OSAMU IYAMA, IDUN REITEN Claire Amiot
email: amiot@math.unistra.fr

AND Gordana Todorov
email: g.todorov@neu.edu

PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS

Let Q be an acyclic quiver and Λ be the complete preprojective algebra of Q over an algebraically closed field k. To any element w in the Coxeter group of Q, Buan, Iyama, Reiten and Scott have introduced and studied in [BIRS09a] a finite dimensional algebra Λw = Λ/Iw. In this paper we look at filtrations of Λw associated to any reduced expression w of w. We are especially interested in the case where the word w is c-sortable, where c is a Coxeter element. In this situation, the consecutive quotients of this filtration can be related to tilting kQ-modules with finite torsionfree class.

Introduction

Attempts to categorify the cluster algebras of Fomin and Zelevinsky [START_REF] Fomin | Cluster algebras. I. Foundations[END_REF] have led to the investigation of categories with the 2-Calabi-Yau property (2-CY for short) and their clustertilting objects. Main early classes of examples were the cluster categories associated with finite dimensional path algebras [BMR + 06] and the preprojective algebras of Dynkin type [START_REF] Geiss | Rigid modules over preprojective algebras[END_REF]. This paper is centered around the more general class of stably 2-CY and triangulated 2-CY categories associated with elements in Coxeter groups [START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF] (the adaptable case was done independently in [START_REF]Partial flag varieties and preprojective algebras[END_REF]), and their relationship to the generalized cluster categories from [START_REF] Amiot | Cluster categories for algebras of global dimension 2 and quivers with potential[END_REF] (see Section 4 for the definition).

Let Q be a finite connected quiver with vertices 1, . . . , n, and Λ the complete preprojective algebra of the quiver Q over a field k. Denote by s 1 , . . . , s n the distinguished generators in the corresponding Coxeter group W Q . To an element w in W Q , there is associated a stably 2-CY category Sub Λ w and a triangulated 2-CY category Sub Λ w . The definitions are based on first associating an ideal I i in Λ to each s i , hence to any reduced word by taking products. This way we also get a finite dimensional algebra Λ w := Λ/I w . Objects of the category Sub Λ w are submodules of finite dimensional free Λ w -modules. The cluster category is then equivalent to Sub Λ w with w = c 2 , where c is a Coxeter element such that c 2 is a reduced expression [START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF][START_REF]Partial flag varieties and preprojective algebras[END_REF]. When Λ is a preprojective algebra of Dynkin type, then the category mod Λ as investigated in [START_REF] Geiss | Rigid modules over preprojective algebras[END_REF] is also obtained as Sub Λ w where w is the longest element [START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF]III 3.5].

Using the construction of ideals we get for each reduced expression w = s u 1 s u 2 . . . s u l a chain of ideals

Λ ⊃ I u 1 ⊃ I u 2 I u 1 ⊃ . . . ⊃ I w ,
which gives rise to an interesting set of Λ-modules:

L 1 w := Λ I u 1 , L 2 w := I u 1 I u 2 I u 1
, . . . , L l w := I u l-1 . . . I u 1 I w which all turn out to be indecomposable and to lie in Sub Λ w . The investigation of this set of modules, which we call layers, from different points of view, including connections with tilting theory, is one of the main themes of this paper, especially for a class of words called c-sortable.

The modules L 1 w , . . . , L l w provide a natural filtration for the cluster-tilting object M w associated with the reduced expression w = s u 1 . . . s u l (see Section 1). These modules can be used to show that the endomorphism algebras End Λ (M w ) are quasi-hereditary [START_REF]2-Auslander algebras associated with reduced words in Coxeter groups[END_REF]. Here we show that these modules are rigid (Theorem 2.3), that is Ext 1 Λ (L j w , L j w ) = 0 and that their dimension vectors are real roots (Theorem 2.7), so that there are unique associated indecomposable kQ-modules (L j w ) Q (which are not necessarily rigid).

The situation is especially nice when all layers are indecomposable kQ-modules, so that L j w = (L j w ) Q . This is the case for c-sortable words. An element w of W Q is c-sortable when there exists a reduced expression of w of the form w = c (0) c (1) . . . c (m) with supp(c (m) ) ⊆ . . . ⊆ supp(c (1) ) ⊆ supp(c (0) ) ⊆ supp(c), where c is a Coxeter element, that is, a word containing each generator s i exactly once, and in an order admissible with respect to the orientation of Q.

Starting with the tilting kQ-module kQ (when c (0) = c), there is a natural way of performing exchanges of complements of almost complete tilting modules, determined by the given reduced expression. We denote the final tilting module by T w , and the indecomposable kQ-modules used in the sequence of constructions by T j w for j = 1, . . . , l. We show that L j w is a kQ-module in this case and that L j w ≃ T j w for all j (Theorem 3.8) and we also show that the indecomposable modules in the torsionfree class Sub (T w ) are exactly the T j w (Theorem 3.11). In particular this gives a one-one correspondence between c-sortable words and torsionfree classes, as first shown in [Tho] using different methods (see also [START_REF] Ingalls | Noncrossing partitions and representations of quivers[END_REF]).

There is another sequence U 1 w , . . . U l w of indecomposable kQ-modules, defined using restricted reflection functors, which coincide with the above sequences. This is both interesting in itself, and provides a method for proving L j w ≃ T j w for j = 1, . . . , l. In another paper [AIRT], we give a description of the layers from a functorial point of view. When the c-sortable word is c m , and c = s 1 . . . s n , then the successive layers are given by P 1 , . . . , P n , τ -P 1 , . . . , τ -P n , τ -2 P 1 , . . . , τ -m P n for the indecomposable projective kQ-modules P i , where τ denotes the AR-translation. In the general case we will give a description of the layers using specific factor modules of the above modules.

The generalized cluster categories C A for algebras A of global dimension at most two were introduced in [START_REF] Amiot | Cluster categories for algebras of global dimension 2 and quivers with potential[END_REF]. It was shown that for a special class of words w, properly contained in the dual of the c-sortable words, the 2-CY category Sub Λ w is triangle equivalent to some C A . We point out that the procedure for choosing A works more generally for any dual of a c-sortable word (Theorem 3.23).

The paper is organized as follows. We start with some background material on 2-CY categories associated with reduced words, on complements of almost complete tilting modules and on reflection functors. In Section 2 we show that for any reduced word w, the associated layers are indecomposable rigid modules, which also are positive real roots. Hence there are unique associated indecomposable kQ-modules. In Section 3 we show that our three series of indecomposable modules {L j w }, {T j w } and {U j w } coincide in the c-sortable case. Section 4 is devoted to examples and questions beyond the c-sortable case. Some of this work was presented at the conferences 'Homological and geometric methods in representation theory' in Trondheim in August 2009, 'Interplay between representation theory and geometry' in Beijing in May 2010, and in seminars in Bonn and Torun in 2010.

Notation. Throughout k is an algebraically closed field. The tensor product -⊗ -, when not specified, will be over the field k. For a k-algebra A, we denote by mod A the category of finitely presented right A-modules, and by f.l. A the category of finite length right A-modules. For a quiver Q we denote by Q 0 the set of vertices and by Q 1 the set of arrows, and for a ∈ Q 1 we denote by s(a) its source and by t(a) its target.

1. Background 1.1. 2-Calabi-Yau categories associated with reduced words. Let Q be a finite connected quiver without oriented cycles and with vertices Q 0 = {1, . . . , n}. For i, j ∈ Q 0 we denote by m ij the positive integer

m ij := ♯{a ∈ Q 1 | s(a) = i, t(a) = j} + ♯{a ∈ Q 1 | s(a) = j, t(a) = i}.
The Coxeter group associated to Q is defined by the generators s 1 , . . . , s n and relations

• s 2 i = 1, • s i s j = s j s i if m ij = 0, • s i s j s i = s j s i s j if m ij = 1.
In this paper w will denote a word (i.e. an expression in the free monoid generated by s i , i ∈ Q 0 ), and w will be its equivalence class in the Coxeter group W Q .

An expression w = s u 1 . . . s u l is reduced if l is smallest possible. An element c = s u 1 . . . s u l is called a Coxeter element if l = n and {u 1 , . . . , u l } = {1, . . . , n}. We say that a Coxeter element c = s u 1 . . . s un is admissible with respect to the orientation of Q if i < j when there is an arrow

u i → u j .
The preprojective algebra associated to Q is the algebra

kQ/ a∈Q 1 (aa * -a * a)
where Q is the double quiver of Q, which is obtained from Q by adding for each arrow a : i → j in Q 1 an arrow a * : i ← j pointing in the opposite direction. We denote by Λ the completion of the preprojective algebra associated to Q and by f.l. Λ the category of right Λ-modules of finite length.

The algebra Λ is finite-dimensional selfinjective if Q is a Dynkin quiver. Then the stable category mod Λ satisfies the 2-Calabi-Yau property (2-CY for short), that is, there is a functorial isomorphism

DHom Λ (X, Y ) ≃ Hom Λ (Y, X[2]),
where D := Hom k (-, k) and [1] := Ω -1 is the suspension functor [AR96, CB00] (see [START_REF] Geiss | Semicanonical bases and preprojective algebras. II. A multiplication formula[END_REF][START_REF] Reiten | Tilting theory and cluster algebras[END_REF] for a complete proof). When Q is not Dynkin, then Λ is infinite dimensional and of global dimension 2. In this case the triangulated category D b (f.l. Λ) is 2-CY [CB00, GLS07, Boc08] (see [Y] for a complete proof).

We now recall some work from [START_REF] Iyama | Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras[END_REF][START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF]. For each i = 1, . . . , n we have an ideal

I i := Λ(1 -e i )Λ in Λ
, where e i is the idempotent of Λ associated with the vertex i. We write I w := I u l . . . I u 2 I u 1 when w = s u 1 s u 2 . . . s u l is a expression of w ∈ W Q . We denote by S i := Λ/I i the simple bimodule corresponding to the vertex i.

We collect the following information which is useful for Section 2:

Proposition 1.1. [START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF] Let Λ be a complete preprojective algebra. By (a) the ideal I w does not depend on the choice of the reduced expression w of w. Therefore we write I w for the ideal I w and Λ w := Λ/I w when w is an expression of w. This is a finite dimensional algebra. We denote by Sub Λ w the category of submodules of finite dimensional free Λ w -modules. This is a Frobenius category, that is, an exact category with enough projectives and injectives, and the projectives and injectives coincide. Its stable category Sub Λ w is a triangulated category which satisfies the 2-Calabi-Yau property [START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF]. The category Sub Λ w is then said to be stably 2-Calabi-Yau.

Recall that a cluster-tilting object in a Frobenius stably 2-CY category C with finite dimensional morphisms spaces is an object T ∈ C such that

• Ext 1 C (T, T ) = 0 • Ext 1 C (T, X) = 0 implies that X ∈ add T .
For any reduced word w = s u 1 . . . s u l , we write M j w := e u j Λ I u j . . . I u 1 .

Theorem 1.2. [BIRS09a, Thm III.2.8] For any reduced expression w = s u 1 . . . s u l of w ∈ W Q , the object M w := l j=1 M j w is a cluster-tilting object in the stably 2-CY category Sub Λ w .

For any reduced word w = s u 1 . . . s u l , we have the chain of ideals

Λ ⊃ I u 1 ⊃ I u 2 I u 1 ⊃ . . . ⊃ I w ,
which is strict by Proposition 1.1 (b). For j = 1, . . . , l we define the layer

L j w := I u j-1 . . . I u 1 I u j . . . I u 1 .
Using Proposition 1.1 (b) it is immediate to see the following Proposition 1.3. We have isomorphisms in f.l. Λ:

L j w ≃ e u j L j w ≃ e u j I u i . . . I u 1 I u j . . . I u 1 ≃ Ker( M j w / / / / M i w ),
where i is the greatest integer satisfying u i = u j and i < j. (If such i does not exist, then we define M i w to be 0.) Therefore the layers L 1 w , . . . , L l w give a filtration of the cluster-tilting object M w .

1.2. Mutation of tilting modules. Let Q be a finite connected quiver with vertices {1, . . . , n} and without oriented cycles.

Definition 1.4. A basic kQ-module T is called a tilting module if Ext 1 kQ (T, T ) = 0 and it has n non-isomorphic indecomposable summands.

For each indecomposable summand T i of T , it is known that there is at most one indecomposable [START_REF] Unger | Schur modules over wild, finite-dimensional path algebras with three simple modules[END_REF], and that there is exactly one if and only if T /T i is a sincere kQ-module [START_REF] Happel | Almost complete tilting modules[END_REF]. We then say that T i (and possible T * i ) is a complement for the almost complete tilting module T /T i . The (possible) other complement of T /T i can be obtained using the following result:

T * i ≇ T i such that (T /T i ) ⊕ T * i is a tilting module [RS91,
Proposition 1.5. [RS91] (a) If the minimal left add (T /T i )-approximation T i f / / B is a monomorphism, then Cokerf is a complement for T /T i .
(b) If the minimal right add (T /T i )-approximation B ′ g / / T i is an epimorphism, then Kerg is a complement for T /T i .

There is a one-one correspondence between tilting modules T and contravariantly finite torsionfree classes F = Sub T containing the projective modules. 1.3. Reflections and reflection functors. Let Q be finite quiver with vertices {1, . . . , n} and without oriented cycles. Let i ∈ Q 0 be a source. Then the quiver Q ′ := µ i (Q) is obtained by replacing all arrows starting at the vertex i by arrows in the opposite direction.

Write kQ = P 1 ⊕ • • • ⊕ P n where P j is the indecomposable projective kQ-module associated with the vertex j. Then using results of [START_REF] Bernšteȋn | Coxeter functors, and Gabriel's theorem[END_REF] and [START_REF] Auslander | Coxeter functors without diagrams[END_REF] we have functors:

mod kQ R i / / mod kQ ′ R - i o o
where R i := Hom kQ (M, -), R - i := -⊗ kQ ′ M and M := τ -P i ⊕ kQ/P i which induce inverse equivalences (recall that in this paper we work with right modules)

(mod kQ)/[e i kQ] R i / / (mod kQ ′ )/[e i DkQ ′ ] R - i o o
, where mod kQ/[e i kQ] (resp. mod kQ ′ /[e i DkQ ′ ]) is obtained from the module category mod kQ (resp. mod kQ ′ ) by annihilating morphisms factoring through P i = e i kQ (resp. e i DkQ ′ ). Since i is a source (resp. a sink) of Q (resp. Q ′ ) we can regard the category mod kQ/[e i kQ] (resp. mod kQ ′ /[e i DkQ ′ ]) as a full subcategory of mod kQ (resp. mod kQ ′ ).

When the vertex i is not a sink or source, a reflection is still defined on the level of the Grothendieck group K 0 (mod kQ). The Grothendieck group is constructed as the group with generators [X] for X ∈ mod kQ and relations

[X] + [Z] = [Y ] if there is a short exact sequence X / / / / Y / / / / Z
R i ([S j ]) = [S j ] + (m ij -2δ ij )[S i ],
where m ij is the number of edges of the underlying graph of Q as before.

This definition is coherent with the previous one. Indeed if i is a source and M is an indecomposable kQ-module which is not isomorphic to P i , then we have

R i ([M ]) = [R i (M )].

Layers associated with reduced words

Throughout this section let w be an element in the Coxeter group of an acyclic quiver Q, and fix w = s u 1 . . . s u l a reduced expression of w. For j = 1, . . . , l we have defined in Section 1 the layer L j w as the quotient

L j w := I u j-1 . . . I u 1 I u j . . . I u 1 .
In this section, we investigate some main properties of these layers. We show that each layer can be seen as the image of a simple Λ-module under an autoequivalence of D b (f.l. Λ). Hence they are rigid indecomposable Λ-modules of finite length, and we compute explicitly their dimension vectors and show that they are real roots. Hence to each layer we can associate a unique indecomposable kQ-module with the same dimension vector [START_REF] Kac | Infinite root systems, representations of graphs and invariant theory[END_REF], but which is not necessarily rigid.

Note that some of the results of this section have been proven independently in [START_REF]Kac-Moody groups and cluster algebras[END_REF] but with different proofs.

2.1. Layers are simples up to autoequivalences. The following easy observation is useful.

Lemma 2.1. Let Q be an acyclic quiver and Q be a full subquiver of Q. For any reduced expression w of w ∈ W Q , the module M j w (respectively, L j w ) for Q is the same as M j w (respectively, L j w ) for Q. Proof. Let Λ := Λ Q and Λ := Λ Q be the corresponding complete preprojective algebras. Let

e := i∈ Q 0 \Q 0 e i .
Then we have Λ = Λ/ Λe Λ and

I i = Λ(1 -e i ) Λ
Λe Λ for any i ∈ Q 0 . Thus the assertions follow.

Proposition 2.2. Let Q be an acyclic quiver and Λ the complete preprojective algebra. Let w = s u 1 . . . s u l be a reduced expression.

(1) For j = 1, . . . , l we have isomorphisms of Λ-modules:

L j w ≃ S u j ⊗ Λ (I u j-1 . . . I u 1 ) ≃ S u j ⊗ Λ I u j-1 ⊗ Λ • • • ⊗ Λ I u 1 . (2) If Q is non-Dynkin, then for j = 1, . . . , l we have isomorphisms in D(Mod Λ): L j w ≃ S u j L ⊗ Λ (I u j-1 . . . I u 1 ) ≃ S u j L ⊗ Λ I u j-1 L ⊗ Λ • • • L ⊗ Λ I u 1 .
Proof. We divide the proof into two cases, according to whether Λ is of non-Dynkin type or of Dynkin type.

non-Dynkin case:

We set w ′ := s u 1 . . . s u j and w ′′ := s u 1 . . . s u j-1 . Since w ′′ is reduced, by Proposition 1.1(e) we have

I w ′′ ≃ I u j-1 ⊗ Λ . . . ⊗ Λ I u 1 ≃ I u j-1 L ⊗ Λ . . . L ⊗ Λ I u 1 ,
and hence we get the second isomorphism. Since w ′ = w ′′ s u j is reduced, we have I w ′ = I u j I w ′′ I w ′′ , and therefore Tor Λ 1 (S u j , I w ′′ ) = 0 by Proposition 1.1 (d). Thus we have

S u j L ⊗ Λ I w ′′ ≃ S u j ⊗ Λ I w ′′ ≃ Λ I u j ⊗ Λ I w ′′ ≃ I w ′′ I u j I w ′′ = L j w .

Dynkin case:

We take a non-Dynkin quiver Q containing Q as a full subquiver. Let Λ be the complete preprojective algebra of Q and I i := Λ(1 -e i ) Λ for i ∈ Q 0 . Using the non-Dynkin case and Lemma 2.1, we have

L j w ≃ S u j ⊗ Λ ( I u j-1 . . . I u 1 ) ≃ S u j ⊗ Λ I u j-1 ⊗ Λ • • • ⊗ Λ I u 1 .
For the idempotent e := i∈ Q 0 \Q 0 e i , the twosided ideal Λe Λ annihilates S u j . Since

I i = I i / Λe Λ holds, we have S u j ⊗ Λ ( I u j-1 . . . I u 1 ) ≃ S u j ⊗ Λ (I u j-1 . . . I u 1 ) and S u j ⊗ Λ I u j-1 ⊗ Λ • • • ⊗ Λ I u 1 ≃ S u j ⊗ Λ I u j-1 ⊗ Λ • • • ⊗ Λ I u 1 .
Thus the assertion follows.

Immediately we have the following result, which implies that L j w is an indecomposable rigid Λ-module of finite length.

Theorem 2.3. For j = 1, . . . , l we have

• if Λ is of non-Dynkin type: dim Ext i Λ (L j w , L j w ) = 1 i = 0, 2, 0 otherwise. • if Λ is of Dynkin type: dim Ext i Λ (L j w , L j w ) = 1 i = 0, 2 (mod 6), 0 i = 1 (mod 6).
Note that one can write down explicitly the dimension for the other i in the Dynkin case by using Ω 3 ≃ ν Λ [START_REF] Erdmann | Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology, Algebras and modules[END_REF]. In the non-Dynkin case, L j w is then 2-spherical in the sense of Seidel-Thomas [START_REF] Seidel | Braid group actions on derived categories of coherent sheaves[END_REF].

Proof. We divide the proof into two cases, according to whether Λ is of non-Dynkin type or of Dynkin type.

non-Dynkin case:

By Proposition 1.1 (c), I w ′′ is a tilting Λ-module with End Λ (I w ′′ ) ≃ Λ. Hence the functor - L ⊗ Λ I w ′′ is an autoequivalence of D(Mod Λ). We have End Λ (S j ) ≃ k and hence Ext 2 Λ (S j , S j ) ≃ k since D b (f.l. Λ) is 2-CY.
Moreover since Q has no loops, Ext 1 Λ (S j , S j ) vanishes and since Λ is known to have global dimension 2, Ext n Λ (S j , S j ) vanishes for n ≥ 3. Hence S j is 2-spherical. Since by Proposition 2.2 the layer L j w is the image of the simple S j by an autoequivalence of D b (f.l. Λ), it follows that L j w is also 2-spherical.

Dynkin case:

We take a non-Dynkin quiver Q containing Q as a full subquiver. Let Λ be the complete preprojective algebra of Q. Then mod Λ can be seen as a full and extension closed subcategory of mod Λ. Using the non-Dynkin case and Lemma 2.1, we get

End Λ (L j w ) ≃ End Λ (L j w ) ≃ k and Ext 1 Λ (L j w , L j w ) ≃ Ext 1 Λ (L j w , L j w ) = 0.
Using the fact that mod Λ is stably 2-CY we get Ext 2 Λ (L j w , L j w ) ≃ k.

Here we state a property about two consecutive layers associated with the same vertex, which gives rise to special non-split short exact sequences in f.l. Λ. Proposition 2.4. Let 1 ≤ i < j < k ≤ l be integers such that u i = u j = u k and such that j is the only integer satisfying i < j < k and u i = u j = u k . Then we have

dim k Ext 1 Λ (L j w , L k w ) = 1.
In order to prove this proposition, we first need a lemma. For 1 ≤ h ≤ l, we denote as before by M h w the Λ-module

M h w := e u h Λ Iu h ...Iu 1 .
Lemma 2.5. Let i < j < k be as in Proposition 2.4.

(1) The map

Hom Λ (M k w , M j w ) → Hom Λ (M k w , M i w ) induced by the irreducible map M j w → M i w is an epimorphism. (2) The image of the map Hom Λ (M i w , M j w ) → Hom Λ (M j w , M j w ) induced by the irreducible map M j w → M i w is Rad Λ (M j w , M j w ) .
Proof.

(1) Since i < j < k, then by Lemma III.1.14 of [START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF], we have isomorphisms

Hom Λ (M k w , M j w ) ≃ e Λ I u j . . . I u 1 e and Hom Λ (M k w , M i w ) ≃ e Λ I u i . . . I u 1 e,
where e is the idempotent e := e u i = e u j = e u k . Then the map

Hom Λ (M k w , M j w ) → Hom Λ (M k w , M i w ) is the epimorphism e Λ Iu j ...Iu 1 e u k → e Λ Iu i ...Iu 1 e u k induced by the inclusion I u j . . . I u 1 ⊂ I u i . . . I u 1 .
(2) It is clear that the image is contained in the radical. By Lemma III.1.14 of [START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF], we have isomorphisms

Hom Λ (M i w , M j w ) ≃ e I u j . . . I u i+1 I u j . . . I u 1 e and Rad Λ (M j w , M j w ) ≃ e I u j I u j . . . I u 1 e.
The map Hom Λ (M i w , M j w ) → Rad Λ (M j w , M j w ) is induced by the inclusion of ideals I u j . . . I u i+1 ⊂ I u j . But since j is the only integer satisfying i < j < k and u i = u i = u k , we have eI u j . . . I u i+1 e ≃ eI u j e and hence the map

Hom Λ (M i w , M j w ) → Rad Λ (M j w , M j w ) is an isomor- phism.
Proof of Proposition 2.4. By the definition of the layers, we have the following short exact sequences (j)

L j w / / / / M j w / / / / M i w and (k) L k w / / / / M k w / / / / M j w Let K be the kernel of the composition map M k w → M j w → M i w .
Then we have a short exact sequence (l) K / / / / M k w / / / / M i w which gives rise to the following long exact sequence in mod End Λ (M w ), where M w = l h=1 M h w :

DExt 1 Λ (M i w , M w ) / / DHom Λ (K, M w ) / / DHom Λ (M k w , M w ) / / DHom Λ (M i w , M w ) / / 0
The space DExt 1 Λ (M i w , M w ) is zero by Lemma III.2.1 of [START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF], and the End Λ (M w )-module DHom Λ (M k w , M w ) is indecomposable injective. Therefore the module DHom Λ (K, M w ) has simple socle, and hence K is indecomposable.

Moreover from the sequences (j), (k) and (l), we deduce that we have a short exact sequence

L k w / / / / K / / / / L j w which is non-split since K is indecomposable. Hence we get dim k Ext 1 Λ (L j w , L k w ) ≥ 1. From (j) we deduce the long exact sequence • • • / / Hom Λ (M k w , M j w ) / / Hom Λ (M k w , M i w ) / / Ext 1 Λ (M k w , L j w ) / / Ext 1 Λ (M k w , M j w ) = 0 .
Hence by Lemma 2.5 (1) we get Ext 1 Λ (M k w , L j w ) = 0. From (j) we also deduce the long exact sequence

0 / / Hom Λ (M i w , M j w ) / / Hom Λ (M j w , M j w ) / / Hom Λ (L j w , M j w ) / / Ext 1 Λ (M i w , M j w ) = 0 .
Hence by Lemma 2.5 (2) we get Hom Λ (L j w , M j w ) ≃ Hom Λ (M j w , M j w )/Rad Λ (M j w , M j w ) which is one dimensional since M j w is indecomposable. Finally using (k) we get the long exact sequence

• • • / / Ext 1 Λ (M k w , L j w ) / / Ext 1 Λ (L k w , L j w ) / / Ext 2 Λ (M j w , L j w ) / / • • •
By the 2-CY property and the previous remarks we have

Ext 1 Λ (M k w , L j w ) = 0 and Ext 2 Λ (M j w , L j w ) ≃ DHom Λ (L j w , M j w ) ≃ k and therefore dim k Ext 1 Λ (L j w , L k w ) ≤ 1.
2.2. The dimension vectors of the layers. In this section we investigate the action of the functor -L ⊗ Λ I w at the level of the Grothendieck group of D b (f.l. Λ) when Λ is not of Dynkin type. We show that this action has interesting connections with known actions of Coxeter groups. We denote by [-

L ⊗ Λ I w ] the induced automorphism of K 0 (D b (f.l. Λ)).
Lemma 2.6. Let Q be a non-Dynkin quiver. For all i, j in Q 0 we have

[S j L ⊗ Λ I i ] = [S j ] + (m ij -2δ ij )[S i ] in K 0 (D b (f.l. Λ))
, where m ij is the number of arrows between i and j in Q.

Proof. Since S i = Λ/I i , we have DS i ≃ S i as Λ-bimodules. Hence we have the following isomorphisms in Mod (Λ op ⊗ Λ):

S j L ⊗ Λ S i ≃ DHom k (S j L ⊗ Λ S i , k) ≃ DRHom Λ (S j , Hom k (S i , k)) ≃ DRHom Λ (S j , DS i ) ≃ DRHom Λ (S j , S i ).
Therefore we have

[S j L ⊗ Λ S i ] = ( t (-1) t dim Ext t Λ (S j , S i ))[S i ] = (2δ ij -m ij )[S i ].
From the triangle S i [-1] / / I i / / Λ / / S i we get a triangle

S j L ⊗ Λ S i [-1] / / S j L ⊗ Λ I i / / S j / / S j L ⊗ Λ S i . Hence we have [S j L ⊗ Λ I i ] = [S j ] -[S j L ⊗ Λ S i ] = [S j ] -(2δ ij -m ij )[S i ].
From Lemma 2.6, we deduce the following results.

Theorem 2.7. Let Λ be the complete preprojective algebra of any type.

(1) For j = 1, . . . , l we have

[L j w ] = R u 1 . . . R u j-1 ([S u j ])
, where the R t are the reflections defined in Section 1. In particular all [L j w ] are positive real roots.

(2) [L 1 w ], . . . , [L l w ] are pairwise different in K 0 (D b (f.l. Λ)).
(3) For j = 1, . . . , l, there exists a unique indecomposable kQ-module

(L j w ) Q such that [L j w ] = [(L j w ) Q ]. Proof.
(1) As in the previous subsection we treat separately the Dynkin and the non-Dynkin case. The non-Dynkin case is a direct consequence of Lemma 2.6 and Proposition 2.2.

For the Dynkin case, we can follow the strategy of the proof of Theorem 2.3 . We introduce an extended Dynkin quiver containing Q as subquiver. Then applying reflection functors associated to the vertices of Q to modules whose support do not contain the additional vertex is the same as applying the reflection functors of Q. Thus using Lemma 2.1, the equality coming from the non-Dynkin quiver gives us the equality for Q.

Hence the [L j w ] are real roots, which are clearly positive since L j w is a module.

(2) By [BB05, Prop. 4.4.4], [S u 1 ], R u 1 ([S u 2 ]), . . . , R u 1 . . . R u l-1 ([S u l ]) are pairwise different.
Thus the assertion follows from (1).

(3) From (1) we know that the dimension vector of the layer L j w is a positive real root, and we get the result applying Kac's Theorem [START_REF] Kac | Infinite root systems, representations of graphs and invariant theory[END_REF].

The layer L j w is always rigid as a Λ-module, but the associated indecomposable kQ-module (L j w ) Q is not always rigid as shown in the following. Example 2.8. Let Q be the quiver 2 ' '

O O O 1 7 7 o o o / / 3
, and w := s 1 s 2 s 3 s 2 s 1 s 3 . Then we have

L 1 w = 1 , L 2 w = 2 1 , L 3 w = 3 1 2 1 , L 4 w = 3 1 , L 5 w = 2 3 3 1 2 1 1
, and

L 6 w = 2 3 3 1 1 .
Thus the associated indecomposable kQ-modules are the following:

(L j w ) Q = L j w for j = 1, . . . 4, (L 5 w ) Q = 3 1 2 3 1 2 1
, and (L 6

w ) Q = 3 1 2 3 1 .
The module (L 6 w ) Q lies in the tube of rank 2, with indecomposable objects 3 1 and 2 on the border of the tube. Since (L 6 w ) Q is not on the border of the tube, it is not rigid. Definition 2.9. [START_REF] Björner | Combinatorics of Coxeter groups[END_REF] Let Q be an acyclic quiver with n vertices, and W Q be the Coxeter group of Q. Let V be the vector space with basis v 1 , . . . , v n . The geometric representation

W Q → GL(V ) of W Q is defined by reflections s i v j := v j + (m ij -2δ ij )v i .
The contragradient of the geometric representation W Q → GL(V ) is then

s i v * j = v * j i = j -v * j + t =j m tj v * t i = j
The Grothendieck group K 0 (D b (f.l. Λ)) has a basis consisting of the simple Λ-modules, and K 0 (K b (proj Λ)) has a basis consisting of the indecomposable projective Λ-modules.

Proposition 2.10. Let Λ be the complete preprojective algebra of non-Dynkin type. (2) This is shown in [IR08, Theorem 6.6]. It is assumed in [START_REF] Iyama | Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras[END_REF] that Q is extended Dynkin, but this assumption is not used in the proof for this statement.

2.3. Reflection functors and ideals I i . In this subsection, we state some basic properties of the first layers. In particular we show that the equivalence -L ⊗ Λ I i , when Q is not Dynkin, can be interpreted as a reflection functor of the category D b (f.l. Λ).

Lemma 2.11. Let Q be an acyclic quiver, and Λ = Λ Q . Let c ∈ W Q be a Coxeter element admissible with respect to the orientation of Q. Let i ∈ Q 0 be a source of Q and R - i : mod kQ → mod kQ ′ be the reflection functor for Q ′ := µ i (Q). Then we have the following :

(1) Λ/I c ≃ kQ in mod Λ. So we view kQ-modules as Λ-modules annihilated by I c .

(2) If w ′ is a subsequence of w, and if w is a subsequence of cw ′ (where w ′ , cw ′ and w are reduced expressions in W Q ), then I w ′ /I w is a kQ-module.

(3) I i /I cs i is a kQ ′op ⊗ kQ-module and isomorphic to τ -1 P i ⊕ kQ/P i = R - i (kQ) as a kQ ′op ⊗ kQ-module, where P i = e i kQ is the indecomposable projective kQ-module associated to i and τ is the AR-translation of mod kQ.

Proof.

(1) This is Propositions II.3.2 and II.3.3 of [START_REF] Buan | Cluster structures for 2-Calabi-Yau categories and unipotent groups[END_REF].

(2) Since I w ′ I c = I cw ′ ⊃ I w , we have that I w ′ /I w is annihilated by I c .

(3) Note that by Proposition 1.1 (b) we have e j I i = e j Λ and e j I cs i = e j I i I c = e j I c if j = i. Therefore by (1) it is enough to prove that e i I i /I cs i ≃ τ -1 (e i kQ).

The projective resolution of e i I i in mod Λ has the form:

( * ) e i Λ / / a∈ Q1
I i I c = e i I i I cs i / / 0 .
Since i is a source in Q, we have the set equality

{a ∈ Q1 , with s(a) = i} = {a ∈ Q 1 , with s(a) = i}.
Therefore by (1) the exact sequence ( * * ) is

e i kQ / / a∈Q 1 ,s(a)=i e t(a) kQ / / e i I i I cs i / / 0 .
Hence we have e i I i

I cs i ≃ τ -(e i kQ).
From Lemma 2.11 we deduce the following result which gives another interpretation of the reflection functor.

Corollary 2.12. Let Q be an acyclic quiver and Λ = Λ Q .

(1

) Let i ∈ Q 0 be a sink of Q. Let Q ′ := µ i (Q). Then the following diagram commutes mod kQ/[e i DkQ] R - i / / _ mod kQ ′ /[e i kQ ′ ] _ f.l. Λ -⊗ Λ I i / / f.l. Λ
, where the vertical functors are the natural inclusions. If Q is not Dynkin, then the following diagram commutes

mod kQ/[e i DkQ] R - i / / _ mod kQ ′ /[e i kQ ′ ] _ D b (f.l. Λ) - L ⊗ Λ I i / / D b (f.l. Λ) ,
where the vertical functors are the natural inclusions.

(2) Let c = s u 1 • • • s un be a Coxeter element and

C -:= R - un • • • • • R - u 1 : mod kQ →
C - / / _ mod kQ/[kQ] _ D b (f.l. Λ) - L ⊗ Λ Ic / / D b (f.l. Λ),
where the vertical functors are the natural inclusions. In particular we have I c l /I c l+1 ≃ τ -l (kQ).

Proof. (1) Denote by c the Coxeter element admissible with respect to the orientation of Q, and by c ′ = s i cs i the Coxeter element admissible with respect to the orientation of Q ′ . We have the following isomorphisms in f.l. Λ.

kQ ⊗ Λ I i ≃ Λ/I c ⊗ Λ I i by Lemma 2.11 (1) ≃ I i /I c I i ≃ I i /I i I c ′ ≃ τ -1 P i ⊕ kQ/P i by Lemma 2.11 (3)
Thus on mod kQ/[e i DkQ], we have -

⊗ Λ I i = -⊗ kQ (kQ ⊗ Λ I i ) ≃ -⊗ kQ (τ -1 P i ⊕ kQ/P i ) = R - i .
The latter assertion can be shown quite similarly since we have kQ

L ⊗ Λ I i ≃ kQ ⊗ Λ I i by Proposition 2.2.
(2) This is a direct consequence of (1).

Tilting modules and c-sortable words

In this section Q is a finite acyclic quiver, Λ is the complete preprojective algebra associated with Q and c a Coxeter element admissible with respect to the orientation of Q. The purpose of this section is to investigate the layers for words w satisfying a certain property called c-sortable. Definition 3.1. [START_REF] Reading | Clusters, Coxeter-sortable elements and noncrossing partitions[END_REF] Let c be a Coxeter element of the Coxeter group W Q . Usually we fix a reduced expression of c and regard c as a reduced word. An element w of W Q is called c-sortable if there exists a reduced expression w of w of the form w = c (0) c (1) . . . c (m) where all c (t) are subwords of c whose supports satisfy t) , by abuse of notation, we will write i ∈ c (t) .

supp(c (m) ) ⊆ supp(c (m-1) ) ⊆ . . . ⊆ supp(c (1) ) ⊆ supp(c (0) ) ⊆ Q 0 . For i ∈ Q 0 , if s i is in the support of c (
Then c-sortability does not depend on the choice of reduced expression of c. It is immediate that the expression w = c (0) c (1) . . . c (m) is unique for any c-sortable element of W Q [START_REF] Reading | Clusters, Coxeter-sortable elements and noncrossing partitions[END_REF].

where Q ′ = µ u 1 (Q). By the induction hypothesis we get that w ′ is c ′ -sortable where c ′ is the Coxeter element admissible for the orientation of Q ′ , i.e. c ′ = s u 1 cs u 1 . We get the conclusion using the following criterion which detects c-sortability: Lemma 3.4. [Rea07, Lemma 2.1] Let c := s u 1 . . . s un be a Coxeter element. If l(s u 1 w) < l(w), then w is c-sortable if and only if s u 1 w is s u 1 cs u 1 -sortable.

3.1. Comparison of three series of kQ-modules. To the c-sortable word w = s u 1 . . . s u l = c (0) c (1) . . . c (m) , we associate two series of kQ-modules T j w and U j w , and show that they coincide with L j w . For j = 1, . . . , l, we define kQ (0) -modules T j w . For 1 ≤ j ≤ l(c (0) ), T j w is the projective kQ (0)module e u j kQ. For j > l(c (0) ), let k be the maximal integer such that k < j and u k = u j . We define T j w as the cokernel of the map f j w : T k w → E where f j w is a minimal left add {T k+1 w , . . . , T j-1 w }-approximation. Example 3.5. Let Q be the quiver 2

% % K K K K 1 / / 9 9 s s s s 3 and w := s 1 s 2 s 3 s 1 s 2 s 1 in Example 3.2.
Let us compute the T j w . For j ≤ 3 the T j w are the projective kQ-modules, thus we have

T 1 w = 1 , T 2 w = 2 1 , and
T 3 w = 3 1 2 1 .
Then we have to compute approximations. We have a short exact sequence 0

/ / 1 / / 2 1 ⊕ 3 1 2 1 / / 2 3 1 2 1 / / 0 ,
where the map 1

/ / 2 1 ⊕ 3 1 2 1
is the minimal left add (T2 w ⊕ T 3 w )-approximation of T 1 w .

Hence we have T 4 w = hence T 6 w = 3 1 . So we have T j w = L j w for any j.

To define the kQ-modules U j w , the following notion is convenient. Definition 3.6. An admissible triple is a triple (Q, c, w) consisting of an acyclic quiver Q, a Coxeter element c admissible with respect to the orientation of Q, and a c-sortable word w = c (0) c (1) . . . c (m) .

We denote by Q (j) the quiver Q restricted to the support of c (j) .

By Proposition 1.1 (b) we have the inclusion

Y I c ′ ⊂ X since u 2 • • • u j is a subword of c ′ u 2 . . . u k .
Thus we have a(y ⊗ pq) = yp ⊗ q = 0 for any y ∈ Y X , p ∈ I c ′ and q ∈ I u 1 . Thus a = 0 and b is an isomorphism. We have isomorphisms

R - u 1 (L j-1 w ′ ) = Y X ⊗ Λ I u 1 I c ′ I u 1 b ≃ Y X ⊗ Λ I u 1 ≃ Y ⊗ Λ I u 1 Im(X ⊗ Λ I u 1 → Y ⊗ Λ I u 1 ) ≃ Y ⊗ Λ I u 1 X ⊗ Λ I u 1 ≃ L j w .
(2) We will now prove that U j w ≃ T j w . For j ≤ l(c (0) ) this is clear because of a basic property of reflection functors.

Assume j > l(c (0) ). Let k be the maximal integer such that u k = u j and k < j. It exists because j > l(c (0) ) and w is c-sortable. We define the subwords w ′′ = s u 1 . . . s u k-1 and w ′ = s u k . . . s u j of w. Let c ′ be s u k . . . s u j-1 , and Q ′ be the quiver

µ u k-1 •• • • •µ u 1 (Q). Then (Q ′ , c ′ , w ′ ) is an admissible triple. We have U 1 w ′ = S u k and U j-k+1 w ′ = R - c ′ (S u k ) = τ -1 kQ ′ (S u k ), thus we have an almost split sequence: 0 → U 1 w ′ → E → U j-k+1 w ′ → 0 Applying the reflection functor R - w ′′ := R - u k-1 • • • • • R - u 1 : mod kQ ′ →
mod kQ to this short exact sequence we still get a short exact sequence:

0 → R - w ′′ (U 1 w ′ ) → R - w ′′ (E) → R - w ′′ (U j-k+1 w ′ ) → 0 which is 0 → U k w → R - w ′′ (E) → U j w → 0 and the left map is a left add {R - w ′′ (U 2 w ′ ), . . . , R - w ′′ (U j-k w ′ )}-approximation, thus a left add {U k+1 w , . . . , U j-1 w }- approximation.
Remark 3.9. The statements L j w ≃ U j w ≃ T j w for j = 1, . . . , l in Theorem 3.8 is also true for non-reduced words w = w ′ s u l = c (0) c (1) . . . c (m) such that w ′ is reduced and that all c (t) are subwords of c whose supports satisfy

supp(c (m) ) ⊆ supp(c (m-1) ) ⊆ . . . ⊆ supp(c (1) ) ⊆ supp(c (0) ) ⊆ Q 0 .
The proof above works without any change. Note that in this situation, the morphism f j w is a monomorphism for j = 1, . . . , l -1, but the morphism f l w may not be a monomorphism and T l w and U l w may be zero. Corollary 3.10. Let w be a c-sortable word, where c is admissible with respect to the orientation of Q. Then the kQ-modules L j w satisfy the following properties: (1) They are non-zero, indecomposable and pairwise non-isomorphic.

(2) The space Hom kQ (L j w , L k w ) vanishes if j > k. Proof. (1) Since w is reduced, L j w is non-zero by Proposition 1.1 (b). Since reflection functors preserve isoclasses, the U j w are indecomposable and pairwise non-isomorphic. (2) Using reflection functors, we can assume that U k w is simple projective, and then this is clear.

Theorem 3.11. Let (Q, c, w = s u 1 . . . s u l ) be an admissible triple. For i ∈ Q (0) 0 , denote by t w (i) the maximal integer such that u tw(i) = i. Let The module T w is by definition T 3 w ⊕ T 5 w ⊕ T 6 w . It is easy to check Theorem 3.11. The module T w is a tilting module over kQ, and we have

T w := i∈Q (0) 0 L tw(i) w ≃ i∈Q (0) 0 T tw(i) w . (b)
Sub T w = { 1 , 2 1 , 3 1 2 1 , 2 3 1 2 1 , 3 1 2 3 1 2 1
, 3 1 }.

3.2. Tilting modules with finite torsionfree class. In this section we establish the converse of Theorem 3.11. Hence we get a natural bijection between tilting kQ-modules with finite torsionfree class and c-sortable elements in W Q .

Let us start with some preparation. To any (not necessarily reduced) word w = s u 1 . . . s u l = c (0) c (1) . . . c (m) such that all c (t) are subwords of c whose supports satisfy

supp(c (m) ) ⊆ supp(c (m-1) ) ⊆ . . . ⊆ supp(c (1) ) ⊆ supp(c (0) ) ⊆ Q 0 ,
we can associate kQ (0) -modules T j w for j = 1, . . . , l and T w in the same way as in the c-sortable case.

Lemma 3.14. Let w = s u 1 . . . s u l = w ′ s u l be as above. Assume that w is non-reduced and that w ′ is reduced. Then the number of indecomposable summands of T w is strictly less than l(c (0) ).

Proof. By Remark 3.9, we have T l w ≃ L l w . Since w is not reduced, this is zero by Proposition 1.1 (b). Since w ′ is reduced, all T j w ≃ L j w (j = l) are indecomposable by Theorem 2.3. Therefore we have the assertion.

Lemma 3.15. Let Q be an acyclic quiver and T be a tilting kQ-module.

(1) The category Sub T has almost split sequences.

(2) If Sub T has finitely many indecomposable modules, then the AR-quiver of Sub T is a full subquiver of the translation quiver ZQ.

Proof.

(1) This is well-known [START_REF] Auslander | Almost split sequences in subcategories[END_REF].

(2) We can clearly assume that Q is connected. Since T is a tilting module, then all indecomposable projectives are in Sub T . The irreducible maps between projectives in Sub T coincide with the irreducible maps between projectives in mod kQ, so that Q is a full subquiver of the AR-quiver of Sub T . Moreover, for any indecomposable module in Sub T , there is a nonzero map from an indecomposable projective module. Since Sub T is of finite type and Q is connected, it follows that the AR-quiver of Sub T is connected.

We now claim that each indecomposable module in Sub T is of the form τ -t P , where τ is the AR-translate in Sub T and P is indecomposable projective. If not, then since Sub T is of finite type, there is some τ -periodic indecomposable X. Then, since the quiver of Sub T is connected, there must be an irreducible map between some periodic indecomposable X and some τ -t P with P indecomposable projective. Applying τ t we can assume that the second module is P . If f : X → P is irreducible, then X is projective, a contradiction. If g : P → X is irreducible, then h : τ X → P is irreducible, so τ X is projective, a contradiction. Thus each indecomposable of Sub T is of the form τ -t P , where P is indecomposable projective.

Then using the fact that Q is a full subquiver of the AR-quiver of Sub T , we deduce that the AR-quiver of Sub T is a full subquiver of ZQ.

From Lemmas 3.14 and 3.15 we deduce a nice consequence.

Theorem 3.16. Let Q be an acyclic quiver. Let c be a Coxeter element admissible with respect to the orientation of Q. Let T be a tilting kQ-module. Assume that Sub T has finitely many indecomposable modules. Then there exists a unique c-sortable word w such that T w ≃ T .

Proof. Without loss of generality, we assume c = s 1 s 2 . . . s n . We denote by τ the AR-translation of Sub T . For any i ∈ Q 0 , we denote by m(i) the minimal number satisfying τ -m(i)-1 (e i kQ) = 0, which exists by Lemma 3.15. Then for t ≥ 0 we look at the set

{i ∈ Q 0 | τ -t (e i kQ) = 0} = {i (t) 1 < i (t) 2 < • • • < i (t)
pt } and define c (t) := s i (t)

1 s i (t) 2 . . . s i (t) p t
. Then the word w := c (0) c (1) . . . c (m) where m := max{m(i

) | i ∈ Q 0 } satisfies supp(c (m) ) ⊆ . . . ⊆ supp(c (1) ) ⊆ supp(c (0) ).
For each expression w = w ′ w ′′ , we define m w ′ (i) + 1 as the number of s i (i ∈ Q 0 ) appearing in w ′ . By using induction on l(w ′ ), we have

T w ′ ≃ i∈Q 0 τ -m w ′ (i) (e i kQ)
by using the almost split sequences in Sub T and the shape of the AR-quiver of Sub T given in Lemma 3.15 (2). In particular the number of indecomposable direct summands of

T w ′ is exactly n since m w ′ (i) ≤ m(i) for any i ∈ Q 0 . Moreover we have T w ≃ i∈Q 0 τ -m(i) (e i kQ) ≃ T since m w (i) = m(i) for any i ∈ Q 0 .
We only have to check that w is reduced. Otherwise we take an expression w = w ′ w ′′ such that w ′ is non-reduced and l(w ′ ) is minimal with this property. By Lemma 3.14, the number of indecomposable direct summands of T w ′ is less than n, a contradiction. Thus w is reduced.

As a consequence we get the following: Corollary 3.17. If T is a tilting kQ-module such that Sub T is of finite type, then all indecomposables in Sub T are rigid as kQ-modules.

Combining Theorem 3.16 with Theorem 3.11 we get the following result which was first proved using other methods in [Tho].

Corollary 3.18. There is 1-1 correspondences (a) {finite torsionfree classes of mod kQ containing kQ} o o 1:1 / / {c-sortable words with c (0) = c} .

(b) {finite torsionfree classes of mod kQ} o o 1:1 / / {c-sortable words} .

3.3.

Co-c-sortable situation. Dually, we can state the defintion. Definition 3.19. Let c be a Coxeter element of the Coxeter group W Q admissible with respect to the orientation of Q. An element w of W Q is called co-c-sortable if there exists a reduced expression w = s u l . . . s u 1 of w of the form w = c (m) c (m-1) . . . c (0) where all c (t) are subwords of c whose supports satisfy

supp(c (m) ) ⊆ supp(c (m-1) ) ⊆ . . . ⊆ supp(c (1) ) ⊆ supp(c (0) ) ⊆ Q 0 .
Note that this definition is equivalent to the fact that w -1 = s u 1 . . . s u l is c -1 -sortable.

We denote by Q (j) the quiver Q restricted to the support of c (j) .

Let w be a co-c-sortable word with c (0) = c. We denote by A w (resp. A w ) the Auslander algebra of the category Fac (T w ) (resp. Fac (T w ) = Fac (T w )/add (DkQ)), that is

A w = End kQ ( l j=1 T w j ) and A = End kQ ( l j=1 T w j ) = End kQ ( l j=l(c)+1 T w j ).
Following [Ami09a, Theorem 5.21] we get the following result:

Theorem 3.23. There exists a commutative diagram of triangle functors:

D b (A w ) Res. / / D b (A w ) - L ⊗ Aw Mw / / D b (Λ w ) C 2 (A w ) F / / Sub Λ w ,
where C 2 (A w ) is the generalized 2-cluster category defined in [Ami09a] associated with the algebra A w of global dimension at most 2, and where F is an equivalence of categories.

The proof in [Ami09a] deals with T w in the preinjective component of the AR-quiver of mod kQ (that is End kQ (T w ) concealed), but the proof only uses the fact that Fac (T w ) is finite and that the AR-quiver of Fac (T w ) is a full subquiver of ZQ.

Note that for any element w, the 2-CY category Sub Λ w is equivalent to a generalized 2cluster category C 2 (A) [START_REF] Amiot | The ubiquity of the generalized cluster categories[END_REF], but the algebra A of global dimension 2 is constructed in a very different way. A link between the construction given here and the construction of [START_REF] Amiot | The ubiquity of the generalized cluster categories[END_REF] is given in [START_REF]A derived equivalence between cluster equivalent categories[END_REF].

Remark 3.24. The proof in [START_REF] Amiot | Cluster categories for algebras of global dimension 2 and quivers with potential[END_REF] does not carry over for c-sortable words. Indeed, for a general co-c-sortable word, the AR-quiver of Fac (T w ) is a subquiver of the quiver of End Λ (M w ), fact which is used in the proof. For a c-sortable word w, the AR-quiver of Sub (T w ) is not a subquiver of the quiver of End Λ (M w ).

Problems and examples

In this section we discuss some possible generalizations of the description of the layers in terms of tilting modules, beyond the c-sortable case. We pose some problems and give some examples to illustrate limitations for what might be true.

Recall from Section 2 that to a reduced expression w of an element w in W Q we have associated a set {L j w } of l(w) indecomposable rigid Λ-modules which we call layers, and which are indecomposable rigid kQ-modules when w is c-sortable, where c is admissible with respect to the orientation of Q. Under the same assumption (i.e. w is c-sortable), we constructed a set {T j w } of l(w) indecomposable kQ-modules via minimal left approximations, starting with the tilting module kQ, and ending up with a tilting module T w . All minimal left approximations were monomorphisms. We showed that the two sets of indecomposable modules coincide. In particular, the module L w := L tw(1) w

⊕ • • • ⊕ L tw(n) w
, where for i ∈ Q 0 = {1, . . . , n} the integer t w (i) is the position of the last reflection s i in the word w, is a tilting module over kQ.

We now consider the case of words w with the assumption that w = cw ′ , where c is a Coxeter element admissible with respect to the orientation of Q. When w = cs u n+1 . . . s u l is a word, we define T w to be a tilting module associated with w if it is possible to carry out the following. Start with kQ = P 1 ⊕ • • • ⊕ P n , where P i is the indecomposable projective kQ-module associated with the vertex i. If possible, exchange P u n+1 with a non-isomorphic indecomposable kQ-module to get a tilting module T ′ = kQ/P u n+1 ⊕ P * u n+1 , then replace summand number i 2 in T ′ by a non-isomorphic indecomposable kQ-module to get a new tilting module T ′′ , etc. If an exchange is possible at each step, we obtain a tilting module T w . We say that a word w = cw ′ starting with a Coxeter element is tilting if T w exists, and w is monotilting if morever T w is obtained by only using left approximations. Hence c-sortable words are examples of monotilting words.

It is natural to ask the following question about tilting and monotilting words. Note that all these questions can also be translated into combinatorial problems for acyclic cluster algebras.

Note that non-reduced words may be monotilting as the following example shows. and w := s 1 s 2 s 3 s 4 s 3 s 1 s 4 . Then w is not reduced.

One can easily check that we have

T 1 w = 1 , T 2 w = 2 1 , T 3 w = 3 1 , T 4 w = 4 2 3 1 1 , T 5 w = 4 2 1
.

Then the minimal left add (T 2 w ⊕ T 4 w ⊕ T 5 w )-approximation of

T 1 w is 1 → 2 1 ⊕ 4 2 3 1 1
. It is a monomorphism whose cokernel is T 6 w = 4 2 3 2 1 1

. The minimal left add (T 2 w ⊕ T 5 w ⊕ T 6 w )approximation of T 4 w is

4 2 3 1 1 → 4 2 3 2 1 1
. It is a monomorphism whose cokernel is T 7 w = 2 . Hence w is monotilting.

Recall that in the c-sortable case, then w is monotilting and Sub T w is of finite type. This is not the case in general. ).

However, it may happen that Sub T w is of finite type for a tilting word w which is not csortable. It follows from Theorem 3.16 that there exists a unique c-sortable word w such that T w = T w. We then pose the following.

  (a) If w = s u 1 . . . s u l and w ′ = s v 1 . . . s v l are two reduced expressions of the same element in the Coxeter group, then I w = I w ′ . (b) If w = w ′ s i with w ′ reduced, then I w ⊆ I w ′ . Moreover w is reduced if and only if I w I w ′ . And for j = i we have e j I w = e j I w ′ . If Λ is not of Dynkin type we have moreover: (c) Any finite product I of the ideals I j is a tilting module of projective dimension at most one, and End Λ (I) ≃ Λ. (d) If S is a simple Λ-module and I is a tilting module of projective dimension at most one, then S ⊗ Λ I = 0 or Tor Λ 1 (S, I) = 0. (e) If Tor Λ 1 (S i , I) = 0, then I i L ⊗ Λ I = I i ⊗ Λ I = I i I for a tilting module I of projective dimension at most one.

( 1 )

 1 The Coxeter group W Q acts on K 0 (D b (f.l. Λ)) by w → [-L ⊗ Λ I w ] as the geometric representation. (2) The Coxeter group W Q acts on K 0 (K b (proj Λ)) by w → [-L ⊗ Λ I w ] as the contragradient of the geometric representation. Proof. (1) This follows directly from Lemma 2.6.

  mod kQ the Coxeter functor. Then the following diagram commutes mod kQ/[DkQ] C - / / _ mod kQ/[kQ] _ f.l. Λ -⊗ Λ Ic / / f.l. Λ , where the vertical functors are the natural inclusions. If Q is not Dynkin, then the following diagram commutes mod kQ/[DkQ]

  Problem 4.1. :(a) Characterize the tilting words w. In particular is every reduced word w = cw ′ starting with a Coxeter element tilting? (b) Characterize the monotilting words. (c) When do two tilting words w 1 and w 2 give rise to the same tilting module? Or formulated differently, for which tilting words w do we have T w ≃ kQ?

  Example 4.2. Let Q be the quiver 2 ' '

⊕ 4 .

 4 Example 4.3. Let Q be the quiver 2 ' ' w := s 1 s 2 s 3 s 4 s 2 s 3 s 4 s 1 . Then one can show that w is monotilting and that T w = Then one can check easily that all the modules of the form

  Problem 4.4. : (a) Characterize the tilting words w with Sub T w finite. (b) For such words w, how can we construct the unique w such that T w = T w?When w is monotilting, we have{T 1 w , . . . , T l(w) w } ⊆ Sub T w = Sub T w = add {T 1 w, . . . , T l( w)w }. Hence l(w) ≤ l( w) and we expect that w is obtained by enlarging some rearrangement of w.

  w := s 1 s 2 s 3 s 4 s 2 s 3 s 1 s 4 . Then w is monotilting and we have T w =

  ,s(a)=i e t(a) Λ / / e i I i / / 0.

	Applying the functor -⊗ Λ	Λ I c	to the exact sequence ( * ), we get an exact sequence
	( * * ) e i	Λ I c	/ /	a∈ Q1 ,s(a)=i e t(a)	Λ I c	/ / e i	I i

  This almost split sequence is an element of Ext 1 Λ (L j w , L k w ), which is the '2-Calabi-Yau complement' of the short exact sequence L j := s 1 s 2 s 3 s 1 s 2 s 1 in Example 3.5.

					w	/ /	/ / K	/ / / / L k w of Proposition 2.4.
	Example 3.13. Let Q be the quiver	1	2 9 9 s s s s	% % K K K K / / 3	and w

3 1 2 1

Acknowledgements. This work was done when the first author was a postdoc at NTNU, Trondheim. She would like to thank the Research Council of Norway for financial support. Part of this work was done while the second author visited NTNU during March and August 2009. He would like to thank the people in Trondheim for their hospitality.

The authors would also like to thank the referee for reading the paper thoroughly, and for several useful suggestions.

All authors were supported by the Storforsk-grant 167130 from the Norwegian Research Council. The second author was supported by JSPS Grant-in-Aid for Scientific Research 21740010 and 21340003. The fourth author was also supported by the NSA-grant MSPF-08G-228.

Let w be an element of W Q , and w = s u 1 . . . s u l a reduced expression. Recall from Section 1 that for j = 1, . . . , l the layer L j w is defined to be the Λ-module:

L j w = e u j I u k . . . I u 1 I u j . . . I u 1 = I u j-1 . . . I u 1 I u j . . . I u 1

where k < j satisfies u k = u j and is maximal with this property.

Example 3.2. Let Q be the quiver 2

3 and w := s 1 s 2 s 3 s 1 s 2 s 1 .

The standard cluster-tilting object M w in Sub Λ w has the following indecomposable direct summands

.

Then we can easily compute the layers L 1 w , . . . , L 6 w . They are the indecomposable summands of the M i w as kQ-modules:

, and L 6 w = 3 1 .

Here is a nice characterization of c-sortable words.

Theorem 3.3. Let w be an element of W Q and w = s u 1 s u 2 . . . s u l be a reduced expression of w.

Then we have the following:

(1) if there exists a Coxeter element c such that w is c-sortable and w is the c-sortable expression of w, then L j w is in mod kQ for all j = 1, . . . , l, where Q is admissible for the Coxeter element c;

(2) if for all j = 1, . . . , l the layer L j w is in mod kQ for a certain orientation of Q, then w is c-sortable, where c is the Coxeter element admissible for the orientation of Q.

Proof. (1) Assume that w = s u 1 . . . s u l is a c-sortable word. Let j ≥ 1, and k be the (possibly) last index such that u j = u k and k < j. Since w is c-sortable, the word s u 1 . . . s u j is a subsequence of cs u 1 . . . s u k . Therefore we have that L j w = e u j I u k . . . I u 1 I u j . . . I u 1 is a kQ-module by Lemma 2.11(2).

(2) We prove this assertion by induction on the length of the word w. For l(w) = 1 the result is immediate. By Lemma 2.1 we can assume that the support of w is Q 0 .

Assume that (2) is true for any word w of length ≤ l -1 and let w := s u 1 . . . s u l be a reduced expression such that L j w is a kQ-module for all j = 1, . . . , l. We first show that u 1 is a source of Q. Assume it is not, then there exists k ≥ 2 such that there is an arrow u k → u 1 in Q. Take the smallest such number. It is then not hard to check that the top of L k w is the simple S u k and that the kernel of the map L k w → S u k contains S u 1 in its top. Thus L k w is not a kQ-module, which is a contradiction. Therefore u 1 is a source of Q.

We have L j w = L j-1 w ′ ⊗ Λ I u 1 for j = 2, . . . , l by Proposition 2.2, where w ′ := s u 2 . . . s u l . By Theorem 2.7 (1) we have [L

) in the Grothendieck group K 0 (D b (f.l. Λ)). By Theorem 2.7 (2) we then have [L j w ] = [S u 1 ] for j ≥ 2. Thus L j w is not isomorphic to the simple projective e u 1 kQ = S u 1 if j ≥ 2. Then by Corollary 2.12, we get

Let (Q, c, w) be an admissible triple with w = s u 1 s u 2 . . . s u l . For j = 1, . . . , l, we define kQ-modules U j w by induction on l. If l = 1 then we define U 1 w = e u 1 kQ (0) , the simple projective kQ (0) -module associated to the vertex u 1 .

Assume l ≥ 2. Then we write w = s u 1 w ′ . It is not hard to check that the triple

is an admissible triple with l(w ′ ) = l -1. Therefore by the induction hypothesis we have

at the source u 1 of Q (0) .

Example 3.7. Let Q be the quiver 2

3 and w := s 1 s 2 s 3 s 1 s 2 s 1 in Examples 3.2 and 3.5.

Let us now compute the U j w 's. By definition

.

And finally we have

where Ri is the reflection functor associated to the quiver 1 / / 2 . Therefore we have

1 . So we have U j w = L j w = T j w for any j. Theorem 3.8. Let w = s u 1 . . . s u l be a c-sortable word where c is admissible for the orientation of Q.

(1) We have L j w ≃ U j w for j = 1, . . . , l.

(2) We have U j w ≃ T j w for j = 1, . . . , l and f j w is a monomorphism for j = l(c

We have the following exact sequence:

(1) T w is a tilting kQ (0) -module.

(2) We have Sub (T w ) = add {L 1 w , . . . , L l w }. Proof. (1) We prove that T w is a tilting kQ (0) -module by induction on l = l(w).

If w = c (0) , then the assertion is clear since T w = kQ (0) by definition. We consider the case w = c (0) . Let k be the maximal integer such that k < l and u k = u l . By the induction hypothesis we know that T w ′ = (T w /T l w ) ⊕ T k w is a tilting kQ (0) -module where w ′ is the word defined by w = w ′ s u l . By definition we have an exact sequence

with a minimal left add {T k+1 w , . . . , T l-1 w }-approximation f . Then f is a minimal left add (T w /T l w )approximation, using the fact that w is c-sortable and Corollary 3.10 (2). Moreover f is a monomorphism by Theorem 3.8. By Proposition 1.5, we have that T w is a tilting kQ (0) -module.

(2) We prove that Sub (T w ) = add {L 1 w , . . . , L l w } by induction on l = l(w). If l(w) = 1, then the assertion is clear. Assume that l ≥ 2 and write w = s u 1 w ′ . Case 1: u 1 is in the support of w ′ : this means that t w (u 1 ) ≥ 2. Thus we have

By definition of the T j w there exists a short exact sequence:

where E is in add {T 2 w , . . . , T j-1 w } and where j is the minimal integer such that u j = u 1 and j > 1. It exists since u 1 is in the support of w ′ .

The approximation map is a monomorphism by Theorem 3.8 (2), thus

Then it is easy to see that

And we get

Remark 3.12. (a) The short exact sequence L k w / / f / / E / / / / L j w in mod kQ is an almost split sequence in the category Sub (T w ).

From a c-sortable word w we define in this subsection, kQ (0) -modules T w j , T w and U w j in a dual manner to the modules T j w , T w and U j w defined in the previous subsections. For j = 1, . . . , l, we define kQ (0) -modules T w j . For 1 ≤ j ≤ l(c (0) ), T w j is the injective kQ (0)module e u j D(kQ (0) ). For j > l(c (0) ), let k be the maximal integer such that k < j and u k = u j . We define T w j as the kernel of the map f w j : E → T w k where f w j is a minimal right add {T w k+1 , . . . , T w j-1 }-approximation. Then we define a kQ-module T w as the direct sum T w = i∈Q (0) 0 T w tw(i) , where t w (i) is the maximal integer such that u tw(i) = i.

A co-admissible triple is a triple (Q, c, w) consisting of an acyclic quiver Q, a Coxeter element c admissible with the orientation of Q, and a co-c-sortable word w = c (m) c (m-1) . . . c (0) .

Let (Q, c, w) be a co-admissible triple with w = s u l s u l-1 . . . s u 1 . For j = 1, . . . , l, we define kQ-modules U w j by induction on l. If l = 1 then we define U w 1 = e u 1 D(kQ (0) ), the simple injective kQ (0) -module associated to the vertex u 1 .

Assume l ≥ 2. Then we write w = w ′ s u 1 . It is not hard to check that the triple

is a co-admissible triple with l(w ′ ) = l -1. Therefore by the induction hypothesis we have

Then a dual version of Theorems 3.8 (2) and 3.11 hold. More precisely we have the following.

Theorem 3.20. Let Q be an acyclic quiver and c be the associated Coxeter element. Then for a co-c-sortable element w the following hold.

(a) For all j = 1, . . . , l, we have T w j ≃ U w j , and f w j is an epimorphism. (b) The subcategory Fac (T w ) is finite, and Fac (T w ) = add {T w 1 , . . . , T w l }. Dually to Theorem 3.16, we also have the following.

Theorem 3.21. For any tilting module T ∈ mod kQ such that Fac T is finite, then the following hold.

(a) The AR-quiver of Fac (T ) is a full subquiver of ZQ.

(b) There exists a (unique) co-c-sortable word w (which can be constructed from the ARquiver of Fac (T )) such that T ≃ T w .

Corollary 3.22. There is 1-1 correspondences (a) {finite torsion classes of mod kQ containing D(kQ)} o o 1:1 / / {co-c-sortable words with c (0) = c} .

(b) {finite torsion classes of mod kQ} o o 1:1 / / {co-c-sortable words} .

Then w is not c-sortable, Sub T w is finite and one can check that w = s 1 s 2 s 3 s 4 s 1 s 2 s 3 s 4 s 1 s 2 s 3 s 4 s 2 s 3 .

When w is c-sortable, w is a monotilting word and T w coincides with L w given by the layers. In general L w is not a kQ-module, but as we have seen there is an indecomposable kQ-module associated with each indecomposable summand of L w , and hence a kQ-module (L w ) Q associated with L w . In this connection we have the following questions: Problem 4.6. :

(1) For which w does the following hold ? (a) each indecomposable summand of (L w ) Q is rigid, (b) (L w ) Q is a tilting module, (c) w is tilting and

As we already saw in Example 2.8, it can happen that (a) fails. In this example, one can check that w is monotilting. To exchange 3 1 we have to use the minimal right approximation g : . The cluster-tilting object M w of Sub Λ w associated with w has the indecomposable summands:

We then see that T w = (L w ) Q , even though w is not a monotilting word.

Example 4.8. Let Q and w be as in Example 4.5. Then we have

⊕ 4 . Each indecomposable summand is rigid, but (L w ) Q is not a tilting module. Therefore we can have (a) without (b).


G Brethouwer

P Schlatter

A V Johansson

Instabilities and transition in three-dimensional flows with rotation TURBULENCE AND INSTABILITIES IN ROTATING CHANNEL FLOW SIMULATIONS

Introduction

In this study, we investigate periodic-like instabilities occurring in turbulent channel flow subject to rapid spanwise rotation. It is now well established that turbulence is enhanced on the so-called unstable side and damped on the stable side of a channel at low to moderate spanwise rotation rates [START_REF] Johnston | Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow[END_REF], but at high rotation rates it is also strongly damped at the unstable side as shown by recent DNS at friction Reynolds number Re τ = 180 [START_REF] Grundestam | Direct numerical simulations of rotating turbulent channel flow[END_REF]. A linear stability analysis showed that beyond a critical rotation number Ro b = 2Ωh/U b all modes with k z = 0 decay for a given Re b = U b h/ν [START_REF] Wallin | Linear stability analysis of rotating channel flow[END_REF]. Here U b is the bulk mean velocity, h the channel half-width, ν the viscosity, Ω the rotation rate and k z the spanwise wavenumber. This stability analysis was convincingly supported by DNS; when Ro b was larger than the critical value a disturbed channel flow became fully laminar after some time. However, the so-called Tollmien-Schlichting (TS) modes with k z = 0 are not directly affected by spanwise rotation. Commonly, the growth of TS waves is inhibited by turbulence in wallbounded flows, but DNS by Wallin et al. [START_REF] Wallin | Linear stability analysis of rotating channel flow[END_REF] indicated that when turbulence is sufficiently weakened by rotation TS waves can grow and cause instabilities.

Numerical studies of rotating channel flow were so far restricted to low Re b . Our aim is to study rotating channel flow at higher Re b since it is likely that turbulence damping and flow relaminarization are sensitive to Re b . In particular, we want to examine whether and when instabilities can occur and get a better understanding of their formation and evolution.

Direct numerical simulations

We have used a spectral code to perform DNS of fully developed spanwise rotating channel flow with Re b fixed at 20000 for Ro b 1.2 and at 30000 for 1.5 Ro b 2.4. The domain size is 8πh × 3πh in the streamwise and spanwise direction in all cases and the spatial resolution in terms of wall units is similar as in other well-resolved DNS. When statistically stationary, Re τ based on the mean of the shear stresses at both walls is 1000 at Ro b = 0 but declines when Ro b is increasing. At the highest rotation rate Ro b = 2.4 and Re b = 30000, Re τ is close to 300, which is the Re τ of a fully laminar flow at this Re b , indicating that turbulent stresses are very weak. The mean velocity profile in this run is accordingly close to the laminar Poiseuille profile.

When the rotation is not too high the turbulent kinetic energy, wall shear stresses and other flow statistics have similar temporal variations as for other wall-bounded flows according to time series, however, when Ro b 0.9 the temporal variations are suddenly much larger. Figure 1(a) shows as an example the time series of the averaged wall-shear stresses τ w at the stable and unstable channel side and the volume integrated streamwise velocity fluctuation u rms at Ro b = 1.2 and Re τ = 420. Both τ w at the stable side and u rms show sudden, synchronous and sharp rises followed by a decay in a periodic-like manner. Figure 1(b) shows for the same simulation the mean profiles of the velocity fluctuations in all the three directions during the sudden bursts (dashed lines) and during the relatively calm periods in between those bursts (solid lines). We see that the turbulence in the unstable (left) part of the channel is hardly affected by the bursts whereas in the stable part the turbulence intensity is low in the period between the burst but that during the burst all three fluctuating velocity components are strongly enhanced. Similar periodic-like features are also observed in the other simulations with Ro b 0.9 although the time period between the bursts and their intensity depend on Ro b .

An explanation of the periodic-like bursts is that at high Ro b turbulence is weak at the stable channel side (figure 1.b) and therefore cannot inhibit the formation of two-dimensional Tollmien-Schlichting (TS) waves. Figure 2 gives evidence for the presence of such TS waves with wave length 2πh in one run. These TS waves can be observed on the stable side long before a burst occurs and grow slowly at first. However, after some time their growth rapidly accelerates and they become unstable producing Λ-shaped vortices. Figure 3(a) shows a visualization of such a Λ-shaped vortex in its early stage and figure 3(b) one in a later stage when it starts to break down and then produce intense turbulence as shown by figure 1(b). The turbulence destroys the TS waves and then it slowly decays so that after some time TS waves can grow again and the cycle repeats itself.

Figure 1 :Figure 2 :

 12 Figure 1: (a) Time series of the volume integrated u rms and the mean wall shear stress τ w at the stable and unstable channel side. (b) Rms values of the velocity fluctuations during a burst (dashed lines) and in the calm period in between the bursts (solid lines). In both cases Ro b = 1.2 and Re τ = 420.

Figure 3 :

 3 Figure 3: Visualizations of a Λ-shaped vortex near the stable wall (a) at Ro b = 1.2 and (b) at Ro b = 1.5.


Yohann Duguet

Geert Brethouwer

Philipp Schlatter

Instabilities and transition in three-dimensional flows with rotation LAMINAR-TURBULENT PATTERNS IN ROTATING PLANE COUETTE FLOW

Numerical simulations

The present investigation focuses on alternately laminar and turbulent oblique patterns observed during the subcritical transition to turbulence in wall-bounded flows. It is shown in the example of plane Couette flow that cyclonic spanwise rotation can delay the formation of such patterns to arbitrary large values of the Reynolds number.

Transition to turbulence in non-rotating plane Couette flow

Plane Couette flow (pCf), the flow between two parallel walls moving in opposite directions, is the simplest canonical example of the effect of a wall-induced shear on a viscous fluid. The only nondimensional parameter ruling the flow is the Reynolds number, here defined as Re = Uh ν , where ±U is the velocity of the two walls, h is the half-gap between them, and ν is the kinematic viscosity of the fluid. We are interested in the way sustained turbulence appears in this system near the onset of transition, knowing that the laminar base flow is linearly stable for all Re. We present here direct numerical simulations in large periodic domains, using a pseudo-spectral code [START_REF] Duguet | Formation of turbulent patterns near the onset of transition in plane Couette flow[END_REF]. Two thresholds value for the Reynolds number characterise this subcritical transition: Re g , below which all perturbations are found to eventually decay, and Re t , above which turbulent motion, once triggered, contaminates the whole domain. In the interval [Re g , Re t ], the turbulent flow features robust coexisting laminar and turbulent domains separated by oblique fronts [START_REF] Prigent | Large-scale finitewavelength modulation within turbulent shear flows[END_REF][START_REF] Barkley | Computational study of turbulent laminar patterns in Couette flow[END_REF][START_REF] Duguet | Formation of turbulent patterns near the onset of transition in plane Couette flow[END_REF].

Effects of a spanwise cyclonic rotation

Cyclonic rotation applied to pCf in the spanwise direction (i.e. where the rotation vector has the same direction as the mean flow vorticity) does not modify the laminar base flow. Furthermore it is known to stabilise the laminar state, thus potentially delaying transition to turbulence in pCf [START_REF] Tsukahara | Flow regimes in a plane Couette flow with system rotation[END_REF]. If F is a parameter measuring the amplitude of this cyclonic rotation (here the opposite of a Rossby number), experimental evidence suggests that the two previously defined thresholds Re g and Re t still exist yet directly increase with F . In particular, the flow just above Re g is still characterised by the stable coexistence of alternatively laminar and turbulent domains. We show here numerical evidence of the laminar/turbulent patterns forming at arbitrary large values of Re in the presence of rotation (see Figure 1). Note that very similar conclusions can be drawn under other stabilizing external forces than rotation, such as a spanwise magnetic field or a stable density stratification.

EUROMECH Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation 2 Scale separation Spectral analysis of the mean flow suggests the existence of two length scales for the patterned flow: a large-scale wavelength Λ z (resp. Λ x ) corresponding to the oblique patterns and a small-scale wavelength λ z (resp. λ x ) related to the presence of near-wall streaks, as in fully developed wall turbulence. It is shown that λ z scales classically with wall units following the relation λ z ≈ 100h + , while Λ z scales directly in outer units of h. It follows that parameter space-tracking of the patterned regime at higher and higher rotation rates widens the associated scale separation. An analysis of the mean flow also reveals its complex three-dimensional structure. A tempting conclusion is that the small-scale (turbulent) fluctuations do not directly contribute to the mechanism for the formation of the patterns, apart throught their active role in sustaining the large-scale circulation.

Computer time provided by SNIC (Swedish National Infrastructure for Computing) is gratefully acknowledged.

Figure 1 :

 1 Figure 1: Streamwise velocity in the mid-plane between the plates. The mean flow direction is horizontal. Left: non-rotating pCf for Re=350. Right: rotating pCf for Re=6000 and Ro=-0.104. Domain sizes: 250h × 125h (top) and 250h × 200h (bottom).