficient to show that, starting with some fixed reduced decomposition of w, if ρ is a reflection which divides w we can get by the Hurwitz action a decomposition of w which begins with ρ.

Since by the Hurwitz action we can bring to the first place any reflection which appears in a reduced decomposition of w, it is enough to show that for any reflection ρ which divides w in W ( Cn ) we can get by the Hurwitz action a reduced decomposition of w involving ρ.

Since all Coxeter elements are conjugate in W ( Cn ) (see [6, §6, n • 1, Lemme 1]) and since the Hurwitz action is compatible with conjugation, we may assume that c = σ 0 σ 2 σ 4 . . . σ 1 σ 3 σ 5 . . . with the same notation as in Section 6. We view W ( Cn ) as W ( Ã2n-1 ) σ as before. We write w = w 1 . . . w r with w i as in 6.3. We get a reduced decomposition of w by concatenation of reduced decompositions of the w i . A reflection which divides w divides some w i , so it is sufficient to show the result for w = w i which we now assume. The element w is either a σ-stable finite cycle, or the product of a finite cycle and its image by σ or a σ-stable pseudo-cycle. In the first case w can be written (a 1 , . . . , a p , 2kn + 1a 1 , . . . , 2kn + 1 -a p ), with k ∈ Z, all a i odd and a 1 < a 2 < • • • < a p < a 1 + 2n, so is a Coxeter element of the Coxeter group of type C p generated by (a i , a i+1 )(2kn + 1 -a i , 2kn + 1 -a i+1 ) for i = 1, . . . p -1 and (a 1 , 2kn + 1 -a 1 ). In the second case w can be written (a 1 , . . . , a r , . . . , a p )(1 -a 1 , . . . , 1 -a r , . . . , 1 -a p ) with a i odd for i ≤ r and a i even for i > r, and the inequalities a 1 < a 2 < • • • < a r < a 1 + 2n and a r+1 > . . . > a p > a r+1 -2n, so that w is a Coxeter element of the Coxeter group of type A p-1 generated by (a i , a i+1 )(1 -a i , 1 -a i+1 ) for i = 1, . . . p -1. In the third case w can be written (a 1 , a 3 . . . , a 2p-1 ) [1] (a 2 , a 4 , . . . , a 2p ) [-1] with the a 2i-1 odd and a i + a 2p+1-i = 2n + 1, so that w is a Coxeter element of the Coxeter group of type Cp generated by (a 1 , a 2p ), the products (a i , a i+1 )(a 2p-i , a 2p+1-i ) for i = 1, . . . , p-1 and (a p , a p+1 ). Hence w is a Coxeter element of a Coxeter subgroup W of type B, C or C; the reflections of W are reflections of W ( Cn ) and a reflection of W ( Cn ) divides w in W ( Cn ) if and only if it divides w in W . A reduced decomposition of w in W ( Cn ) is thus a reduced decomposition of w in that Coxeter subgroup.

Since we know that the Hurwitz action is transitive on the reduced decomposition of an element in groups of type A or C (see [2, Proposition 1.6.1]), the only case which remains is the case of a Coxeter element of a group of type C: we have only to study the case where w is c itself. We start with the reduced decomposition of c given by (σ 0 , σ 2 , σ 4 , . . . , σ 1 , σ 3 , σ 5 , . . .). We remark that if we delete σ 0 (resp. σ n ) from this decomposition we get a reduced decomposition of a Coxeter element of the Coxeter group W ′ of type C n generated by σ 1 , σ 2 , . . . , σ n (resp. of the Coxeter group W ′′ of type C n generated by σ 0 , σ 1 ,. . . ,σ n-1 ). Using the transitivity of the Hurwitz action for Coxeter groups of type C n , we know that by the Hurwitz action we can make appear any reflection of W ′ or W ′′ in some reduced decomposition of c.

We remark that if c = ρ 1 . . . ρ n+1 is a reduced decomposition, the Hurwitz orbit of (ρ 1 , . . . , ρ n+1 ) contains (ρ 2 , . . . , ρ n+1 , c -1 ρ 1 c) so that by the Hurwitz action for any n ∈ Z and any i ∈ {1, . . . , n + 1} we can get a reduced decomposition of c involving c n ρ i c -n . So we will get our result if we prove that any reflection appearing in a reduced decomposition of c is conjugate by some power of c to a reflection of W ′ or of W ′′ .

The action of c by conjugation on permutations of Z is induced by the translations 2i → 2i-2 and 2i + 1 → 2i + 3 for all i. The reflections of

W ′ are (a, b)(1 -a, 1 -b) with 1 ≤ a < b ≤ 2n and a + b = 2n + 1, (a, 2n + 1 -a) with 1 ≤ a ≤ 2n. The reflections of W ′′ are (a, b)(1 -a, 1 -b) with 1 -n ≤ a < b ≤ n and a + b = 1, (a, 1 -a) with 1 -n ≤ a ≤ n.
)( 1-a-b 2 , 3-a-b 2 ) 2
which is a reflection of W ′ . Last of all we can conjugate (a, 2kn + 1 -a) where k is arbitrary to (kn, kn + 1) which is equal either to (n, n + 1) which is in W ′ or to (0, 1) which is in W ′′ . Remark 7.2.

(i) One can conjecture that the Hurwitz action is transitive on the reduced decompositions of a Coxeter element in any Coxeter group. (ii) We have seen in the proof of Theorem 7.1 that a divisor of a Coxeter element of W ( Cn ) is a Coxeter element of a Coxeter subgroup which is a direct product of groups of type A, C or C (there can be at most one component of type C and there cannot be at the same time a component of type C and a component of type C).

Using Theorem 7.1 we can simplify the relations in the presentations of M(P σ ) and G(P σ ) given in Corollary 6.5(ii). As in Corollary 6.5(ii) and in its proof we denote by w the element of P σ corresponding to an element w ∈ W ( Cn ) dividing c. From this presentation we deduce one of our main result, which is that the Artin-Tits group G( Cn ) is isomorphic to G(P σ ). We use the notations R and R as defined in Corollary 6.5. Proof. By Corollary 6.5(ii) the monoid M(P σ ) has a presentation with set of generators R and relations given by the equalities between the lifts of any two reduced decompositions of c. We have to prove that one can pass from a reduced decomposition of c to another one by applying only relations of the form r.t = (rtr).r. But this is precisely the transitivity of the Hurwitz action, whence (i). Since all the simple reflections of W ( Cn ) are in R, we have elements σ i ∈ R for i = 0, . . . , n, where as in Section 6 the simple reflections of W ( Cn ) = W ( Ã2n-1 )

σ are σ i = s i s 2n-i for i = 2, . . . , n -1, σ n = s n and σ 0 = s 2n . The natural morphism M(P σ ) → M(P ) σ extends to a group morphism f : G(P σ ) → G(P ) σ . We know that G(P ) is the group G( Ã2n-1 ) so that G(P ) σ is the group G( Cn ) by Theorem 5.2. The morphism f maps σ i to σ i for i = 2, . . . , n -1 and maps σ n to σ n and σ 0 to σ 0 . We claim that σ 0 ,. . . , σ n satisfy the braid relations of type Cn . Indeed, for any pair of distinct simple reflections σ i and σ j , we have σ i σ j W ( Cn) c, up to swapping i and j; if σ i and σ j , commute, then σ i σ j W ( Cn) c so that σ i σ j = σ j σ i in P σ ; if σ i and σ j satisfy a braid relation of length 3 we have σ j .σ i .σ j = σ j .σ i σ j σ i .σ i = σ j σ i σ j σ i σ j .σ j .σ i = σ i .σ j .σ i ; if σ i and σ j satisfy a braid relation of length 4 we have σ i .σ j .σ i .σ j = σ i .σ j .σ i σ j σ i .σ i = σ i .σ j σ i σ j σ i σ j .σ j .σ i = σ i σ j σ i σ j σ i σ j σ i .σ i .σ j .σ i = σ j .σ i .σ j .σ i . Hence there exists a morphism g : G( Cn ) → G(P σ ) which maps σ i to σ i for i = 0, . . . , n. We have then f • g = Id since G( Cn ) is generated by {σ i , i = 0 . . . , n}.

On the other hand the transitivity of the Hurwitz action and the presentation of G(P σ ) imply that one can express any r where r W ( Cn) c as a conjugate of some σ i by a product of elements of the form σ ±1 j , so that G(P σ ) is generated by {σ i , i = 0, . . . , n}. This implies that g is surjective, so that f and g are isomorphisms.

8. Some consequences Among consequences of our results we can recover the known fact that the center of B( Cn ) is trivial. The Garside structure gives also a solution to the word problem in this group. As an illustration of the use of a Garside structure we compute the centralizer of powers of a Coxeter element in an Artin-Tits group of type Cn . Proposition 8.1. With the notation of section 5, let c = σ 0 σ 1 . . . σ n ; then for any h ∈ Z-{0}, the centralizer of c h in the Artin-Tits group G( Cn ) is isomorphic to the Artin-Tits group of type C gcd(h,n) .

Proof. It is a general result (see e.g., [4, Proposition 2.26]) that the monoid of fixed points M ϕ under an automorphism ϕ of a Garside monoid M fixing the Garside element ∆ is a Garside monoid with same Garside element (this can also be seen as a particular case of Corollary 2.7): if P is the set of divisors of ∆ in M then the set of divisors of ∆ in M ϕ is P ϕ . In this situation the group of fixed points of ϕ in the group G of fractions of M is the group of fractions of M ϕ . We apply this to M = M(P σ ) and G = G(P σ ), taking for ϕ the conjugation by c h . As in section 4 we identify the elements of P σ with the elements of W ( Cn ) corresponding to the 2n-periodic, non-crossing σ-stable partitions of X ∪ Ξ, where X = 1 + 2Z and Ξ = 2Z. Recall that c is the permutation given by x → x + 2 for x ∈ X and ξ → ξ -2 for ξ ∈ Ξ.

Let p be a σ-stable non-crossing partition stable by the action of c h . If A is a part of p contained in X or in Ξ, then A + 2kh is also a part of p for all k ∈ Z. If A is a part of p which

Figure 4 .

 4 Figure 4. A non-crossing product of a cycle and its image by σ in Ã9

If a and b have

  same parity with a < b ≤ a + 2n, we can conjugate (a, b)(1 -a)(1 -b) ∈ P σ by a power of c to (1, b + 1 -a)(0, a -b) which is a reflection of W ′ . If a and b have different parities and a + b = 1 mod 2n we can conjugate (a, b)(1 -a, 1 -b) by a power of c to ( a+b-1 2 , a+b+1

Theorem 7. 3 .

 3 (i) The monoid M(P σ ) has the following monoid presentation by generators and relations:M(P σ ) =< R | r.t = rtr.r if r, t ∈ R and rt W ( Cn) c > + (ii)The Artin-Tits group of type Cn is isomorphic to G(P σ ); in particular it is the Garside group of the Garside monoid M(P σ ) and has the group presentation G( Cn ) =< R | r.t = rtr.r if r, t ∈ R and rt W ( Cn) c >.

This work was partially supported by the "Agence Nationale pour la Recherche" project "Théories de Garside" (number ANR-08-BLAN-0269-03). 1

intersects both X and Ξ and is not c h -stable, then A crosses its image under the action of c h , which is impossible since two distinct parts of p are non-crossing. Hence any part A of p has to be invariant by the action of c h which is equivalent to A = A + 2kh for all k ∈ Z. Hence a part of a partition p w associated to an element w ∈ (P σ ) c h (see Proposition 3.9) is either an infinite part both 2n and 2h periodic (and there is at most one such part in p w ) or a finite part A contained in X or in Ξ and such that A + 2kh and A + 2kn are parts of p w for any k. Conversely if all the parts of a σ-stable non-crossing partition p w are of one of these form then w is centralized by c h .

From the above study we see that the centralizer of c h is equal to the centralizer of c gcd(n,h) . Hence to prove the proposition we are reduced to the case where h divides n.

We first study the case h = n.

Lemma 8.2. The centralizer of c n is an Artin-Tits group of type C n .

Proof. From the above discussion we see that an element w ∈ (P σ ) c n is a product of pairwise commuting and non-crossing elements of the form either (2a 1 + 1, 2a 2 + 1, . . . , 2a r + 1)(-2a 1 , -2a 2 , . . . , -2a s , . . . , -2a r ) with a 1 < a 2 < . . . < a r < a 1 + n, or (2a 1 + 1, 2a 2 + 1, . . . , 2a r + 1) [START_REF] Bessis | Complex reflection arrangements are K(π, 1)[END_REF] (-2a 1 , -2a 2 , . . . , -2a r ) [-1] with 0 < a 1 < a 2 < . . . < a r ≤ n. Hence all elements of (P σ ) c n can be written x. σ x where x is a permutation of the set X of odd integers and σ x is a permutation of the set Ξ of even integers. Since a permutation of X and a permutation of Ξ are always non-crossing, two such elements x. σ x and y. σ y in (P σ ) c n are non-crossing if and only if x and y are non-crossing. In [10, Proof of 5.10] we have described the germ for the dual presentation of an Artin-Tits group of type C n as the set of n-periodic, non-crossing permutations of Z. From this description we see that under the bijection 2k + 1 → k from X to Z, the map x. σ x → x defines an isomorphism of germs from (P σ ) c n to the dual germ for the Artin-Tits group of type C n .

We deduce the proposition using the fact that the centralizer of c h for a divisor h of n is the centralizer of c h in the centralizer of c n so that it is the centralizer of c h in an Artin group of type C n , which is known to be an Artin-Tits group of type C h (see [START_REF] Bessis | Springer theory in braid groups and the Birman-Ko-Lee monoid[END_REF]).


L Biadala

Ph. Tamarat Y Louyer

B Lounis
email: blounis@u-bordeaux1.fr

Band-edge exciton fine structure of single CdSe/ZnS nanocrystals under external magnetic fields

We report a spectroscopic study of the two lowest-energy exciton levels of individual CdSe/ZnS nanocrystals under applied magnetic fields. Field-induced coupling between the bright and the dark excitonic states is directly observed in the low temperature photoluminescence spectrum and decay, and allows the determination of the angle between the nanocrystal c-axis and the field. Orientation dependent Zeeman splittings of the dark and bright exciton sublevels are measured and provide the corresponding exciton Landé factors, as well as spin-flip relaxation rates between Zeeman sub-levels.

Over the past decade, significant efforts have been spent on magneto-optic properties of semiconductor quantum dots, motivated by the need to understand the fundamental spin structure of the quantum dots electronic states and by potential applications in solid-state spintronics [1] and quantum information processing [2]. Among quantum dots, chemically synthesized CdSe nanocrystals (NCs) have already shown suitable properties for nanoscale electronics [3][4][5], laser technology [6], and biological fluorescent labelling [7,8]. They are particularly attractive due to their bright luminescence which is size tunable across the visible spectrum.

It is well established that the photoluminescence (PL) of spherical or nearly spherical NCs at low temperature arises from the two lowest levels of the band-edge exciton fine structure, namely the long-lived ground excitonic state F (also called the "dark" state, with a total spin projection J = ±2 on the crystal c-axis), and the lowest-energy optically active state A (J= ±1 L ) which lies a few meV above the ground state [9][10][START_REF] Klimov | Semiconductor and Metal Nanocrystals:Synthesis and Electronic and Optical Properties[END_REF]. Indeed, recent studies based on temperature dependence of PL spectra [START_REF] Biadala | [END_REF]13] and PL decay [START_REF] Biadala | [END_REF], as well as polarizationresolved spectroscopy [14], showed that luminescence of individual CdSe NCs with cylindrical symmetry display two sharp zero-phonon lines (ZPLs) which are attributed to purely radiative recombination from the two lowest (bright and dark) exciton states. Yet, the origin of the F ZPL remains an open question.

Magneto-optical methods largely contributed to the understanding of the NCs band edgeexciton fine structure. Most of the magneto-optical studies have been applied to ensembles of NCs [9,[15][16][17][18], where averaging over orientations and size distributions reduces the amount of available quantitative information. As a consequence, Zeeman splittings of fine structure exciton levels are not resolved, and the corresponding Landé g-factors are not well known. A recent low temperature study of single NCs under magnetic fields allowed measurements of Zeeman shifts within an exciton state doublet attributed to a distortion-induced splitting of the bright state [19].

In this letter, we report a cw and time resolved spectroscopic study of individual CdSe NCs in an external magnetic field (up to 7 T). Besides the direct observation of the fieldmixing between the two lowest fine structure states, we measure the Zeeman energy splitting of the dark and bright exciton states, extract their g-factors and determine the spin relaxation rates between dark exciton Zeeman sublevels. Furthermore we observe an unexpected lifetime lengthening of the dark excitonic state with increasing fields which suggests that the dark exciton ZPL takes its origin in an intrinsic magnetic coupling of bright and dark exciton states.

Samples of CdSe/ZnS core-shell NCs (Qdot655 from Quantum Dot Corp., peak emission at 655 nm at room temperature) are prepared by spin coating clean glass coverslips with a nanomolar solution of NCs in polyvinyl alcohol. A homebuilt scanning confocal microscope is used to image single NCs excited with the 532 nm line of a cw frequency doubled Nd:YAG laser. It is based on a 0.95 NA objective which is inserted in a magnetic cryostat together with the sample and a piezoscanner. The objective axis is perpendicular to the magnetic field, allowing magneto-optical studies in the Voigt configuration. The emitted photons are filtered from the scattered excitation light by a bandpass filter (60 nm FWHM) and sent to a singlephoton-counting avalanche photodiode and a spectrometer. The PL decay measurements are performed with a conventional time correlated single photon counting setup, using a pulsed laser source (optical parametric oscillator at 570 nm, 6 ps pulse width, 76 MHz repetition rate) and a pulse picker which reduces the repetition rate. At 2 K, the luminescence intensity of most of the single NCs is remarkably stable and did not show any blinking. This high photostability allows the acquisition of resolution-limited PL spectra of single NCs. The NCs that we studied display two ZPLs in their zero-field emission spectrum [START_REF] Biadala | [END_REF][13][14] as shown in the lowest spectrum of Fig. 1(a). These ZPLs, named F and A, have been attributed to the radiative recombination from the two lowest levels of the band-edge exciton fine structure. Tens of single NCs were studied as a function of external magnetic field. We observed various spectroscopic behaviors which result from different relative orientations between the NC c-axis and the magnetic field.

Fig. 1 (a) shows the evolution of a single NC PL spectrum as a function of external magnetic field. This NC does not display a resolved Zeeman splitting of the ZPLs, but only a broadening of the F ZPL at large magnetic fields. Moreover, the weight of this ZPL dramatically increases upon raising field from zero to 7 T [Fig. 1 (c)]. These observations suggest that the magnetic field is nearly perpendicular to the NC c-axis and induces a mixing of the bright and dark exciton states, enhancing the oscillator strength of the latter. In order to determine the relative angle between field and crystal axis, we recorded the field-dependent PL decay of this NC at 2 K. The decay is biexponential with a fast decay occurring in the ns range and a much slower one with a lifetime of hundreds of ns at zero field [START_REF] Biadala | [END_REF]20,21]. The long decay time, which is assigned to the dark exciton radiative recombination, drastically shortens with increasing fields (from ~500 ns at 0 T to ~100 ns at 7 T, in Fig. 1(b)). Indeed, the magnetic coupling opens an additional channel for ground exciton recombination via admixture in the F state of the A state. The decay rate of the dark exciton in a magnetic field B is given by [10]:

  0 4 3 cos 2 1 cos 1 cos 2 1 1 1 0 2 2 1            F F            with    3 / B g e B  , (1)
 being the exchange interaction energy ( ~ 0.28 meV for 4 nm radius NCs [10]), B  the Bohr magneton and e g ~ 0.9 the electron g-factor [22][23][24].

  A markedly different spectroscopic behavior is obtained for small angles between the magnetic field and the NC c-axis. As displayed in Fig. 2 (a), a well resolved Zeeman splitting of the dark exciton sublevels J=±2 is indeed observed in the emission spectra of single NCs, leading to 2 ZPLs named F + and F -. The energy splitting is linear with field [Fig. 2 ~3 ( h g being the hole g-factor) [22], and larger than the value ~ 0.9 deduced from PL polarization measurements on ensembles of 4 nm-radius NCs at 1.45 K [17]. At high magnetic fields (B  5 T) a weak Zeeman splitting of the bright exciton sublevels shows up in the PL spectra. The bright exciton g-factor A g is then estimated to A g ~ 0.5 for this NC. This value is smaller than A g ~ 1 deduced from time-resolved Faraday rotation studies performed on NC ensembles [22]. Theoretical calculations led to disparate g-factors which depend on the NC shape and size, as well as surface boundary conditions [22][23][24]. The g-factors are also expected to display anisotropy [23,26]. This could explain the large difference between the A g of our NCs with cylindrical symmetry and that obtained on non symmetric NCs [19].

An interesting feature of the emission spectra of Fig. 2 (a) is the field-dependent intensity of the dark exciton Zeeman components. The weight of the higher energy ZPL (F + )

clearly drops with increasing fields. This is a signature of a spin-flip relaxation between dark exciton Zeeman sub-levels which occurs with absorption and emission of acoustic phonons.

Indeed, the existence of such spin-flip relaxations has previously been demonstrated from polarization-resolved magneto-photoluminescence studies of the dark exciton ground state on NC ensembles [START_REF] Klimov | Semiconductor and Metal Nanocrystals:Synthesis and Electronic and Optical Properties[END_REF]17]. The spin-flip rates can be determined with a simple thermal mixing model [START_REF] Biadala | [END_REF]13] of , where

    1 exp 1    T k E N B F B
is the Bose-Einstein number of phonons whose energy matches the Zeeman splitting

     ) ( 1 2 2 ) ( 1 1 B N B N F B F B       
. We can estimate the spin-flip rate at different energy splittings: is likely due to the decrease of the phonon density of states [27]. Indeed, a field induced coupling between J=+2 and J=-2 states is unlikely for these nanocrystals with c-axis nearly parallel to B  , although a high order effect cannot be completely ruled out. It is interesting to compare values of 1  to the zero-temperature spin-flip rate 0  between bright and dark exciton states in zero field. Typical values of the order of 0  ~ 1 ns -1 have been measured on those NCs [START_REF] Biadala | [END_REF], where the energy splitting between bright and dark states lies in the meV range. Such a large difference (~ two order of magnitudes) between 1  and 0  is related to the fact that high order phonon processes might be involved in

2 2      J J spin-flip mechanisms.
In Fig. 3 we present the magnetic field dependence of the dark exciton energy splitting and decay rate of several NCs having different orientations. For comparison, decay rates are normalized to their zero-field values. The orientation  of each NC is obtained from the fit of the decay rate evolution with magnetic field for B  1 T, taking 0  =2 ns in expression (1) [see Fig. 3 (a)]. For all the NCs, the Zeeman splitting [see Fig. 3 (b)] can be well reproduced using the same value of F g = 2.7. Note that varying the value of 0  in the range 1-4 ns also gives acceptable fits of the decay rates and leads to a confidence interval of 2.2-2.9 for F g values. Interestingly, we observe that NCs with small orientation angles  display a lengthening of the dark exciton lifetime with increasing fields for B  1 T. This observation indicates that the longitudinal component of magnetic field decreases the F exciton oscillator strength by reducing an intrinsic contamination of the dark state by the optically active state. It also suggests that this intrinsic admixture takes its origin from an interaction with magnetic defects in the NC which can generate strong internal magnetic fields, enabling zero-phonon recombination to occur [10]. The external field tends to orientate the intrinsic magnetic moment of these defects along its direction. If this direction is close to the c-axis, the effect of a moderate external field is therefore a reduction of the contamination of F by A . At large fields and for 0   , the normal component of external field will then strengthen the mixing between A and F states. This behavior can be qualitatively reproduced with a simple orientation model of an intrinsic magnetic moment in a field which In the same vein, a longitudinal spin relaxation time lengthening with magnetic field followed by shortening has been reported by Gupta et al. in a time-resolved Faraday rotation experiment performed on NCs ensembles [22]. This observation was tentatively explained with a mechanism involving the hyperfine interaction with unpolarized nuclear spins.

However, the hyperfine field contribution is most likely negligible in CdSe [28] since only a small fraction of ions (25% of Cd ions) have nuclear spins (and these spins are only 1/2). Instead, magnetic defects which potentially generate strong effective internal magnetic fields might be involved [10]. Magnetic properties have indeed been reported for CdSe NCs and attributed to a variety of sources, including dangling bonds [29], surface ligands [30,31], and defects in the nanocrystals [32]. Further magneto-optical investigations and theoretical models are needed to elucidate the physical origin of these large internal magnetic fields.

In summary, various spectroscopic behaviors were found when applying an external magnetic field to single CdSe NCs, depending on the relative orientation of the NC c-axis in the field. For the first time, coupling between the two lowest energy excitonic levels, as well as Zeeman splittings of these levels were clearly identified on single NCs and allow estimations for the bright and the dark excitons g factors. In further investigations, Zeeman splittings of exciton sublevels will be precious tools to probe the density of states of acoustic phonons as a function of energy, without the need to change the NC size. The ability to tune The decay rates are normalized to the zero-field decay rate. The data relative to the NC presented in Fig. 1 (squares), and that presented in Fig. 2 (circles), are included. The angle  between the magnetic field and the NC c-axis is derived for each NC from fits of the decay rates with expression (1) at fields larger than 1 T, taking

  3  10 -3 ns -1 is the zerofield radiative decay rate of the dark exciton, and 1 0   the decay rate of the J=0 fine structure state of the band-edge exciton. Taking

  ns[10,25], and using  as a fitting parameter, the field dependence of the long component decay rate is reproduced by the expression (1), for  = (75 ± 5)° [see Fig 1(c)].

  Fig.2 (c)]. From the linear Zeeman splitting we further deduce the dark exciton g-factor

1 

 1 the zero-temperature relaxation rate of the transition    F F . Assuming identical pumping rates to the states  at a magnetic field B, we use rate equations to derive the populations of the states  F and  F in the stationary regime and at low saturation. The relative intensity of the luminescence arising from  F in the dark exciton emission is given by 

1 

 1 ~1

  is the superposition of the external field ext B  and a weak internal field 0 B  responsible for the alignment of the defects magnetic moments. In the NC, the field i B  created by this magnetic moment adds to the external field, resulting in an effective field which writes this expression for B in equation (1), the field evolution of the dark exciton lifetime is reproduced with 0 B = 0.15 T oriented perpendicularly to the c-axis, 10 -3 ns -1 (see the solid curve of Fig. 2 (c)).

Fig. 1 :

 1 Fig. 1: (a) PL spectra of a single NC in a magnetic field nearly perpendicular to the c-axis. The lines labelled A and F are the recombination ZPLs of the bright and the dark excitons, respectively. (b) PL decay of this NC for various intensities of the magnetic field. (c) Circles: Field-induced shortening of the PL long decay time. The solid curve is a fit which leads to the relative angle  = (75 ± 5)° between the NC c-axis and the field, taking 2 0  

Fig. 2 :

 2 Fig. 2: (a) PL spectra of a single NC at 2 K, for various intensities of the magnetic field which is almost parallel to the c-axis. (b) Zeeman energy splittings of the dark (F) and bright (A) exciton sublevels. Linear fits of Zeeman splittings yield

Fig. 3 :

 3 Fig. 3: (a) Evolution of the long decay rate as a function of magnetic field, for five NCs.

  splitting of the dark exciton sublevels, for the same NCs. The solid lines are the calculated Zeeman splittings, taking the angles from Fig.3(a), and F g = 2.7 for all NCs.

energy levels and spin relaxation rates within the band-edge exciton fine structure of single NCs has also potential applications in quantum information processes. This work was funded by the Agence Nationale de la Recherche, Région Aquitaine, the French Ministry of Education and Research, and the European Research Council.


Yann Louyer

Louis Biadala

Philippe Tamarat

Brahim Lounis
email: blounis@u-bordeaux1.fr

Spectroscopy of Neutral and Charged Exciton States in Single CdSe/ZnS Nanocrystals

We report a low temperature spectroscopic study of the lowest-energy exciton states in single CdSe/ZnS colloidal nanocrystals. In their neutral charge state, the zero-phonon lines of the lowest two levels of the band-edge exciton fine structure are directly observed in the photoluminescence spectrum. The attribution of these lines is confirmed by polarization-resolved studies of the photoluminescence spectrum and decay. We also identified the emission from charged nanocrystals which displays a single zero-phonon line ~17 meV red-shifted with respect to the neutral nanocrystal emission lines. This bright trion state has a lifetime which ranges from 1.5 to 4.5 ns.

Quantum-confined nanoparticles have been increasingly investigated over the past decade due to the superior efficiency and tunability of their emission wavelength from the ultraviolet to the near infra red. Among those nanoparticles, colloidal CdSe nanocrystals (NC) are particularly attractive for many applications such as nanoscale electronics [1][2][3], laser technology [4], and quantum cryptography [5,6].

These applications require a detailed understanding of both the spectral and polarization properties of NC emission and, specifically, the structure of the lowest-energy band-edge exciton states.

In neutral and spherical colloidal CdSe NCs, the band-edge exciton is split by the intrinsic electron-hole exchange interaction and by the crystal field of the hexagonal lattice into five energy levels [7]. The exciton ground state F , which is often referred to as the "dark" exciton (J=±2), is located a few meV below the lowest-energy optically active exciton state A . Recent investigations of low temperature PL spectra from individual CdSe NCs led to the observation of a zero-phonon-line (ZPL)

doublet with an energy splitting in the meV range [8][9][10]. Different attributions of these sharp emission lines were proposed. Htoon at al. [8] found that the two lines are characterized by linearly and orthogonally polarized emission dipoles, and attributed them to emission from two linear combinations of the J=±1 L bright exciton sublevels. They suggested that the splitting originates from an anisotropic electron-hole exchange interaction resulting from a breakdown of cylindrical symmetry in the NCs. This distortion-induced splitting is often referred to as the X-Y splitting and has been well known for epitaxial quantum dots [11][12][13][14]. On the other hand, the ZPL doublet reported in Refs [9,10] has been attributed to the radiative recombination from the lowest bright and dark states of the band-edge exciton fine structure. In this letter, we report single CdSe/ZnS NCs polarization-and time-resolved PL studies which fully support the latter assignment. For some NCs, we also identified transitions to a charged exciton state (trion), and measured for the first time this trion lifetime on single NCs. ). This temperature dependence shows evidence for a thermal mixing between two excitonic states having a two orders of magnitude difference in their decay rates [15]. It can be reproduced within the frame of a three-level model taking into account the ground state and the two thermally mixed emitting states [9]. The fit (solid curve) of the experimental data gives the bright state radiative decay rate 0.22 ns -1 , and the bright-to-dark spin flip rate 1.0 ns -1 . This reduced spin relaxation rate is due the small phonon density of the states at the scale of the inter-level splitting [9]. The weight of the long component represents ~90% of the PL decay at 2 K. One can deduce that it arises from the dark exciton radiative recombination since the F line and its phonon replica correspond to the same fraction (~90% ) in the luminescence spectrum. Such observations cannot be reproduced using a model with two X,Y bright states having oscillator strengths of the same order of magnitude.

Samples of

We also performed polarization-resolved studies of the PL spectra and the PL decay for several single NCs, using a half-wave plate and a Glan polarizer on the PL beam pathway. As examplified in Fig. 2, we observed various behaviors which probably result from different relative orientations between the NC hexagonal axis and the laser field. For ten NCs studied, we could not find a single NC for which the A and F lines are linearly and orthogonally polarized, as one would expect if these lines stemmed from the X,Y states. Furthermore, we could not attenuate the intensity of the F line by more than a factor of 2 upon rotating the half wave plate, whereas the intensity of the A line could be modified by a larger factor (e.g. ~4 for the NC of Fig. 2 (a), ~ 10 for the one of Fig. 2 (b)). According to theoretical calculations using the pseudopotential configuration-interaction method [16], the emission from the bright exciton is polarized perpendicular to the hexagonal NC axis, whereas the degree of linear polarization of the dark exciton decreases with the NC size as a result of an increasing polarization component along the crystal axis. Our polarization measurements therefore support the attribution of the F line to the long-lived dark exciton state. Moreover, polarized decay measurements show that the weight of the short time component follows that of the A line of the polarized emission spectrum.

Another important result obtained with these NCs is the spectroscopic signature of a charged excitonic state. Among hundreds of single NCs studied, we found 17 NCs for which the emission switched between the A, F doublet and a red-shifted single ZPL X* (Fig. 3 (a)). We attribute such spectral jumps to transitions between neutral and charged excitonic states. A trion state is indeed prepared when an exciton is created in the presence of an unpaired charge carrier in the NC core and a charge carrier trapped at the surface or in a lattice defect of the NC. As displayed in the inset of Fig. 3 (a), the distribution of spectral shifts between the dark exciton ZPL and the trion ZPL ranges from 15.5 to 18.5 meV. Similar spectral jumps have been previously reported for single CdSe nanorods with a ~2.5

nm radius and a ~ 25 nm length [17], as well as for self assembled CdSe quantum dots [18]. Yet, much smaller jumps (a few meV) have been recently observed by Fernée et al. on single NCs with a ~4 nm radius [10]. The bright trion ZPL is accompanied by a weak LO-phonon replica (weight lower than 7%, data not shown) and confined acoustic phonon sidebands. As shown in Fig. 3 (a), one can distinguish two LA-phonon replica, 1.1 meV and 1.8 meV redshifted from the X* ZPL, which we attribute to acoustic radial breathing modes of the NC [19][20][21].

We also recorded the PL decay of single NCs which were permanently charged. As shown in Fig. 3 (b), the decay is monoexponential with a relaxation time in the range 1.5-4.5 ns with mean value 3.0 ns. These decay times are shorter than the radiative lifetime of the neutral bright exciton (lying in the range 4.5-7.4 ns with a mean value of 6.2 ns, data not shown). This suggests that Auger recombination processes contribute to the charged exciton decay rate. According to pseudo-potential configurationinteraction calculations [16], a single red-shifted ZPL associated to a short lifetime emission (in the nanoseconds range) is a signature of the negative trion state. Recent ensemble measurements on CdSe NCs with a comparable size provided a value of ~ 1 ns for the negative trion lifetime, using an indirect deconvolution method to extract the trion lifetime from multiexponential decays [22].

In summary, our spectroscopic investigations of single CdSe/ZnS NCs demonstrated that the ZPL doublet observed in the low temperature PL spectra correspond to the radiative recombination from the two lowest levels of the band edge exciton fine structure. We also determined the characteristic features of a bright charged state attributed to the negative trion state. Further investigation will aim at determining the optical coherence lifetime of the trion ZPL emission for applications in quantum optics, such as the realization of indistinguishable single photon sources.

Figure 1 (

 1 a) shows an example of individual NC spectra displaying two sharp resolution-limited ZPLs. At 2 K, most of the luminescence (~90 %) stems from the low energy ZPL (denoted F) with its longitudinal acoustic (LA) and optic (LO) phonon replica. This ZPL is attributed to the dark exciton radiative recombination. When raising the temperature from 2 K to 10 K, the high energy ZPL (A) largely prevails over the F line. This striking observation supports a phonon-mediated thermal mixing between two emitting levels which have very different oscillator strengths, and therefore different spin natures (namely J=1 and J=2). Furthermore, one can clearly notice at 3.5 K (Fig.1 (b)) that the relative weights of the LO-phonon sidebands with respect to the ZPLs are significantly different for the two states. This supports the assumption that radiative recombination from the dark state occurs preferentially with emission of phonons.

Figure 1 (

 1 Figure 1 (e) shows the PL decay of this single NC recorded at 2 K. The decay shows a

FIG 1 .FIG. 2 .FIG. 3 .

 123 FIG 1. (a)-(d) Non polarized PL spectrum of a single CdSe/ZnS NC recorded at various temperatures, from 2 K to 10 K, at a CW excitation intensity of 50 Wcm -2 . The peaks labelled F and A are attributed to the ZPLs of the dark and bright excitons, respectively. The peaks LO-F and LO-A are LO-phonon replica of the peaks F and A. The structures LA-F and LA-A are acoustic phonon replica of the peaks F and A. (e) Biexponential PL decay of the same NC at 2 K (peak excitation intensity 1 MW cm -2 ). The decay time of the long component is ~ 700 ns. The evolution of the long component decay rate withtemperature is shown in the inset (square dots). The solid curve is a fit using the model of reference[9].

  MHz repetition rate) and a pulse picker which reduces the repetition rate.At 2 K, most of the single NCs are photostable and do not show any blinking.

	excited with the 532 nm line of a cw frequency doubled Nd:YAG laser. It is based on a 0.95 NA
	objective (diffraction-limited laser spot size of ~ 0.4 µm 2 ) which is inserted in a helium cryostat together
	with the sample and a piezoscanner. The emitted photons are filtered from the scattered excitation light
	by a bandpass filter (60 nm FWHM) and sent to a single-photon-counting avalanche photodiode and a
	spectrometer. The PL decay measurements are performed with a conventional time correlated single
	photon counting setup, using a pulsed laser source (optical parametric oscillator at 570 nm, 6 ps pulse
	width, 76
	CdSe/ZnS core-shell NCs (Qdot655 from Quantum Dot Corp., peak emission at 655
	nm at room temperature) are prepared by spin coating clean glass coverslips with a nanomolar solution
	of NCs in polyvinyl alcohol. A homebuilt scanning confocal microscope is used to image single NCs

Acknowledgement. This research was funded by the Région Aquitaine, the Agence Nationale pour la Recherche (ANR), the French Ministry for Education and Research (MESR) and the European Research Council (ERC).