semicanonical bases and preprojective algebras · 2006-12-11 · semicanonical bases and...

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS

CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHROER

Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday.

ABSTRACT. We study the multiplicative properties of the dual of Lusztig’s semicanonical basis.The elements of this basis are naturally indexed by the irreducible components of Lusztig’s nilpotentvarieties, which can be interpreted as varieties of modules over preprojective algebras. We provethat the product of two dual semicanonical basis vectors �� and �� is again a dual semicanonicalbasis vector provided the closure of the direct sum of the corresponding two irreducible components��

and��

is again an irreducible component. It follows that the semicanonical basis and thecanonical basis coincide if and only if we are in Dynkin type � with �� . Finally, we provide adetailed study of the varieties of modules over the preprojective algebra of type �� . We show that inthis case the multiplicative properties of the dual semicanonical basis are controlled by the Ringelform of a certain tubular algebra of type �� and by the corresponding elliptic root system oftype � �"!�# !%$& .

RESUME. Nous etudions les proprietes multiplicatives de la base duale de la base semi-canoniquede Lusztig. Les elements de cette base sont naturellement parametres par les composantes irreduc-tibles des varietes nilpotentes de Lusztig, qui peuvent etre interpretees comme varietes de modulessur les algebres preprojectives. Nous demontrons que le produit de deux vecteurs �� et �� dela base semi-canonique duale est encore un vecteur de la base semi-canonique duale si la sommedirecte des composantes irreductibles

� �et� � �

est encore une composante irreductible. Il en resulteque les bases canonique et semi-canonique ne coıncident que pour le type de Dynkin � avec�� '� . Finalement, nous etudions en detail les varietes de modules sur l’algebre preprojective detype �� . Nous montrons que dans ce cas les proprietes multiplicatives de la base semi-canoniqueduale sont controllees par la forme de Ringel d’une algebre tubulaire de type ��(�)��)�*� et par lesysteme de racines elliptique de type � �+!�# !,$& qui lui est associe.

CONTENTS

1. Introduction 22. Varieties of modules 63. Preprojective algebras 84. Constructible functions 95. Semicanonical bases 96. Comultiplication 127. Multiplicative properties of the dual semicanonical basis 148. Embedding of -/. into the shuffle algebra 159. A Galois covering of 0 for type 132 1910. From Schur roots to indecomposable multisegments 2311. Cases 154 , 176 , 198 : the component graph 2512. Cases 154 , 176 , 198 : the graph of prime elements of : . 26

Mathematics Subject Classification (2000): 14M99, 16D70, 16E20, 16G20, 16G70, 17B37, 20G42.C. Geiss acknowledges support from DGAPA-UNAM for a sabbatical stay at the University of Leeds. J. Schroer wassupported by a research fellowship from the DFG (Deutsche Forschungsgemeinschaft). He also thanks the LaboratoireLMNO (Caen) for an invitation in Spring 2003 during which this work was started. B. Leclerc is grateful to the GDR2432 (Algebre non commutative) and the GDR 2249 (Groupes, geometrie et representations) for their support.

1

2 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHROER

13. End of the proof of Theorem 1.2 2614. Case 1�� : the tubular algebra

�and the weighted projective line � 27

15. Case 1 � : the root system 3016. Case 1�� : parametrization of the indecomposable irreducible components 3517. Case 1�� : the component graph 3618. Proof of Theorem 10.3 3719. Concluding remarks 3920. Pictures and tables 41References 51

1. INTRODUCTION

1.1. Let � be a simple Lie algebra of simply-laced type 1�� , and let � be a maximal nilpotentsubalgebra. Let :� be the canonical basis of the quantum enveloping algebra �� [29, 37] and: . the basis dual to :� . When � tends to 1, these two bases specialize to bases : and : . of ��and �� , respectively. Here � stands for a maximal unipotent subgroup of a complex simple Liegroup � with Lie algebra � .

Let denote an indexing set for the simple roots of � . Given a finite-dimensional -gradedvector space ! with graded dimension " !#" , we denote by 0�$ the corresponding nilpotent variety,see [38, % 12]. This variety can be seen as the variety of modules over the preprojective algebra 0attached to the Dynkin diagram of � , with underlying vector space ! [46].

For a variety & let ')(�(*�+&,� be the set of irreducible components of & . Lusztig has shown thatthere are natural bijections

'-(.(*� 0�$/��0�1 :2 3��" !#"4� (resp. : . ��" !#"4� )576198;: (resp. 8 .: )

where < ��" !#"4� (resp. < . ��" !#"4� ) is the subset of : (resp. : . ) consisting of the elements ofdegree " !#" . Kashiwara and Saito [30] proved that the crystal basis of �� 3�� can be constructedgeometrically in terms of these irreducible components (this was a conjecture of Lusztig).

This paper is motivated by several problems about the bases :� and : . and their relations withthe varieties 0 $ and the preprojective algebra 0 .

1.2. One problem, which was first considered by Berenstein and Zelevinsky [4], is to study themultiplicative structure of the basis : . . Two elements 8 . = and 8 .4 of : . are called multiplicativeif their product belongs to : . up to a power of � . It was conjectured in [4] that 8�. = and 8 .4 aremultiplicative if and only if they � -commute. We refer to this as the BZ-conjecture. The conjecturewas proved for types 1 4 and 1 6 [4], and it also holds for 1 8 [51].

More recently, Marsh and Reineke observed a strong relationship between the multiplicativestructure of : . and properties of the irreducible components of the varieties 0>$ . They checked[42] that for � of type 1 2?�+@BADCE� , if the irreducible components

5 =GF 0�$ ! and5 4 F 0�$�H are

the closures of the isomorphism classes of two indecomposable 0 -modules I = and I 4 , then 8 .: !and 8 .:JH are multiplicative if and only if K2LNM =O �+I = ��I�4P�/QSR . This was verified by a case-by-casecalculation, using the fact that for type 1 2,�+@#ATCE� the preprojective algebra is of finite represen-tation type, that is, it has only a finite number of isomorphism classes of indecomposable modules[15]. They also calculated many examples in type 138 and conjectured that this property still holdsin this case (note that 0 is again representation-finite for 1 8 ). But a conceptual explanation wasstill missing.

Let5 =VU 5 4 denote the subset of 0W$ !-X $ H consisting of all 0 -modules I isomorphic to Y =ZU Y 4

with Y =\[ 5 = and Y 4 [ 5 4 . It follows from a general decomposition theory for irreducible

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS 3

components of varieties of modules developed in [14] that the condition K2LNM =O �+I = ��I�4*��Q R forsome �+I = ��I�4*� [ 5 = � 5 4 is equivalent to

5 = U 5 4 being an irreducible component of 0 $ ! X $�H .In [31] counterexamples to the BZ-conjecture were found for all types other than 1 2 with@ A �

. In particular in type 1�� , for a certain ! of dimension 8 one can find an irreduciblecomponent

5of 0�$ such that

(1) � 8 .: � 4 Q7�� 4 � 8 .: �� 8 .: � � �where

5�� Q 5 U 5 and5� �

are two irreducible components of 0�$ X $ , see also [25]. This seemsto be the smallest counterexample to the BZ-conjecture in type 1 . Moreover, it also shows thatthe result of Marsh and Reineke does not generalize to 1W� . Note however that the BZ-conjecturewas proved for large families of elements of : . [8, 9, 10, 33]. For example, in type 1 it holds forquantum flag minors, and the reformulation in terms of direct sums of irreducible components isalso valid [49].

So one would like to get a better understanding of the relationship between multiplicativityof elements of : . and direct sum decompositions of irreducible components of varieties of 0 -modules.

1.3. Another interesting problem concerns the singular supports of the simple perverse sheavesused by Lusztig [37] to define the canonical basis : . Let be a Dynkin quiver, which is obtainedfrom the Dynkin diagram of � by choosing an orientation. Let (�� ! � be the affine space ofrepresentations of with underlying finite-dimensional -graded vector space ! . This is a finiteunion of isomorphism classes (or orbits) � . In Lusztig’s geometric construction, the elementsof :2 3��" !#"4� are given by the perverse extensions �� of the constant sheaves �� on the orbits � .In [38] Lusztig considered the singular supports �� of these sheaves and showed that theyare unions of irreducible components of 0 $ (independent of the chosen orientation of the Dynkindiagram of � ). He conjectured that in fact each �� is irreducible, equal to the closure 0 � ofthe conormal bundle of � . Unexpectedly, Kashiwara and Saito [30] produced a counterexampleto this conjecture. They exhibited two orbits � � �� for type 1�� such that

�� Q 0 � �� 0 � � ��The corresponding vectors 8 � � and 8 � � � of :2 have principal degree 16, and apparently this is thesmallest counterexample in type 1 .

It turns out that this counterexample is dual to the counterexample above for : . , in the sensethat 0 � � Q 5�

and 0 � � � Q 5� �, see [31, Remark 1]. One motivation for this paper was to find an

explanation for this coincidence.

1.4. What makes these problems difficult is that, although the canonical basis reflects by defini-tion the geometry of the varieties �� (�� ! � , we want to relate it to the geometry of someother varieties, namely the irreducible components of the nilpotent varieties 0 $ . It is natural tothink of an intermediate object, that is, a basis reasonably close to the canonical basis, but directlydefined in terms of the varieties 0 $ . Lusztig [41] has constructed such a basis � Q��! J:#" andcalled it the semicanonical basis. This is a basis of �� (not of the � -deformation � 3�� ) whichgives rise, like : , to a basis in each irreducible highest weight ��N� -module. Let � . Q$�&% : "denote the basis of � � � � dual to � . Our first main result is the following:

Theorem 1.1. If5 = F 0�$ ! and

5 4 F 0�$WH are irreducible components such that5 Q 5 = U 5 4

is an irreducible component of 0�$ ! X $WH , then %N: ! %J:JH Q'%J: .

In other words, the dual semicanonical basis � . satisfies the multiplicative property which wasexpected to hold for the dual canonical basis : . .


An irreducible component5 [ ')(�(*� 0�$ � is called indecomposable if

5contains a dense subset

of indecomposable 0 -modules. By [14], every irreducible component5

of 0>$ has a canonicaldecomposition 5 Q 5 = U��3U 5 �where the

5�� F 0�$�� are indecomposable irreducible components. Our theorem implies that

%J: Q'%J: ! �� %J:� �Hence � . has a natural description as a collection of families of monomials in the elements indexedby indecomposable irreducible components. Such a description of � . resembles the descriptionof : . for type 1 2 �+@?A � � obtained by Berenstein and Zelevinsky.

1.5. So a natural question is how close are the bases � . and : . ? In type 1 , Berenstein andZelevinsky [4] proved that all minors of the triangular matrix of coordinate functions on � belongto : . . We prove that they also belong to � . . Hence using [33, 49], it follows that � .� : . containsall multiplicative products of flag minors. However the two bases differ in general. More preciselywe have:

Theorem 1.2. The bases � . and : . coincide if and only if � is of type 1 2 with @?A �.

For example in type 1 � , we deduce from Equation (1) and Theorem 1.1 that

(2) % : � Q 8 .: � � 8 .: � �(where for simplicity we use the same notation 8 .: � and 8 .: � � for the specializations at �\Q � ).Nevertheless, since � . and : . have lots of elements in common, we get an explanation why theBZ-conjecture (or rather its reformulation in terms of irreducible components of varieties of 0 -modules) holds for large families of elements of : . .

Of course, by duality, these results also allow to compare the bases � and : . In particular,returning to the example of [30], we can check that

(3) 8 � � � Q O�� O�� and this is probably the smallest example in type 1 for which the canonical and semicanonicalbases differ. One may conjecture that, in general, the elements : occurring in the � -expansion of8 � [ : are indexed by the irreducible components

5of �� , so that �� is irreducible

if and only if 8 � Q O � . (There is a similar conjecture of Lusztig [40] for the “semicanonicalbasis” of the group algebra of a Weyl group obtained from the irreducible components of theSteinberg variety.) Assuming this conjecture we get an explanation of the relationship betweenthe counterexamples to the conjectures of Berenstein-Zelevinsky and Lusztig.

1.6. In the last part of the paper, we consider the first case which is not well understood, namelytype 1 � . In this case, the preprojective algebra 0 is representation-infinite, but it is still of tamerepresentation type [15]. Motivated by our description of � . in terms of indecomposable irre-ducible components of varieties of 0 -modules, we give a classification of the indecomposableirreducible components for the case 1�� . We also give an explicit criterion to decide when the clo-sure of the direct sum of two such components is again an irreducible component. These resultsare deduced from [24], in which a general classification of irreducible components of varieties ofmodules over tubular algebras is developed. They are naturally formulated in terms of the Ringelbilinear form � 0�P0�� of a convex subalgebra

�of a Galois covering of 0 . The algebra

�is a

tubular algebra of type ��J�.CJ��E� and the corresponding 10-dimensional infinite root system � is an

elliptic root system of type �� =��4=�� in the classification of Saito [50], with a 2-dimensional lattice ofimaginary roots. Note that the irreducible component

5of Equation (1) corresponds to a generator

of this lattice. (This is an a posteriori justification for calling 8�.: an imaginary vector in [31].) TheRingel form � 0�P0�� allows to define a distinguished Coxeter matrix � of order � acting on � . Weprove the following:


Theorem 1.3. There is a one-to-one correspondence � 61 5 �� between the set of Schur rootsof � and the set of indecomposable irreducible components of the nilpotent varieties of type 1>�which do not contain an indecomposable projective 0 -module. Moreover

5 �� = � U 5 ��4P� is anirreducible component if and only if the Schur roots � = and ��4 satisfy certain conditions which areall expressible in terms of � 0�P0�� and � .

We also explain how to translate from the language of roots to the language of multisegments,which form a natural indexing set of canonical and semicanonical bases in type 1 .

1.7. The paper is organized as follows. In Section 2 we recall the general theory of varieties ofmodules. We explain a general decomposition theory for irreducible components of such varieties.This is followed in Section 3 by a short introduction to preprojective algebras. Then we recall theconcept of a constructible function in Section 4. Following Lusztig [41], we review in Section 5 thedefinition of the semicanonical basis of �� , which is obtained by realizing �� as an algebra- of constructible functions on the nilpotent varieties. In order to study the dual semicanonicalbasis and its multiplicative properties we also need to describe the natural comultiplication of�� in terms of - . This was not done in [41], so we provide this description in Section 6. InSection 7 we introduce the dual semicanonical basis � . of - . and prove Theorem 1.1. Notethat for this theorem we do not restrict ourselves to types 1 � � � , and only assume that � is thepositive part of a symmetric Kac-Moody Lie algebra. We end this section with the proof of the“only if” part of Theorem 1.2. In Section 8 we embed - . into the shuffle algebra. This gives apractical way of computing elements of � . . We use this to prove that in type 1 all nonzero minorsin the coordinate functions of � belong to � . . In the rest of the paper we focus on the Dynkincases 192 �+@ A � � . In Section 9 we consider a Galois covering

�0 of the algebra 0 , with Galois

group � , and we use it to calculate the Auslander-Reiten quiver of 0 for @?A �. We also introduce

an algebra�

whose repetitive algebra is isomorphic to�0 . For @ A �

,�

has finite representationtype, while for @DQ �

it is a tubular algebra of tubular type ��J�.CJ��E� . In Section 10 we recallfrom [24] that the indecomposable irreducible components of 0 are in one-to-one correspondencewith the � -orbits of Schur roots of

�0 . We also describe the map which associates to such a Schur

root the multisegment indexing the corresponding indecomposable irreducible component. Thecomponent graphs for the representation-finite cases 1 4 , 156 and 158 are constructed in Section 11,and the corresponding graphs of prime elements of : . are described in Section 12. In Section 13we prove the “if” part of Theorem 1.2. All the remaining sections are devoted to the case 1 � .In Section 14 we relate the category of

�0 -modules to the category �� of modules over

the tubular algebra�

and to the category �� +�� of coherent sheaves on a weighted projectiveline � of type ��J�.CJ��E� in the sense of Geigle and Lenzing [23]. In Section 15 we consider theGrothendieck groups �� +�� = � . They are naturally endowed with a(non-symmetric) bilinear form � 0/�P0�� (the Ringel form) and a Coxeter matrix. This gives rise to

an elliptic root system of type � = �4=�� . We give an explicit description of its set of positive rootsand of the subset �� of Schur roots. In Section 16, we show that �� naturally parametrizes the� -orbits of Schur roots of

�0 , hence also the indecomposable irreducible components of 0 . Then

Section 17 describes the component graph of 0 for type 1 � , thus making precise the statementsof Theorem 1.3. Section 18 consists of the proof of Theorem 10.3. We conclude by noting theexistence of similar results for type �38 and by pointing out some possible connections with thetheory of cluster algebras of Fomin and Zelevinsky (Section 19). Section 20 contains a collectionof pictures and tables to which we refer at various places in the text.

1.8. Throughout, we use the following conventions. If �� = 1!� 4 and "#� � 4 1$� 6 aremaps, then the composition is denoted by "� %�� = 1&� 6 . Similarly, if '(� � 1 � and )#� � 1 Care arrows in a quiver, then the composition of ' and ) is denoted by )*' .

Modules are always assumed to be left modules.


All vector spaces are over the field � of complex numbers.We set � � Q �*� [ � "�� R�" , � � Q �*� [ � " �� R�" and �� Q�� " . We also set

� � Q �� [ � "� � R�" and �TQ � � � �*R�" .

2. VARIETIES OF MODULES

2.1. A quiver is a quadruple Q � �� = �� where and = are sets with non-empty, and�E�� = 1 are maps such that � � = ��)� and � � = ��)� are finite for all � [ . We call the set ofvertices and = the set of arrows of . For an arrow ' [ = one calls � � ' � the starting vertexand � � ' � the end vertex of ' .

A path of length � in is a sequence � Q ' = ' 4 �� '�� of arrows such that �J� ' � ��Q�� ' � � = � for�A��A�� 0 � . Set �J��Z��Q�� '�� and � ��Z��Q�� ' = � . Additionally, for each vertex � [ let � � be apath of length 0. By � we denote the path algebra of , with basis the set of all paths in andproduct given by concatenation. A ��! #"�$&%('*) for is a linear combination

�+�-, =/.

� � �where .

� [ � and the � � are paths of length at least two in with �J�� Q0�J��21*� and � �� Q0� ��21 �for all � A��43 A�� . Thus, we can regard a relation as an element in �� .

An ideal 5 in � is admissible if it is generated by a set of relations for . Note that this differsfrom the usual definition of an admissible ideal, where one also assumes that the factor algebra� 7685 is finite-dimensional.

2.2. A map 9 �� #1 � such that ;:<9 � = � R � is finite is called a dimension vector for . Wealso write 9 � instead of 9 ��)� , and we often use the notation 9 Q �&9 � � ��=> . By � � > � we denote thesemigroup of dimension vectors for .

Let ?A@CB � J� be the category of finite-dimensional -graded vector spaces. Thus, the objects of?A@CBN� N� are of the form ! QED ��=>�F � where the F � are finite-dimensional vector spaces, and onlyfinitely many of the F � are nonzero. We call " !#"EQ ��HG � � F � �� => the dimension vector or degree of! . The morphisms in ?�@CBN� N� are just linear maps respecting the grading. Direct sums in ?I@CB � J�are defined in the obvious way.

A representation of with underlying vector space ! [ ?J@CB � J� is an element

I Q �+ILK��MK =*N ! [ ( �� ! ��QPOK =*N !Q � � R � FTS � K � � FHU � K � � �

For a representation I Q �+I K � K =�N ! [ (�� ! � and a path � Q ' = ' 4 �� ' � in set

I2V QTILK ! ILK H �� ILK�W �Then I satisfies a relation X ��#, = .

� � � if X ��#, = .� I8V �/Q R . If � is a set of relations for , then

let ( �� ! � be the set of all representations I [ (�� ! � which satisfy all relations in � .This is a closed subvariety of (�� ! � . Let Y be the algebra ��Z685 , where 5 is the admissibleideal generated by � . Note that every element in (�� ! � can be naturally interpreted as anY -module structure on ! , so we shall also write

�� &Y ��! ��QT(�� ! � �This is the affine variety of Y -modules with underlying vector space ! . A dimension vectorfor Y is by definition the same as a dimension vector for , that is, an element of � � > � . ForI [ �� &YG��! � we call [I\&]B�+IV��Q " !#" the dimension vector of I .


2.3. Define � $ Q�� =>�� F � � � This algebraic group acts on �� &Y ��! � as follows. For" QD� " � � ��=*> [ � $ and I Q �+I K��MK =*N ! [ �� &YG��! � define

" � I Q �+I � K �MK =�N ! where I � K Q " U � K � ILK�" � =S � K � �The � $ -orbit of an Y -module I [ �� &Y ��! � is denoted by � �+IZ� . Two Y -modules I ��Y [�� &YG��! � are isomorphic if and only if they lie in the same orbit.

For a dimension vector 9 for Y set

!��>Q O ��=> �� &YG��9J��Q �� &Y ��!�� &9N��Q � $ �Thus � �� &YG� " !#"4��Q �� &Y ��! � for all ! [ ? @CB � J� . For this reason, we often do not distinguishbetween �� &Y � " !#"4� and �� &Y ��! � . Any problems arising from this can be solved via theexistence of an isomorphism between these two varieties which respects the group actions and thegradings.

2.4. By abuse of notation, we identify � [ with the dimension vector mapping � to 1 and 3 �Q �to R . If ! is an -graded vector space with " !#"EQ � , then the variety � �� &YG��! � consists just of asingle point and is denoted by

5 � � � . The corresponding 1-dimensional Y -module is denoted by � � .An element I [ �� &Y ��! � is said to be nilpotent if there exists an � [ � � such that for any

path � of length greater than � we have I V�Q R . By �� &YG��! � we denote the closed subsetof nilpotent elements in �� &YG��! � . The nilpotent elements are exactly the Y -modules whosecomposition series contains only factors isomorphic to the simple modules � � , � [ .

2.5. An irreducible component5 [ '-(.(*�� &YG��! �� is called indecomposable if it contains a

dense subset of indecomposable Y -modules. Let G�� +')(�(*�� &Y ��! �� be the set of indecompos-able irreducible components of �� &Y ��! � . Let

')(�(*�&Y ��Q �� =�� )$

'-(�(*�� &YG��9J��be the set of all irreducible components of varieties of Y -modules, and set

G�� V�+')(�( �&Y/��Q �� =�� $

G�� Z�+')(�(*�� &YG��9J��

2.6. Let5 � [ '-(�(*�� &YG��! � �� , � A � A�� be irreducible components of Y -modules, and

set ! QS! = U �� U ! � . Let5 = U �� U 5 � be the set of modules in �� &Y ��! � , which are

isomorphic to I =�U �� U I � with I � [ 5 �for all � . The closure

5 = U ��U 5 � is irreducible,but in general it is not an irreducible component. According to [14] any irreducible component5 [ ')(�(*�&Y/� has a decomposition 5 Q 5 = U��3U 5 �for certain indecomposable irreducible components

5 � [ G�� +')(�(*�&Y/�� . Moreover the components5 = � �� 5 � are uniquely determined up to reordering. This is called the canonical decompositionof5

.For irreducible components

5 �and

5 � �of Y -modules define

�;LNM =� � 5 � � 5 � � ��Q � G�� HG �\K�LJM =� �+I � ��I � � ��"N�+I � ��I � � � [ 5 � � 5 � � " �This is the dimension of the extension group K�LJM =� �+I � ��I � � � for all �+I � ��I � � � in a certain dense opensubset of

5 � � 5 � �. For irreducible components

5�� [ ')(�(*�&Y � � � A � A�� it is known that5 = U ��U 5 � is an irreducible component if and only if �;LJM =� � 5 � � 5 1 ��Q7R for all ��Q�3 , see [14].


The component graph ��&Y � of Y is defined as follows. The vertices of ��&Y � are the elementsin G�� +')(�(*�&Y/�� . There is an edge between vertices

5 �and

5� �if and only if

5 � U 5 � � is again anirreducible component, or equivalently if

�;LJM =� � 5 � � 5 � � ��Q'�;LJM =� � 5 � � � 5 � ��Q7R �

3. PREPROJECTIVE ALGEBRAS

3.1. Assume that SQ � �� = �� is a finite quiver without loops. (A loop is an arrow ' with� � ' � Q � � ' � , and is finite if is finite. Note that this implies that = is finite as well.) Thedouble quiver Q � �� = � � � �3� of is obtained from by adding for each arrow ' [ = anadditional arrow '9. . Define �J� ' �/Q �J� ' � , � � ' �/Q � � ' � , � � '9.P� Q � � ' � and � � ' .P� Q �J� ' � . Forany � [ let

� � Q +K =*N !�� S � K � , �

' . ' 0 +K =*N !�� U � K � , �

'*' .

be the Gelfand-Ponomarev relation associated to � . This is a relation for . The preprojectivealgebra � �� corresponding to is defined as

� ��G��Q7� 6 5where 5 is generated by the relations � � , � [ . These algebras were introduced and studied byGelfand and Ponomarev, compare also [46].

3.2. For a preprojective algebra 0\Q�� and some ! [ ?J@CB � J� set

0 $ Q �� G� ��! � �Lusztig proved that 0W$ has pure dimension �TG � �+(�� ! �� , i.e. all irreducible components of0�$ have dimension �HG � �+(�� ! �� . Usually the varieties 0 $ are called nilpotent varieties. If is a Dynkin quiver, then one might call them just preprojective varieties, since the nilpotencycondition follows automatically in these cases, as shown by the next proposition.

Proposition 3.1. For a preprojective algebra � ��G� the following are equivalent:

(a) � �� is finite-dimensional;(b) �� ! ��Q �� G� ��! � for all ! [ ? @!B � J� ;(c) is a Dynkin quiver.

The equivalence of (a) and (c) is classical (see for instance [44]). The implication (c) Q�� (b) isproved by Lusztig [38, 14.2], and the converse by Crawley-Boevey [13].

3.3. The following remarkable property of preprojective algebras was proved in [12].

Proposition 3.2. For finite-dimensional modules & and over a preprojective algebra 0 we have

�HG � K�LJM =O �+&�� 2Q �TG �\K2LNM =O ��&,� �Most preprojective algebras are of wild representation type. The following proposition lists the

exceptional cases. We refer to [15] and [24] for further details.

Proposition 3.3. Let � ��G� be a preprojective algebra. Then the following hold:

(a) � �� is of finite representation type if and only if is of Dynkin type 1 2 with @?A �;

(b) � �� is of tame representation type if and only if is of Dynkin type 1�� or �78 .


4. CONSTRUCTIBLE FUNCTIONS

4.1. Let & be an algebraic variety over � , endowed with its Zariski topology. A subset Y of &is said to be constructible if it is a finite union of locally closed subsets. It is easy to see that if &is irreducible and if we have a partition & Q Y =�� Y�� into a finite number of constructiblesubsets, then there exists a unique Y � containing a dense open subset of & .

A function � & 1 � is constructible if it is a finite � -linear combination of characteristicfunctions � � for various constructible subsets Y . Equivalently, 2�+& � is finite and � = � � � is aconstructible subset of & for all � [ � . The set of constructible functions on & is denoted by�S�+& � . This is a � -vector space.

4.2. If5

is an irreducible component of & and [ �S�+& � , then5 Q �

� = R �5 � = � � ��

is a finite partition into constructible subsets, hence there is a unique � [ � such that5 � = � � �

contains a dense open subset of5

. In other words, a constructible function has to be constanton a dense open subset of each irreducible component

5of & . We denote by

%J: � � �+&,��1 �the linear form associating to its value on this dense open subset of

5.

4.3. By ��&Y � we denote the Euler characteristic of a constructible subset Y . If Y < Q� , then�W�&Y � < � Q��&Y � � �W� < � . Hence � can be regarded as a “measure” on the set of constructiblesubsets of & . For [ � �+&,� it is then natural to define� � =�� 2�+IV�2Q +

� = R ��W� � = � � �� [ � �This is a linear form on � �+&,� . More generally, for a constructible subset Y of & we write� � = � 2�+IV�2Q +

� = R �� = � � � Y � �5. SEMICANONICAL BASES

5.1. In this section we assume that 9Q � �� = �� is a finite quiver without loops, and asbefore for ! [ ? @CB � N� let 0�$ Q � �� G� ��! � be the corresponding nilpotent variety. Wedenote by � � 0�$/�� the subspace of � � 0W$ � consisting of the constructible functions which areconstant on the orbits of �/$ .

5.2. For ! ��! � ��! � � such that " ! � " � " ! � � "EQ " !#" , Lusztig [41] defines a bilinear map� � �S� 0 $ � � � � � � �S� 0 $ � � � � � � � 0�1 � � 0 $ � ��as follows. Let I [ 0W$ . Define F

�to be the variety of all -graded subspaces � of ! such that

I �� and " � "�Q " ! � � " . In other words, F�

is the variety of all 0 -submodules of I withdimension vector " ! � � " . For such a � let I � [ 0 $�� and I � � [ 0�� be the elements induced byI , and let

�I � [ 0 $ � and�I � � [ 0 $ � � be the elements obtained by transporting I � and I � � via some

isomorphisms ! 6�� 0 1 ! � and � �0 1 ! � � . For � [ � � 0 $ � �� and � � [ � � 0 $ � � �� define �

�� F

�1 � by �

�� ;��Q � � �I � � � � � �I � � � �

Following [41] define

� � � � � �;�+IZ��Q �� = �"! �

�� ;��


5.3. Let � � F 0 $ � (resp. � � � F 0 $ � � ) be a � $ � -orbit (resp. a � $ � � -orbit). For I [ 0W$ let� �� IZ� be the constructible subset of F�

consisting of all 0 -submodules Y of I of isomor-phism type � � � and such that I 6 Y has isomorphism type � � . The above definition yields

�� ;�+IZ��Q �� IZ�� Note that in general the variety 0 $ has infinitely many orbits. (Indeed, by Proposition 3.3 thealgebra 0 has in general infinite representation type, hence, by the validity of the second Brauer-Thrall conjecture (see [2]), it has in general an infinite number of non-isomorphic representationsof a given dimension). Therefore the support of the function � � � � � � � � may consist of an infinitenumber of orbits.

5.4. Let�- Q D $ �S� 0 $ �� where ! runs over the set of all isomorphism classes of vector

spaces in ? @CB � N� . (For example, we can take ! [ �P! � " 9 [ � � > � " .) The operation � defines thestructure of an � � > � -graded associative � -algebra on

�- .

For � [ , we recall that5 � � � denotes the variety 0 $ where " !#" Q � . (

5 � � � is just a single point.)Following [41] define - to be the subalgebra of �

�- � � � generated by the functions � :�� , � [ .

We set - $ Q - � � 0�$/�� 5.5. For two distinct vertices �.�43 [ , let � � 1 denote the negative of the number of arrows ' [ =such that � �J� ' � �� ' � " Q � ��43 " . Set also � � � Q � �� [ J� . Let � be the symmetric Kac-MoodyLie algebra over � with Cartan matrix �� 1 � � � 1 => . Let � be a maximal nilpotent subalgebra of � ,and let �� be its enveloping algebra. We denote by � � �� [ N� the Chevalley generators of �� .The defining relations of � �� are

�� =+� , � � 0 �*� � � � � �� 1/� � �� = � � �� Q7RN�

where � � � �� Q0� �� 6�� .The algebra � �� is � � > � -graded by assigning to � � the degree � . It is known that the number

of irreducible components of Lusztig’s nilpotent variety 0 $ is equal to the dimension of the ho-mogeneous component of �� of degree " !#" . This was proved by Lusztig [38, 39] when � is offinite or affine type and by Kashiwara and Saito [30] in general.

5.6. Lusztig has proved that there is an algebra isomorphism� � � ��1 -

given by� �(� � ��Q � :�� . To do this he constructed for every -graded vector space ! a � -basis

�! :#" 5 [ ')(�(*� 0�$/� "of - $ , naturally labelled by the irreducible components of 0 $ . Using the notation of Section 4.2,it is characterized by

(4) % : � � : �2Q�� : � : � � � 5 � 5 � [ ')(�(*� 0�$ �� In other words, the function : is the unique element of - $ equal to � on a dense open subsetof5

and equal to R on a dense open subset of any other irreducible component5 �

of 0�$ , see [41,Lemma 2.5].

The basis of �� obtained by transporting via� � = the collection

�$�! : " 5 [ ')(�(*� 0 $ � "E�

where ! ranges over the set of all isomorphism classes of vector spaces in ? @CB � J� , is called thesemicanonical basis of �� and is denoted by � .


Example 1. Let be the quiver with two vertices � and � and one arrow ' � � 1 � . Thus is aDynkin quiver of type 134 .

(a) Let ! Q F = U F 4 with �HG � � F = ��Q �HG � F 4 Q � . Then

0�$ Q �PI Q ��.8 � [ � � � "��N8 Q7R�" �The variety 0W$ has two irreducible components

5 Q � �� R � " � [ ��"E� 5 � Q � � RN�.8;� "�8 [ ��" �The group � $ Q7�5. � �7. acts on 0W$ with three orbits � � RN��R � "E� 5 0 � � RN��R � "E� 5 � 0 � � RN��R � " . Wehave

:�Q � :,Q � :�� = � � � :�� 4 � � : � Q � : � Q � :�� 4 � � � :�� = � �(b) Let ! Q F = U F 4 with �HG � � F = ��Q �HG � � F 4*��Q � . Then

(�� ! ��Q � 4 � �� 4 � ��and 0�$ F (�� ! � has dimension

�. The variety 0�$ has three irreducible components

5 Q �PI [ 0�$D" ( � �+ILK��WA �� ( � �+ILK��P� A � "E� 5 � Q �PI " I K��Q7R�"E� 5 � � Q �PI?" ILK Q7R�" �We have:

:�Q � : � �� Q �� :�� = � � � :�� 4 � � � :�� 4 � � � :�� = � �2Q �

� �� :�� 4 � � � :�� = � � � :�� = � � � :�� 4 � � � : � Q � : � Q �� :�� = � � � :�� = � � � :�� 4 � � � :�� 4 � � � : � � Q � : � � Q �� :�� 4 � � � :�� 4 � � � :�� = � � � :�� = � � �

Note that : �Q � : and � : �[ - .

5.7. Next, we consider composition series of modules over preprojective algebras. Let � denotethe set of pairs �(\-� � where \ZQ �� = � �� is a sequence of elements of and �Q �� = � �� [�*RN� � " � . The integer � is called the length of �(\-� � .

Given �(\-� � [ � such that X � � �� Q " !#" , we define a flag in ! of type �(\-� � as a sequence Q�� ! QT! �� ! = � �� ! � Q7R��

of graded subspaces of ! such that

" ! � � = 6 ! � " Q�� for � Q �� . Thus �HG � ! � � = 6 ! � is equal to R or � . So these are complete flags, with pos-sible repetition of some subspaces. (It will be convenient below to allow such flags with repeatedsubspaces.) We denote by �� the variety of flags of type �(\-� � . When �� = � �� Q � �� *�(flags without repetition), we simply write �� .

Let I [ 0�$ . A flag

is I -stable if I �+! � � � ! � for all � . We denote by ��

(resp. �� ) the

variety of I -stable flags of type �(\-� � (resp. of type \ ). Note that an I -stable flag is the same as acomposition series of I regarded as a 0 -module.

5.8. For �(\-� � [ � with X � � �� Q " !#" , define

9�� Q �� !:�� ! � � �� :�� [ - $ �If � � Q � for all � , we simply write 9�� instead of 9�� . In general, 9�� Q 9�� where � is the subwordof \ consisting of the letters �� for which � � Q � . By definition, the functions 9 � , where � runs overall words, span - . The following lemma results immediately from the definition of the product �of constructible functions.

Lemma 5.1. Let I [ 0W$ . We have

9�� +IZ��Q09!�*�+IZ��Q �W� �"� ��


Example 2. Retain the notation of Example 1 (b). Let I Q �� .8 � [ 0�$ be given by the followingmatrices (with respect to some fixed bases of F = and F 4 )

� Q�� R RR R�� [ Q � � R � F = � F 4 � � 8�Q�� R

R R�� [ Q � � R � F 4 � F = � �Let us calculate 9 � 4 �4= � 4 �4=�� +IV��Q �� 4 �4= � 4 �4=��

�� . To construct a flag

Q �+!��\! = �\! 4 �\! 6 � R � [ � � 4 �4= � 4 �4=�� we first have to choose a line ! 6 in the 2-dimensional vector space F = � �� +IV��Q F = . We maytake ! 6 to be

(a) the 1-dimensional image of I , or(b) any line except this one.

In case (a) the module I 6 induced by I in the quotient ! 6 ! 6 is the semisimple module. So weget �� 4 �4= � 4 ��

� ��2Q � � � � � �

In case (b), I 6 Q �� 6 �.8 6 � where

� 6 Q � RR�� [ Q � � R � F = 6 ! 6 � F 4*� � 8 6 Q � � R � [ Q � � R � F 4�� F = 6 ! 6 � �

and at the next stage ! 4 6 ! 6 must be taken as the kernel of I 6 (no choice), and ! = is also com-pletely determined. Thus, in case (b) we get�� 4 �4= � 4 ��

� ��2Q � � � � � �

So finally, �� 4 �4= � 4 �4=��Q � � �>Q C �

5.9. In this section we assume that � is a simple finite-dimensional Lie algebra. Equivalently, is a Dynkin quiver. Then �� has a PBW-basis < N associated to this quiver . The image� � < N � is easy to describe. Let ! [ ?�@CB�� J� . The affine space (�� ! � can be regarded asa subset of 0W$ by identifying it to the set of I [ 0�$ with I KTQ R for every ' [ = 0 = .Clearly this is an irreducible component of 0�$ . Moreover by our assumption it has finitely many� $ -orbits.

Lemma 5.2. Let � be a �/$ -orbit in ( �� ! � . There exists a unique � � [ - $ whose re-striction to (�� ! � is the characteristic function of � . The collection of all � � where � runsthrough all � $ -orbits in (�� ! � is equal to

� � < N � - $ .

Proof. By [38, 10.19, 12.12], the map from - $ to � �+(�� ! �� sending to its restrictionto (�� ! � is a vector space isomorphism. Moreover, the space� N Q�O

$� �+(�� ! ��

(where ! ranges over all isoclasses of vector spaces in ?I@CBN� N� ) endowed with the image of theproduct � is the geometric realization of the Hall algebra of over � due to Schofield (see [38,10.19]). In this setting, the PBW-basis is the basis of

� Ngiven by the characteristic functions of

the � $ -orbits in (�� ! � . Thus the lemma is proved. �6. COMULTIPLICATION

The algebra � �� is in fact a bialgebra, the comultiplication being defined by

@ 61 @�� @2� �+@ [ �� In this section we describe the comultiplication

�of - obtained by transporting this comultipli-

cation via�

.


6.1. For ! ��! � ��! � � [ ? @!B � J� such that " ! � " � " ! � � " Q " !#" , define a linear map� �� $$ � � $ � � � � � 0�$/� �� 0�1 � � 0 $ � � 0 $ � � � � � ��

as follows. We have ! �0�1 ! � U ! � � , so given I � [ 0 $ � and I � � [ 0 $ � � we can regard I � U I � �as an element of 0W$ . Here, I � U I � � denotes the direct sum of I � and I � � as endomorphisms of ! �and ! � � , or equivalently as modules over the preprojective algebra 0 . For [ � $ , I � [ 0 $ � andI � � [ 0 $ � � set

� � �� $$ � � $ � � ��+I � ��I � � ��Q 2�+I � U I � � � �This is clearly a constructible function on 0 $ � � 0 $ � � which is constant on � $ � � � $ � � -orbits.

6.2. Let \ Q �� = � �� with X � � � Q " !#" . Let ! Q ! � U ! � � , I � [ 0 $ � , I � � [ 0 $ � � andI QTI � U I � � [ 0�$ .

Lemma 6.1. We have

� � �� $$ � � $ � � 9�� +I � ��I � � ��Q +� � � � � � � �

9 � � � � �+I � �29 � � � � � �+I � � �where the sum is over all pairs �� of sequences in �*RN� � " � such that

(5)

�+� , � � � � � �/Q " ! � " ��+� , � � � �� /Q " ! � � " � � � � � � � �� Q � � R A � A � � �

Proof. By definition we have

� � �� $$ � � $ � � 9�� +I � ��I � � ��Q09�� +IV��Q ��

To Q �+!��+� �� [ ��

�, we associate the pair of flags

� Q �+! � 6 ! � � � �� Q �+! � ! � � � �� We regard

� �as a flag in ! � � , and

�as a flag in ! � by identifying ! � with ! 6 ! � � . Clearly, we have �� [ � � � � � �

�� and

� � [ � � � � � � �� for some sequences � , � � in �*RN� � " � satisfying the conditions (5).

Let = denote the set of pairs �� satisfying (5). For �� [ = , let � � �� be the

subset of ��

consisting of those

for which � � � �� [ � � � � � ��

�� . Then clearly we have a

finite partition

� � ��Q ��

=� H � � ��

where 4 �� = consists of the pairs �� such that �� is nonempty.

Now for �� [ 4 , the map '�� 0 1 � � � � � �

��

�� sending

to

� �� is a vector bundle, see [38, Lemma 4.4]. Hence,�� Q ��

��

�� Q ��

��

��

and� � �� $$ � � $ � � 9�� +I � ��I � � �2Q ��

��Q +

� � � � � � � �= H �W� � � � � � �

�� W� � � � � � � �

��

On the other hand, +� � � � � � � �

= !9�� +I � �M9�� +I � � �2Q +

� � � � � � � �=�

!�W� � � � � � �

�� W� � � � � � � �

��

and it only remains to prove that = Q� 4 . Clearly 4 �� = , so let �� [ = . Let �� [ �� and

�� [ � � � � � � �� . The assumption I Q7I � U I � � implies that ��

�� is nonempty.

Indeed, the flag

in ! whose � th subspace is the direct sum of the � th subspaces of �

and � �

isI -stable and by construction

[ � � �� . So �� [ 4 and the lemma is proved. �


6.3. By Lemma 6.1, the map� �� $$ � � $ � � induces a linear map from - $ to - $ � � - $ � � , given

by 9�� 61 +� � � � � � � �

9�� 9�� where the pairs �� satisfy (5). Taking direct sums, we obtain a linear map

- $ 0 1 O$ � � $ � � - $ � � - $ � � �

the sum being over all isomorphism types ! � and ! � � of -graded vector spaces such that " ! � " �" ! � � " Q " !#" . Taking direct sums over all isomorphism types ! , we get a linear map� � - 0 1 - � - �Since �+I � U I � � � U I � � � Q I � U �+I � � U I � � � � , � is coassociative. Since I � U I � � � I � � U I � , � iscocommutative.

Lemma 6.1 shows that�

is multiplicative on the elements 9 � , that is, for \ZQ �� = � �� (6)

� �&9��2Q � �&9 � ! � � �� &9 � � ��Q �&9 � ! �� 9 � ! � � �� &9 � � �� 9 � � � �where the product in - � - is defined by

� = � �4P� � � " = � " 4P��Q � = � " = � � � �4 � "�4*� �Proposition 6.2. Under the algebra isomorphism

� � = � - 0 1 �� , the map�

gets identifiedwith the standard comultiplication of �� .Proof. Equation (6) shows that

�is an algebra homomorphism � - � � ��1 � - � - � � � . Moreover

the generators 9 � � � Q � :�� Q � �(� � � are clearly primitive. The result follows. �7. MULTIPLICATIVE PROPERTIES OF THE DUAL SEMICANONICAL BASIS

7.1. The vector space - is � � > � -graded, with finite-dimensional homogeneous components. Let- . denote its graded dual. Given an -graded space ! and an irreducible component

5of 0>$ ,

we have defined a linear form %�: on �S� 0�$ � , see Section 4.2. We shall also denote by %�: theelement of - . obtained by restricting % : to - $ and then extending by R on all - $ � with" ! � " �Q " !#" . Note that by (4), the basis of - . dual to the semicanonical basis �! : " is nothing but�&%J: " . From now on � .�Q �&%N: " will be called the dual semicanonical basis of - . .Lemma 7.1. For

5 [ ')(�(*� 0W$/� there exists an open dense �/$ -stable subset � : F 5such that

for all [ - $ and all I [ � : we have %N:�� Q 2�+IV� .Proof. For a given , this follows from Section 4.2. Moreover, there exists such an open setsimultaneously for all because - $ is finite-dimensional. �7.2. For I [ 0W$ define the delta-function �

�[ - . by �

�� 2Q 2�+IZ� � � [ - � � We then have

(7) ��Q'%N: � �+I [ � : � �

The next Lemma follows immediately.

Lemma 7.2. Let5 [ '-(�(*� 0W$/� and suppose that the orbit of I [ 5 is open dense. Then % : Q��

�.

Let � denote the multiplication of - . dual to the comultiplication�

of - .

Lemma 7.3. Let I = [ 0�$ ! and I 4 [ 0�$�H . We have ��!� ��H�Q �

�! X

�H �

Proof. For [ - , one has

� ��!� ��H �V� �2Q � �

�! � �

�H �V� � � ��Q � � �;�+I = ��I 4*�2Q 2�+I = U I 4P��Q��

�! X

�H � � � �


Lemma 7.4. Suppose that5 Q 5 = U 5 4 is an irreducible component of 0�$ . Then there exists

I [ � : of the form I QTI = U I 4 with I = [ �>: ! and I 4 [ �>:JH .Proof. The direct sum

5 = U 5 4 is the image of the morphism� � � $ � 5 = � 5 4/0�1 5

defined by � � "��I = ��I�4P��Q�" � �+I = U I 4P� �Since � : ! (resp. � :JH ) is open dense in

5 = (resp. in5 4 ), the subset � $ � � : !

� � :JH is opendense in � $ � 5 = � 5 4 . Now, since

�is a dominant morphism between irreducible varieties, the

image under�

of �/$ � � : !� � : H contains a dense open subset of

5, hence it has a nonempty

intersection with � : . Since both� � � $ � � : !

� �>:JH � and � : are � $ -stable we can find I intheir intersection of the form I QTI = U I 4 with I =�[ � : ! and I�4 [ � :JH . �7.3. We can now give the proof of Theorem 1.1.

Proof of Theorem 1.1. Choose I ��I = ��I�4 as in Lemma 7.4. Then Lemma 7.1 and Lemma 7.3yield

%J: ! � %N: H Q��!� ��H Q��

�! X

�H Q��

�Q'%N: � �

Corollary 7.5. Let5 Q 5 = U �� U 5 � be the canonical decomposition of the irreducible com-

ponent5

of 0 $ . The dual semicanonical basis vector % : factorizes as

% : Q'% : ! �� % :� �Proof. For � Q � this follows from Theorem 1.1. Assume that � � � . By [14]5 � Q 5 = U ��3U 5 � � =is an irreducible component. Moreover 5 � U 5 � Q 5 �so by Theorem 1.1 we get % : Q'% : � � %N: � . The result follows by induction on � . �

The factorization given by Corollary 7.5 will be called the canonical factorization of %V: .

7.4. We shall now deduce from Theorem 1.1 the proof of the “only if” part of Theorem 1.2.

Theorem 7.6. Let � be of type 1 2 �+@�� 72��+@�� or � . Then, the bases : . and � .do not coincide.

Proof. Assume first that � is of type 1W� or �58 . Then the preprojective algebra 0 is of tamerepresentation type. In this case, we have �;LNM =O � 5 � 5 � Q R for any irreducible component

5of

0�$ , see [24, 25]. Therefore by Theorem 1.1 and [14] the square of any vector of � . belongs to � . .On the other hand, it was shown in [31] that for the cases 1 � and � 8 there exist elements of : .

whose square is not in : . . They are called imaginary vectors of : . . This shows that : . and � .are different in these cases.

Now if � is not of type 1 2 with @?A �, then the Dynkin diagram of � contains a subdiagram of

type 1�� or � 8 , and the result follows from the cases 1W� and � 8 . �In the next sections we shall prepare some material for the proof of the “if” part of Theorem 1.2,

to be given in Section 13.

8. EMBEDDING OF -/. INTO THE SHUFFLE ALGEBRA

We describe a natural embedding of - . into the shuffle algebra. This is then used to describea certain family of elements of � . in type 152 .


8.1. Let � Q � � � be the free associative algebra over � generated by . A monomial in � iscalled a word. This is nothing else than a sequence \VQ �� = � �� in . Let� �� 0 1 -

be the surjective algebra homomorphism given by � ��)��Q � :�� , and more generally by � �(\ ��Q 9 � .Let � . denote the graded dual of � . We thus get an embedding of vector spaces� . � - . 0�1�� . �Let �� \ � " denote the basis of � . dual to the basis � \ " of words in � . Let � [ - . . We have� . �� Q +

�� . �� ;�(\ �� \ � Q +

�� (\ �� \ � Q +

��&9�� \ � �

By Lemma 5.1, we obtain in particular

(8) � . � � � ��Q +�

�W� � � �� \ � �

8.2. Denote by the multiplication on � . � - . � obtained by pushing � with � . , that is, for�Z� [ - . set � . �� . ��Q � . �� Lemma 8.1. The product is the restriction to � .�� - .P� of the classical shuffle product on � .defined by � � � = � �� = � �� 2 � Q + S � � � S � =�� S � � � 2 � � �where the sum runs over the permutations � [�� 2 such that

�J� �*� � �� and � � � � �*�� J� � � @ � �Proof. This follows easily from Lemma 6.1 and the duality of � and

�. Indeed,� . �� Q +

�� ;�&9�� \ �

Q +�� ;� � �&9��+�� \ �

Q +�

+� � � � � � � �

��&9�� &9�� \ � �where the pairs �� are as in Lemma 6.1. Now it is clear that the coefficient of � � \ � in this lastsum is the same as the coefficient of � � \ � in the shuffle product�� +

� ��&9 � � �� +

� � � �&9 � � � ��

and the lemma is proved. �8.3. Suppose that � is of type 132 . Then - . � �� . � �� , where � is the group ofunitriangular �+@ � �*� � �+@ � �*� -matrices.


8.3.1. Let us construct an explicit isomorphism

' �E�� 1 - . �Let � � 1 denote the coordinate function assigning to @ [ � its entry @ � 1 . Then

� �4�� Q7�� 1 " � A�� 3 A\@ � � � �It is known that in the isomorphism �� . , the natural basis of �� consisting ofmonomials in the � � 1 gets identified to the dual of the PBW-basis of �� associated to the quiver

2�� K !oo ��KEHoo @K �� !oo

(see for example [33, 3.5]). The �$ -orbits of (�� 2 ��! � are naturally labelled by the multiseg-ments of degree " !#" , and if we denote by � � .� " the dual in - . of the PBW-basis � � � " in - ,then more precisely the above isomorphism maps the monomial � � ! 1 ! �� 1�� to the element � .�indexed by the multisegment � Q � � = �43 = 0 � � � �� 43�� 0 � � �For ��A 3 , let I � ��43 � denote an indecomposable representation of 2 with socle � � and top �H1 (upto isomorphism there is exactly one such representation). Then the orbit of I � ��43 � is open denseso �

�� 1 � belongs to - . by Lemma 7.2. On the other hand, by Lemma 5.2

�� 1 � � � � �2Q

� if � Q � ��43��)�R otherwise.

Hence ��. � � � 1 � Q �� 1 � and ' is the algebra homomorphism determined by '�� 1 � = ��Q �

�� 1 � .

8.3.2. If we regard the functions � � 1 as entries of a unitriangular �+@ � �*� � �+@ � �*� matrix wemay consider some special elements of � �4�� given by the minors of this matrix.

Proposition 8.2. The images under ' of all nonzero minors of the matrix belong to � . .Proof. We shall use the embedding � . . First note that since I � �.�43�� has a unique composition series,� . '�� 1 � = �2Q �� 3 �43�0 �� Let � � 1 be the � � � -minor taken on the sequence of rows � Q �� = � �� and the sequenceof columns 3 Q �#3 = � �� 3"� � . Since is a unitriangular matrix with algebraically independententries � � 1 above the diagonal, the function � � 1 is nonzero if and only if �� A 3 � for every � . Weshall assume that this condition is satisfied. Let

. Q �#3"� �43"� � = � �� 43 = � � 0 �*� � � Q �� = � �� = � ��0 �*� �Then . 6�� is a skew Young diagram. We identify it with the following subset of � � � � �

. 6�� Q � ��.8 � " � AB8 A � �� A . ��" �Each pair ��.8 � is called a cell of . 6�� . Let Y be a standard Young tableau of shape . 6�� , that is, atotal ordering � = � �� of the cells of . 6�� which is increasing both on the rows and on thecolumns. We associate to Y the element� � Y �VQ �� 0 8 �� = 0?8 = �of � . , where � � Q �� .8 � �� A � A �� . Before we continue with the proof of Proposition 8.2 weneed the following lemma.

Lemma 8.3. � . '�� 1 ��Q +�� Y � �where Y runs over the set of standard Young tableaux of shape . 6�� .


Proof. Set � � 1 Q � . 'W�� 1 � and ��Q + � � � Y � �We shall prove that

� � 1 Q$�� by induction on the number � of cells of . 6�� . If � Q � , then� � 1 Q � � � �ZQ �� for some � , and the statement is clear. So suppose � � � .For � Q �� @ define � � [ K � � R �� . � by

� � � �� = � �� S �+�2Q � � � = � �� S � = � if � = Q ��R otherwise �

It is immediate to check that � � is a derivation with respect to the shuffle product, i.e.

� � � "N��Q�� " � � � � "N� � � � " � [ � . � �It is also clear that Q " if and only if � � � �WQ� � � " � for every � . Note that

� � 1 is the minor onrows � and columns 3 of the matrix

Q

�� J� � � �� @2��@ 0 �� R � � � � � �� @2��@ 0 �� ...

......

...R R R ��

��where in the expansion of the determinant the shuffle product of the entries is used. It follows that,if 3 � � [ 3 and 3 �[ 3 then � 1 �

� � 1 � Q� � � where � is obtained from 3 by replacing 3 � � by 3 ,

and otherwise � 1 �� 1 �2Q7R .

On the other hand,

� 1E�� Q +� ��

where ranges over the Young tableaux whose shape is a skew Young diagram L6�� obtained from

. 6�� by removing an outer cell � Q �� .8 � with � 0B8 Q 3 . It is easy to check that there is onesuch diagram only if 3 � � [ 3 and 3 �[ 3 , and that this diagram then corresponds to the pair �� above. So, by induction

� 1E�� Q �� Q� � � Q�� 1E� � � 1 �

in this case, and � 1 �� Q R Q�� 1 �� 1 � , otherwise. Therefore �� Q

� � 1 . This finishes theproof of Lemma 8.3. �

We continue with the proof of Proposition 8.2. Let� Q � � = �43 = 0 � � � �� 43"� 0 � �be the multisegment corresponding to the pair �� 43 � . (Here we leave out � � � �43 � 0 � � in case � � Q�3 � .)Following [47] this parametrizes a laminated 0 -module I � � � , that is, a direct sum of indecom-posable subquotients of projective 0 -modules. Let ! be the underlying -graded vector space ofI � � � . It is known that the � $ -orbit of I � � � is open dense in its irreducible component, hence thefunction �

�� belongs to the dual semicanonical basis.

Now it is easy to see that the types \ of composition series of I � � � are in one-to-one correspon-dence with the standard Young tableaux of shape . 6�� , and that for each tableau, the correspondingflag variety � � �

�� is reduced to a point. Therefore, comparing with Lemma 8.3 we see that� . � � � � � � �2Q � . 'W�� 1 � �

Hence '�� 1 ��Q�� belongs to the dual semicanonical basis. This finishes the proof of Proposi-

tion 8.2. �


9. A GALOIS COVERING OF 0 FOR TYPE 152In order to prove the “if” part of Theorem 1.2 we need to study the canonical decomposition of5 [ ')(�(*� 0 � for type 172��+@ A � � . Our main tool for this will be the Auslander-Reiten quiver of 0 ,

which we will calculate by using a Galois covering�0 of 0 . This covering is in fact important for

all @ , and it will also play an essential role in our investigation of type 1�� in the last sections ofthe paper. So we shall work in type 132 for general @ in the next two sections, and we shall specifywhich results are only valid for @BA �

. We will also exclude the trivial case 1 = and assume that@ � � .9.1. For @ � � , let again 2 be the quiver

� �K !oo ��K Hoo @K �� !oo

of Dynkin type 152 . Let 032 Q � �� 2N� be the preprojective algebra corresponding to 2 . Thus0 2 Q7� 2 6 5 2 where the double quiver 2 of 2 is

�K �! // �K !oo

K �H// ��KEHoo

K �� ! // @2�K �� !oo

and the ideal 5 2 is generated by

' = ' . = � ' .2 � = '�2 � = � ' .� ' � 0#' � � = ' .�� = � � �A��WA\@ 0 �E� �

9.2. Next, let�0 2 Q � � 2 6 �5 2 where

� 2 is the quiver with vertices � � 1 "�� A ��AT@2�43 [ � " and

arrows' � 1�� *�&1 1 � 1�� ' .� 1 �� 1>1 �� *�&1 � = � � �A��WA\@ 0 ��43 [ � � �

and the ideal�5 2 is generated by

' = 1 ' . = � 1 � = � ' .2 � = � 1 ' 2 � = � 1�� ' .� 1 ' � 1�0#' � � = � 1 � = ' .� � = � 1 � � �A ��A\@ 0 �J�43 [ � � �For @ Q �

we illustrate these definitions in Figure 1.Denote by

� � � � � �2 the full (and convex) subquiver of

� 2 which has as vertices the set

� � 1 [ � 2 "�� C A � � �C3 AB8 � � "E�

and denote by� � � � � �2 the restriction of

�0 2 to

� � � � � �2 , see 9.10 for an example.

9.3. The group � acts on�032 by � -linear automorphisms via

� � 1 Q � 1 � � � � ' � 1 Q ' � � 1 � � � � ' .� 1 Q ' .� � 1 � � �This induces an action

� � � �� 032N��0�1 �� 0 2 ��&�� 61 � � � � �

where � � � � denotes the�032 -module obtained from � by twisting the action with . Roughly

speaking, � =�� is the same�0 2 -module as � , but “translated one level upwards”.

If we consider�032 and 032 as locally bounded categories we have a functor � �

�032B0�1 032

defined by ��1 61 �� ' � 1 61 ' � � ' .� 1 61 ' .� �This is a Galois covering of 0 2 in the sense of [22, % 3.1], with Galois group � . It provides us withthe push-down functor [22, % 3.2]

�� 032N��0 1 �� 032N�which we also denote by � . It is defined as follows. Let I [ �� 0 2 ��! � be a

�0 2 -module with

underlying graded vector space ! Q D � � 1 F � . Then � �+IZ� has the same underlying vector space


�O � � =�

K �!

��@

@@@

... 6�HKEH H}}{{{{{

K ��H""E

EEEE

... � !K � !||yyyy

y

4�HK ! H~~||

||K �H H""D

DDD

8 !K�!{{ww

www K ��

!##G

GGGG

= H K �! H BBB

B6 !K H !||zz

zzK ��!##G

GGGG

��K � �{{ww

www

4 !K !�!~~}}}}

K �H !!!D

DDD

8��K��{{xx

xxx K ��

##FFFF

F

=! K �

!�! A

AAA

6��KEH��}}{{{{

K ��""F

FFFF

� � !K � # � !||xxxx

4��K ! �~~}}

}}K �H��!!D

DDD

8 � !K�# � !||xx

xxK ��# � !""F

FFF

= �...

6 � ! ...

� � H�

��

O � � =K �! // 4K !oo

K �H// 6KEHoo

K ��

// 8K�

oo

K ��// �K �oo

FIGURE 1. The Galois covering

with the grading ! Q D � F � where F � Q D 1 F � , and � �+IV� has maps � �+IV� K � Q D 1 I K � and� �+IZ� K �� Q D 1 I K �� .The next lemma follows from [22, % 3.6] for @BA �

, and from [16] for @ Q �, as noted in [15,

% 6.3].

Lemma 9.1. Let ��A\@?A �. Any finite-dimensional indecomposable 0 2 -module is isomorphic to� �+IZ� for some indecomposable

�032 -module I , which is unique up to a translation I 61 � � � I by the

Galois group � .

9.4. For a dimension vector [,Q �&9 � � = � � � 2 � 1 =�� for�0 2 define 9�Q �&9 = � �� 9 2 � where

9 � Q +1 =�� 9

� �

We have a morphism of varieties�� &9N� � �� 032��[2� �

0 1 � �� 032 ��9N�where the push-down morphism �� is defined by � � � "�� D��Q�" � � � �D� .

For [ � let � Q � � � [ be the th shift of [ , that is,

� � Q 9 � �� A��WA\@2�43 [ � � �Thus, if [ is the dimension vector of a

�032 -module � , then � is the dimension vector of the shifted

module � � � � . In this case we write [ � � . This defines an equivalence relation � . To simplifyour notation, we shall not always distinguish between [ and its equivalence class. However, itwill always be clear from the context which one is meant. To display a dimension vector (or itsequivalence class) for

�0 2 we just write down the relevant entries, all other entries are assumed to

be 0.


9.5. Recall that to any finite-dimensional algebra Y over a field � (and more generally to anylocally bounded � -algebra or locally bounded category, see e.g. [48, p. 54]) is associated a trans-lation quiver

� � called the Auslander-Reiten quiver of Y . It contains a lot of information aboutthe category �� &Y � . In particular if Y is representation-finite and standard, one can recover�� &Y/� from

� � .The vertices of

� � consist of the isomorphism classes of indecomposable finite-dimensionalY -modules. If � and � are two such modules, then there are �HG � ��,� � �� 6�� 4 � � �� arrowsfrom � to � in

� � , where � denotes the radical of the category �� &Y � (compare [45, Chapter2]). This means that there is an arrow from � to � if there exists a nonzero irreducible homo-morphism from � to � . Here, by abuse of notation, we do not distinguish between a module andits isomorphism class. The quiver

� � is endowed with an injective map , the translation, definedon the subset of vertices corresponding to non-projective modules. If � is indecomposable andnon-projective then � Q � where R 1 � 1 � 1 � 1 R is the Auslander-Reiten sequence(or almost split sequence) ending in � .

The stable Auslander-Reiten quiver� � of Y is obtained from

� � by removing all translates � 2 � and 2 �+@ [ �� of the projective vertices � and the injective vertices as well as thearrows involving these vertices. Thus the translation induces a permutation on the vertices of� � .

We refer the reader to [1, Chapter VII] for more details on Auslander-Reiten theory, or to [21]and [45, Chapter 2] for � algebraically closed as we assume here. Note however that these papersuse slightly different conventions.

9.6. In Section 20.1 we display the Auslander-Reiten quivers of 0 4 , 0 6 and 038 . To calculatethem one first determines the Auslander-Reiten quivers of their coverings

�0 4 ,

�036 and

�038 . Indeed

the algebras�032 are directed, that is, there is no sequence of indecomposable

�032 -modules of the

form

� 1 � = 1 �� 1 � � 1 �with all homomorphisms being nonzero and non-invertible. It follows that the Auslander-Reitenquiver of

�032 can be calculated by a combinatorial procedure known as the knitting procedure, see

for example [21, % 6.6]. By applying the push-down functor � to this quiver, one then obtains theAuslander-Reiten quiver of 0 2 [22, % 3.6].

In our pictures, each indecomposable 0 2 -module � is represented by the dimension vector [of a

�032 -module � � such that � � � � �>Q � . One has to identify each vertex in the extreme left

column with the vertex in the extreme right column represented by the same dimension vector upto a shift by the Galois group � . The Auslander-Reiten quiver of 0 6 is shaped like a Moebiusband, and for 0 4 and 038 one gets a cylinder.

In particular, we see that 0 4 has 4 isoclasses of indecomposable modules, 0 6 has 12, and 038has 40.

We should point out that for @ � �there are 0 2 -modules which are not obtained from a

�032 -

module via the push-down functor.

9.7. For an algebra Y let �� &Y � be the stable category of finite-dimensional Y -modules ([48,p. 55]). By definition, the objects of �� &Y/� are the same as the objects of �� &Y � , and the mor-phism space

Q � � � � � �� is defined asQ � � � � � �� modulo the morphisms factoring through

projective modules. The isomorphism classes of indecomposable objects in �� &Y/� correspondnaturally to the isomorphism classes of non-projective indecomposable Y -modules.

The stable category �� &Y � is no longer abelian, but if Y is a Frobenius algebra then �� &Y �has the structure of a triangulated category ([26, % I.2]) with translation functor � � = , the inverse ofHeller’s loop functor.


Moreover, in this situation we may identify the quiver of the stable category �� &Y/� (see forexample [26, % I.4.8]) with the stable Auslander-Reiten quiver

� � defined in 9.5 - just remove theprojective-injective vertices from

� � .

9.8. To a finite-dimensional algebra < one associates its repetitive algebra�

< ([26, p. 59], [48,p. 57]). This is an infinite-dimensional algebra without unit. However it is locally bounded, andits indecomposable projective modules coincide with its indecomposable injective modules, thatis,

�< is a Frobenius algebra.Hence for a finite-dimensional algebra < , �� < � is a triangulated category, see 9.7. More-

over if < has finite global dimension, there is an equivalence of triangulated categories between�� < � and the derived category

� � �� < �� of bounded complexes of < -modules (Happel’sTheorem [26, % II.4]). Under this equivalence the functor � � = corresponds to the translation func-tor � 61 �S� � � .9.9. If < is a finite-dimensional algebra of finite global dimension we have by [27, % 3.2]

(9) � � �� < ��Q � � �� < �� U � � �< � �

where� � �< � is the subgroup of � � ��

�< �� generated by the classes of projective�< -modules.

Thus we may identify � � �� < �� with � � �� < �� . This coincides with the Grothendieck

group of � ��

< � viewed as a triangulated category.For a

�

< -module � we denote by � � � its class in � � �� < �� Q � � ��

< �� . In par-ticular, if � is projective, then � � �2Q R . Notice that � � � depends only on the dimension vector[J\&]B� �D� [ ��

�

< �� and (9) provides an efficient method for calculating � � � . In this contextwe have

(10) � � �ZQ �/� � �where � is the Coxeter transformation of < and is the self-equivalence of ��

�

< � inducedby the Auslander-Reiten translation of ��

�

< � . This follows from Happel’s Theorem 9.8, theconstruction of the Auslander-Reiten translate in [26, % I.4.6] and the definition of the Coxetertransformation.

9.10. In case � A\@ A �we have in � �� 032 � a functorial isomorphism

(11) � =�� Q � � = � = �which is proved along the lines of [15, % 6.4]. In particular this implies � � =�� ZQD0�� = � � � .

Moreover, if we set� 2 Q � � � � 2 � 8

�2 it is easy to see that

�032 Q �� 2 . For example

� � is thealgebra given by the following quiver with relations:

� ��Q

4 H~~||

| B

BB

��

8 !~~||

|""E

EEE

��

= H B

BB(�

�

6 !~~||

| B

BB

��

��||yyy

y ��(

4 !~~}}

} A

AA8��

""DDDD

~~}}}

=! 6�� !

Here the dotted lines indicate zero relations and commutativity relations.Thus by Happel’s Theorem 9.8 we have �� 0 2�� Q

� � �� 2 �� . In particular we canapply 9.9. Note that

� 2 is hereditary of type 1 = (resp. 1 6 ) for @\Q � (resp. @ Q C ), and tiltedof type � � in case @DQ �

. Happel’s description of the Auslander-Reiten quiver of the derivedcategory of a (piecewise) hereditary algebra of Dynkin type [26, % I.5.6, % IV.4.3] together with (11)yields the shape of the stable Auslander-Reiten quiver in these cases. In case @ Q �

the algebra


� � is tubular and we can derive the structure of �� 0 � � from the known structure of the derivedcategory of a tubular algebra [27]. We will discuss this case in Section 14.

10. FROM SCHUR ROOTS TO INDECOMPOSABLE MULTISEGMENTS

10.1. Let � �032N� be the set of dimension vectors of the indecomposable�0 2 -modules. By � � �032 �

we denote the set of Schur roots of�032 , i.e. the set of dimension vectors of the (indecomposable)�

0 2 -modules � with K� � �O � �D� �Q � . For [ [ � � �032N� let

5� Q � � [ �� 032 ��[2��"3K � � �O � � � �Q � " �

For � A\@?A �this is an irreducible component of �� 0 2 ��[ � . We write

5� � 5��

if [ � � .For an irreducible component

5of �� 032��[2� define

� � 5 ��Q � � � � � �&9J� � 5 � �Clearly, � � 5 � is an irreducible subset of �� 0 2��9J� , but in general � � 5 � is not an irreduciblecomponent. The following result was shown in [24].

Theorem 10.1. Assume that � A @BA �. For [ [ � � �032 � , the variety � � 5 �� is an indecompos-

able irreducible component of �� 0 2 ��9J� . Moreover, � induces a bijection from � � �0 2 � 6 � toG�� V�+'-(.(*� 0 2�� .10.2. Let � �+@ � be the set of multisegments supported on � �� @ " , that is, of the form� Q +

= � � � 1 � 2 �� 1 � ��43 �

where � � 1 [ � . The degree 9 Q �&9 = � �� 9 2�� of � is given by

9��/Q +� � � � 1 � � 1�� A � A\@ � �

There is a one-to-one correspondence � 61 � � between the set � � �+@ � of multisegments ofdegree 9 in � �+@ � and the set of � � �&9J� -orbits in � �� 2 ��9N� .

Let �� &9J� be the multisegment labelling the unique dense orbit of �� 2 ��9J� . It can becomputed recursively by �� &9J��Q � ��.8.� � �� &9G0 �3�where

��Q � G�� "*9 � �Q R�" and 8�Q ��3L � 3 "*9 � �Q7R for all � A��WA 3 "E�and � � Q � if � [ � � �.8.� and � � Q7R , otherwise. For example,�� J�.CJ� ��E��Q � �� J�.C � � �4CJ�.C � � � � � � � �

10.3. Let � � be the projection morphism from �� 0 2 ��9J� to � �� 2 ��9N� obtained by forget-ting the maps labelled by the arrows ' .� . Lusztig [38] proved that the irreducible components of�� 032��9N� are the closures of the sets � � =� �� , � [ � � �+@ � . Define

5 � Q � � =� �� A multisegment � is said to be indecomposable if

5 � is indecomposable. By G�� Z� � �+@ �� wedenote the set of indecomposable multisegments in � �+@ � .


10.4. For a dimension vector [ QD�&9 � � = � � � 2 � 1 =�� for�0 2 set�� &[2��Q +

1 = �� &9 �#3 ��

where 9 �#3 ��Q �&9 = ��9 4 � �� 9 2 3� . The variety �� 0 2 ��9 �#3J�� is irreducible and, in fact, isomor-

phic to �� 2 ��9 �#3J�� . There is a�032 -module � � � 1 � such that the orbit � � � � � 1 � � is dense in

�� 032��9 �#3 �� , and � � � 1 � is uniquely determined up to isomorphism.

Let� �&[2� be the set of modules � [ �� 0 2 ��[ � which have submodules � � 1 � 1 =�� such that

� 1 � = � � 1 and � 16�� 1 � = �Q � � � 1 � for all 3 [ � . Thus � 1 Q R if and only if 9 ��)� Q R for all��A 3 , and � 1 � = Q � 1 if and only if 9 �#3J��Q7R . Set5 �� &[ ��Q � �&[ � �

For5 [ '-(�(*� �0 2 � let �� 5 � be the unique multisegment in � �+@ � such that

� � � : � Q� � � � � 5 ��

This defines a map � �E'-(.(*� �0 2 ��0�1 � �+@ � �If � � 5 � is an irreducible component, say

5 � , of �� 0 2 ��9N� , then �� 5 �2Q � .

Lemma 10.2. For all dimension vectors [ for�0 2 the set

5 �� &[2� is an irreducible component of�� 032��[2� , and we have �� 5 �� &[2��Q �� &[2� �Proof. One can easily see that

K2LNM =�O � � � �� 1 � ��Q7R for all �WA 3 �

Then [14, Theorem 1.3] yields that5 �� &[2� is an irreducible component. The second part of the

lemma follows directly from the definitions. �10.5. Assume that � A\@?A �

. Define� �E � � �0 2 ��1 � �+@ � by

� �&[2��Q�� J�� J� � � � �4CJ�.C � � � � � � � if @�Q �

and [ Q � = � �= == 4 == = � �� J� � � � �4CJ�.C � � � � � � � � � � � � � � if @�Q �

and [ Q � = == 4 == =� � = � �� &[2� otherwise �

Theorem 10.3. Let � A @,A �. The map

�establishes a bijection from � � �032N� 6 � to G�� Z� � �+@ �� .

Moreover the following diagram commutes:

� � �032N� 6 � � � �� : //

��

G�� V�+')(�(*� �0 2�� 6 �� : � � � �� : ��

��G�� Z� � �+@ �� :�� // G�� +')(�(*� 032N��The proof of Theorem 10.3 will be given in Section 18.


11. CASES 1 4 , 1 6 , 178 : THE COMPONENT GRAPH

11.1. In the case that 0 2 is of finite representation type, the results of the previous sectionssimplify greatly.

Theorem 11.1. Assume that @ Q �J�.CJ� � . Then for each [ [ � �0 2 � there exists (up to iso-morphism) exactly one indecomposable

�0 2 -module � � with dimension vector [ . Furthermore,

K � � �O � � �� Q � for all [ , i.e. � � �0 2 ��Q7 � �0 2 � � Therefore,

G�� V�+')(�( � 032N��Q � � � 5 � � "*[ [ � �032 � 6 � "and G�� Z� � �+@ ��Q � �� &[ � "*[ [ � �032 � 6 � " �Proof. The first two statements follow from the general theory of directed (simply connectedrepresentation-finite) algebras [45, % 2.4.(8)]. Then we apply Theorem 10.1 and Theorem 10.3. �11.2. In Section 20.2 we list the 40 indecomposable multisegments in � � � � labelling the 40indecomposable 0 8 -modules, and we redisplay the Auslander-Reiten quiver of 0 8 with verticesthese multisegments. The translation can be read off by going horizontally two units to the left,for example � � = �2Q � = �� 4P��Q � 4��3� � � 6P��Q � 4 � �and so on. Note that � 6 � � � 6 � � � 6�� 8 � have no -translate because they are the projective ver-tices.

11.3. We shall use� O to describe the pairs �+& � � of indecomposable 0 2 -modules such that

K2LNM =O �+&�� /Q R . By Proposition 3.2, K2LNM =O �+&�� /Q R if and only if K2LNM =O ��&,�Q R , sothis is a symmetric relation.

Recall, that for any finite-dimensional algebra 0 the Auslander-Reiten formula [1, Proposition4.5] gives us

�K�LJM =O �+&�� Q Q � � O �� = ��&,� . Now, if 0 is selfinjective, representation-finite

and admits a simply connected Galois covering�0 with Galois group � , we get�

K2LNM =O �+& � � �Q Q � � O �� = ��&,� �Q O�= �Q � � �O � � � � ��

� � �& � �

Here & and are indecomposable 0 -modules, and�& and

� are indecomposable

�0 -modules

which under push-down give & and , respectively. In this situation it is easy to determine thedimensions of the summands in the last term using additive functions on the stable Auslander-Reiten quiver

� �O Q � �for some Dynkin quiver

�, see [21, % 6.5].

As we have seen, this is exactly the situation for 0 Q 0 2 �+@#A � � , and by (11) there will be atmost one � [ � with

Q � � �O � ��

� � �&?� �Q7R .Since induces a self-equivalence of �� 0 � we have to do this calculation only for one

representative & of each -orbit.The stable Auslander-Reiten quivers of 0 4 , 0 6 and 0 8 have 1, 2 and 6 non-trivial -orbits,

respectively.For @ Q � there are only two indecomposable non-projective 0 2 -modules, say & and , and

K2LNM =O H �+& � � �Q R .In Sections 20.3 and 20.4 we display several copies of the stable Auslander-Reiten quivers of

0 6 and 0 8 . In each copy we pointed a representative & � of a -orbit, and we marked with the sign� all indecomposable 0 2 -modules � such that

K2LNM =O � � ��& � � �Q7R �For example, there are 21 indecomposable 0 8 -modules � such that

K�LJM =O � � � ��& 4 � �Q7R �


11.4. We note that the previous description shows in particular that for @ A �every indecom-

posable 032 -module & satisfiesK�LJM =O �+&��&,��Q7R �

This was first observed by Marsh and Reineke. It follows that the orbit closures of the indecompos-able 032 -modules are the indecomposable irreducible components of the varieties of 0 2 -modules.Therefore the results of 11.3 give a complete description of the component graph �� 0 2 � for @?A �

.

12. CASES 174 , 176 , 158 : THE GRAPH OF PRIME ELEMENTS OF : .In this section we consider the dual canonical basis : . of �� for type 152��+@?A � � . Using the

bijection � 61 5 � we may label its elements by multisegments � or irreducible components5

.We will write : . Q � 8 .� " � [ � �+@ � " or : .�Q � 8�.: " 5 [ ')(�(*� 0 2�� " depending on the context.

12.1. An element 8 . [ : . 0 � � " is called prime if it does not have a non-trivial factorization8�. Q 8 . = 8 .4 with 8 . = �.8 .4 [ : . . Let

� 2 be the set of prime elements in : . .Let ��2 be the graph of prime elements. The set of vertices of � 2 is

� 2 , and there is an edgebetween 8 . = and 8 .4 if and only if the product 8(. = 8�.4 is in : . . These graphs give a complete descriptionof the basis : . [4]. Indeed, : . is the collection of all monomials of the form�� =�� B�� 2 � � � 8 .� � � �

� �where the �V� � � [ � satisfy for � �Q � �

� �V� � � �V� � � � �Q7R � Q�� [ ��2N� �Note that for all 8 . in : . , the square of 8(. also belongs to : . . This is a particular case of theBZ-conjecture which holds for 1 2 �+@ A � � . Therefore there is a loop at each vertex of � 2 . Simi-larly, the @ vertices labelled by an irreducible component containing an indecomposable projectivemodule are connected to every other vertex (because the corresponding elements of : . belong tothe � -center). Let ��2 be the graph obtained from � 2 by deleting these @ vertices and all the loops.Clearly � �2 contains all the information.

12.2. The graphs ��4 and ��6 have been determined by Berenstein and Zelevinsky [4]. They arerespectively dual to an associahedron of type 1 = and 176 in the terminology of [11, 19, 20]. In 20.5we display the graph ��8 , which has C � vertices (corresponding to the multisegments � = � �� 6 �in the list of 20.2) and 330 edges. We have calculated it by computer using the BZ-conjecture.As suggested by Zelevinsky, ��8 is dual to an associahedron of type � � . The maximal completesubgraphs of � �2 all have the same cardinality, namely � , C and � for @�Q � , C and

�, and are called

clusters. There are respectively � , � � and � � � clusters.

12.3. The following theorem was proved by Marsh and Reineke for @BA C and conjectured for@ Q �

[42].

Theorem 12.1. For @?A �the graph � 2 is isomorphic to �� 0 2N� via the map 8 .� 61 5 � .

Proof. This is checked by using the explicit descriptions of both graphs. For example, from thefirst quiver of Section 20.4 we get that the vertex labelled by � 8 in �� 038 � is connected to all othervertices except � 63� � � � � � � � =�= � � = 4�� = 63� � = 8 � � = �� 6 = � � 6 4 �The same happens in the graph � 8 , as can be seen from Section 20.5. �

13. END OF THE PROOF OF THEOREM 1.2

In this section we prove the “if” part of Theorem 1.2.


13.1. Let � be of type 1 8 . For brevity set % � Q'%J:�� for � [ � � � � .Proposition 13.1. For every indecomposable multisegment � of � � � � we have

8 .� Q % � �Proof. For 34 multisegments � � out of the 40 elements of G�� V� � � � �� the vector 8 .� � is a minor,and the result follows from Proposition 8.2.

The six elements which are not minors are 8�.� � where C � A �>ADC � . Denoting by � the multi-segment duality of Zelevinsky [52], we have

�� 6 = �2Q � 6 4�� 6 6P��Q � 6 8�� 6��P�2Q � 6 � �Denote also by � the linear involution on � . given by

�� = � �� *�+��Q �� J� �� = � �It follows from [38, 41] that

� � � . � : . ��Q � . � : . � and �� . � � . ��Q � . � � . � �Moreover it is known [52] that for any multisegment � we have

� � � . � 8 .� ��Q � . � 8 .� � � � � �Hence it is enough to prove the lemma for � Q C ��.C CJ�.C � . This can be checked by an explicitcomputation in � . . The calculation of � . � 8 .� � is easy to perform using the algorithm of [32].On the other hand, for I � a point in the dense orbit of the irreducible component

5 � � , we have% � � Q �

�� and � . � � � �)� can be computed via (8) in Section 8. Thus, for � Q C � we obtain the

following expression for both � .�� 8 .� � � and � . � % � � � :� � � � ��J�.CJ�.CJ� �� J� � �.CJ� ��.CJ�� J� � �.CJ�.CJ� �� J� �� .CJ��J�.C �� J� ��.CJ� � �.CJ�� J� �� .CJ�.CJ�� J�.CJ� � � ��.CJ�� J� ��.CJ��J�.C �� J�.CJ� � �.CJ� �� .CJ��J� ��.CJ�� .CJ��J�.CJ� �� J� ��.CJ�.CJ�� J� � �.CJ� ��J�.C � � �� J� � � ��.CJ��J�.C � � � � � �J� � � ��.CJ�.CJ�� J�.CJ� �� .CJ�� J�.CJ� ��.CJ�� J�.CJ� ��J�.C �

The calculations for �2Q C C and � Q C � are similar and we omit them. �13.2. We can now finish the proof of Theorem 1.2. Assume that � is of type 1 8 . Let5 Q 5 = U ��U 5 �be the canonical decomposition of an irreducible component

5 [ ')(�(*� 0 8 � . By Corollary 7.5

% : Q'% : ! �� % : � �All components

5 � , �A � A � are of the form5 �/Q 5 � ��for some indecomposable multisegment � � � . By Proposition 13.1 we thus have %�: � Q 8 .: � .Moreover, using Theorem 12.1 we get

8 .: ! �� 8 .: � [ : . �Hence %N: belongs to : . . Thus for Dynkin types 1 2 with @\A �

the dual canonical basis and thedual semicanonical basis coincide. �

14. CASE 1 � : THE TUBULAR ALGEBRA�

AND THE WEIGHTED PROJECTIVE LINE �For the rest of this article we set 0BQ 0 � , �0 Q �

0W� , Q � , � � � � � � Q � � � � � �� ,� Q � � Q � � � �4= ��

and � Q � � � � . For our convenience we define moreover� � �

�Q � � � � �

�,� ��Q � � � � , � � Q � � = �

and� .�Q � � � = � �

�. Note that

� . �Q � op.


14.1. Almost all components of the Auslander-Reiten quiver of�

are tubes. This plays a crucialrole in our results, so we shall recall the definition of a tube (see [1, p. 287], [45, p. 113]).

Let � 1 � be the quiver with vertices � � 1 " � [ � �43 [ � � " and arrows

� � 1 1 � 1 � = "� [ � �H3 � � " � � � 1>1 �� *�&1 � = " � [ � � 3 � ��" �Define a map on the set of vertices by �� *� 13�2Q ��1 . For a vertex I in � 1/� and � � � let

� I�� Q � � � �+IZ��" � [ � " �Then � Q�� 1 � 6N�� is the quiver with vertices � I�� , and having an arrow � I�� 1 � Y � � if andonly if there is an arrow I � 1 Y � for some I � [ � I�� and some Y � [ � Y � � . The vertex � ��1;� � in � issaid to have quasi-length 3 . The map induces a map on � again denoted by , given by �� I�� Q � �+IZ� � �and called the translation. In this way, � becomes a translation quiver in the sense of [45, p. 47].One calls � a tube of rank � . A tube of rank 1 is called a homogeneous tube. The mouth of atube is the subset of vertices of quasi-length � . Sometimes we also consider as a tube a translationquiver

�whose stable part

�, obtained by deleting the translates of the projective and injective

vertices, is a tube.

14.2. For an algebra Y , we call a class of indecomposable Y -modules a tube if the vertices of� � that belong to that class form a tube [43, % 3.1]. A family of tubes � Q �� =��

is called atubular & -family.

In our situation the index set & will always be the projective line � = � �� . Such a tubular familyis said to be of type � � = � �� 2 � if for certain points I = � �� I 2 the corresponding tubes haverank � = � �� 2 respectively, and for all remaining points the tubes are homogeneous. Becauseof these exceptional points, a better index set is provided by the weighted projective line � inthe sense of Geigle-Lenzing [36], with exceptional points I = � �� I 2 having respective weights� = � �� 2 . Below we will point out some strong relations between the representation theory of0 and the weighted projective line � of weight type � � = � � 4�� 6P��Q ��J�.CJ��E� (see 14.6, 15.6).

14.3. The algebra� � is a tame concealed algebra of type

�� . This means that� � is obtained

from a hereditary algebra of type�� by tilting with respect to a preprojective tilting module [45,

% 4.3]. The tame concealed algebras have been classified by Happel and Vossieck [28], and onecan check that

� � belongs to one of the frames in this list.Similarly

� � is a tame concealed algebra of type�� . Its tubular type is � � �.CJ��E� (see [45,

p. 158]). There are two indecomposable� � -modules � and � �

completely determined (up toisomorphism) by their respective dimension vectors� = = �= =� = =�� = == == = � � �It can be shown that � and � �

lie at the mouth of a tube of the Auslander-Reiten quiver of� � ,

and that this tube has rank�. Moreover, the tubular extension

� � � � �-� � � � (see [45, % 4.7]) isisomorphic to

�. It follows that

�is a tubular algebra of tubular type ��J�.CJ��E� (see [45, % 5]).

Similarly,�

can be regarded as a tubular coextension of� � .

Note that the algebras� � 4 �� and

� � 4 � � =�

are isomorphic to� � and

� � , respectively, so theyare tame concealed. Similarly the algebras

� � 4 � � 4 � � =�

and� � 4 � � = � 4

��are isomorphic to

�and

� . ,respectively, hence they are tubular algebras. Thus we may speak of preprojective, regular andpreinjective

� � �� -modules or� � � � � � =

�-modules.


14.4. We are going to define some tubular families of�0 -modules. Following [27, % 2] we intro-

duce the following classes of modules. Let � � �� be the class of indecomposable� � � � = �

�� =�-modules

� such that the restriction of � to� � �� is regular and nonzero. Similarly, let - � � � � � =

�be the class

of indecomposable� � � � � � =

�-modules � such that the restriction of � to

� � �� is preinjective andthe restriction to

� � � � =�is preprojective.

Clearly, we may also interpret the classes � � �� and - � � � � � =�

as classes of�0 -modules, on which

the Galois group � acts by

(12) � 1 � � � �� Q � � � � 4 1 � � � 1 � - � � � � � =�Q - � � � 4 1 � � � 4 1 � = � � �#3 [ � � �

The classes � � Q � � � � and �H� Q � � = � are tubular families of type ��J�.CJ��E� . In Section 20.6and 20.7 we display the dimension vectors of the non-homogeneous tubes in these families. Theremaining tubes consist only of the homogeneous

� � �� -modules.The classes - � � � � � =

�are the regular modules of the tubular algebra

� � � � � � =�which do not belong

to the first or last tubular family. We may decompose - � � � � � =�

into a disjoint collection of tubularfamilies �� =

�� = �� of type ��J�.CJ��E� (see [45, % 5.2] and also 15.8 for more details).

14.5. Define � Q - � � = � �� - � � �4= � � � � = �

and �Q ��[I\&] � �D� [ � � �� 0 ��" � [ � " F � �0��

The next proposition shows that�

is a transversal of the action of the Galois group on the set ofindecomposable

�0 -modules.

Proposition 14.1. For each indecomposable�0 -module � there is a unique 3 [ � such that

� 1 � � [ � . Moreover for each [ [ � �0 � there is a unique � [ � with �� [ [ � .

Proof. By repeating the argument in [27, % 1] we find that each indecomposable module is con-tained in � ��= � � - � � � = �

�� . By (12) it only remains to show the unicity of 3 . This follows

from the fact that the classes - � � = � �� - � � �4= � � � � = � are pairwise disjoint. Indeed � � � � and � � = �

are disjoint as we can see from the dimension vectors of their objects, see 20.6 and 20.7. On theother hand - � � = � �

�and - � � �4= � are disjoint because a

� � �� -module can not be preprojective andpreinjective at the same time.

The second claim follows from the first. Indeed, we can determine the class � � �� or - � � � � � =�

which an indecomposable module belongs to by means of [I\�] � � � only (see [45, 5.2] and 15.8below). �14.6. Proposition 14.1 shows in particular the (known) fact that

�0 is locally support finite. More-

over the additive closure � � � � � � (i.e. the full subcategory of � �� 0�� consisting of modules whichare direct sums of modules of

�) is closed under Auslander-Reiten sequences. Thus by [17]

and [22, % 3.6] we may identify the Auslander-Reiten quivers of � � � � � � and of �� 0�� .These considerations together with (11) imply that the image

�of�

in �� 0�� is a transversalfor the indecomposable objects in �� 0 � under the action of � by powers of the translationfunctor � � = .

On the other hand, since�

is a tubular algebra of tubular type ��J�.CJ��E� , we know from [23] thatwe have equivalences of triangulated categories

� �� 0�� Q� � �� Q � � �� +��

where � is a weighted projective line of weight type ��J�.CJ��E� . Since �� +�� is a hereditary cate-gory we have the useful decomposition� � �� +��WQ�� = � �� +� �;� � � �


This is similar to the decomposition

�� 0��Q�� = � � � � � � � ��;� � �that we have just explained. Although a lot of our intuition on �� 0�� comes from the comparisonwith �� +�� , we prefer to avoid using this machinery.

15. CASE 1�� : THE ROOT SYSTEM

15.1. Write � � � � � for the Grothendieck group of �� . We have � � � � � Q � = � by identi-fying the class of a

�-module � with its dimension vector [I\&] � �D� [ � = � .

Let � 0/�P0�� = � � � = � 0 1 � be the Ringel bilinear form of�

, given by

�&[J\&] � � � ��[I\&]T� � � ��Q +�� 0 �*� � �HG �\K2LNM �� ,� � � � �� [ ��

see [45, p.70, 71]. The algebra�

being tubular has global dimension 2, so K�LJM 1 � � � �� Q R for3 � � , and this infinite sum is in fact finite. Explicitly for

[ Q�� H H � � !� ! H �

�! � �� H ! � � �� ! ! �� ! �� Q

� U H H U �!U

! HU �!

U ��U H!

U � �U! !

U ��

U � � !��we have

�&[�� Q + 9 � � � �0 +� �� 9 � �"�� +

� � �� 9 � �� where the first sum runs over all indices � 1 , the second one over all pairs ��(1E� � � � such that there isan arrow from � 1 to � � , and in the third sum � � � � � is the number of relations from � 1 to � � , i.e.

� � � ��ZQ � if �� 1E� � � � [ � ��4 �� = � �*� � = � � � � �*� � 4 � � = � �*� C = �.C � � �*� � � � � � = � " �R otherwise �

15.2. Let � be the matrix giving the Ringel form:

�&[�� Q [ � ��where [ and � are interpreted as column vectors in � = � and [ means transposition. Define theCoxeter matrix of

�by � QD0 � � = � (see [45, p.71]). It is easy to check that

(13) �&[�� Q 0 �� &[ � ��Q � �/�&[2� � �/��N� � � �&[�� [ � = � � �The data �\Q �� = � � � 0/�P0 � � �� is called a bilinear lattice [34].

15.3. Let [ QS�&9 � � = � � � � � 1 =�� be an element of � � � �0 ��Q � � �� 0�� . The support � � � �D�of � is defined as the set of vertices �(1 of

� such that the ��1 th component of [I\&]B� �D� is nonzero.

For �WA 3 it will be convenient to identify the Grothendieck group � � � � �� 1 � � with the subgroup of

� � � �0�� of elements with support in� �� 1 � , see 9.2. We shall denote by � 0/�P0��

� � 1 � the Ringel formon � � � � �

� � 1 � � , so that � 0/�P0��Q � 0�P0�� 4=�.

Let � = � � 6�� (resp. � 4�� 8 ) be the dimension vectors of the indecomposable projective�0 -

modules with support in� � � � 4

�(resp. in

� � � = �4=

�), that is,� = Q � = � �= �� = �� =� � = � � � 6 Q � � = �= == = == =� = � � � � � Q � � � =� =� = �= �= � � � �

� 4 Q � = �= = �= =� = =� = � � � 8 Q � � =� = == == = �= � � �


Up to shift these are the dimension vectors of all indecomposable projective�0 -modules. Note

that these are precisely the dimension vectors of the projective�0 -modules that belong to

�. The

following result is verified by a direct calculation.

Lemma 15.1. The map � � � �0 � 0 1 � � �� 0��Q � � � � � ��[I\&] � �D� 61 � � � (see 9.9) inducesisometries �

� � � � �� = � �

��0�1 � � � � � �� 61��0 I 8 � ! � 4 0 I 4�� 8�

� � � � �� = � 4 � ��0�1 � � � � � �� 61��0 I =

� ��=�0 I 6�H � 6 0 I � ! � �of bilinear lattices, that is,

��2�� = � � � Q ��

�� 2�� = � 4 � Q �

��

��

15.4. Let � �� 1 � � � � � � � � � 1 � � 0 1 � be the quadratic form � �

� � 1 � �&[2� Q �&[��[ � � � � 1 � . For simplicitywe write � �

� �Q7� �

� � ��and �Q7� � � �4=

�. Define � � �� to be the positive generator of the one-dimensional

radical of � ��

(recall that� � � � is tame concealed). This means for example that the support of � � � �

and of � � = � is contained in� � � �4=

�. Explicitly

� � � � Q�� = 4 =6 6= 4 = � � � � = � Q�� = == 4 == =� � � � �Moreover � � 4 �� Q �

� � � � � � and � � 4 � � = � Q �� = � . Since � �

� � � � =�is positive semidefinite of corank 2 it

is easy to see that ( � � � � �� =

��2Q �� U �� = � . Notice that (Lemma 15.1)

(14)

�� =

��QD0� � = � �

��

��2Q��

It will be convenient to set � � Q�� and �J� Q�� = � . These are the two generators of ( � � � � � .15.5. Let

� Q ��[ [ � = � "3�N�&[2��Q7RN� �� [ �Q7R�"be the set of roots of � . A root [ [ � is called imaginary if �N�&[2� Q R , and real otherwise. Let��

(resp. � �� ) be the set of imaginary (resp. real) roots of � .The form � being positive semidefinite, �

� �consists of the nonzero elements of ( � � � �E� . Note

that, by (13), [ is a radical vector if and only if �/�&[2��Q [ . Indeed,

�&[�� [ ��Q �&[,0 �/�&[2� �� and the Ringel form � 0/�P0 � is nondegenerate.

Since�

has tubular type ��J�.CJ��E� , the form � induces a positive definite form of type � on� = � 6 ( � � � �E� . (Note that the Dynkin diagram of type

� � is a star with three branches of lengths

�J�.CJ�� .) Thus, � is an elliptic root system of type � = �4=�� in the sense of Saito [50].

15.6. By [45, p.278], the elements of � � = � are precisely the dimension vectors of the inde-composable

�-modules. Note however that there exist elements in � whose coordinates are not

all nonnegative or all nonpositive. For example� � = �� = �� [ � �Following [36] and [24] we will introduce another notion of positive root in � . Define the subsetof positive roots of � as

�� Q �� [ � " �� J� � �BR or � �� J� ��Q R and �� BR � "E�and set � � Q � 0 [ "[ [ � � " .

In our setting � � is the set of classes in � �E� � � Q � � �� 0 �� of the non-projective (in-decomposable)

�0 -modules which belong to

�as we will see in Corollary 15.3. Then it follows


from 14.6 that � Q�� , see also [36]. One could also derive this fact from a careful analysisof � .

Our definition of � � is partially inspired by the interpretation of the bilinear lattice � as theGrothendieck group of the category �� +�� , see 14.6. Indeed, since �� +�� and �� arederived equivalent [36], they have isomorphic Grothendieck groups [26, p.101]. The linear forms� 0/��/� � and �� P0 � correspond to the functions “rank” and “degree” on � � �� +� �� , and � �corresponds to the set of classes of indecomposable sheaves.

15.7. Since for [ [ � � we have �&[��J� ��Q7R or �� [ ��Q R , the quotient �� [ � 6 �&[��J� � is awell-defined element of � � . For . [ � � let � � Q ��[ [ � � "�� [ � 6 �&[��J� ��Q . " be the setof positive roots of slope . . Thus

�� Q �� = �� For � [ � and 8 [ � such that � � R or ( � Q7R and 8<� R ) set

� � � ��Q�� 8 �/� �We have � � �

� � Q � � � � � " 8C6 � Q . " � This follows immediately from the equalities

�� /� ��Q 0 ��/� �� Q �J� �� Q ��J� ��/� ��Q R �Set

� � � Q �� = � � �� Q �� = � � � � �

Thus � � Q�� 15.8. The above decomposition of � � is motivated by the following fact. For . [ � � let � � � � � � =

��

be the class of indecomposable� � � � � � =

�-modules � with

�� [I\�]T� �D� � �� =

�

�&[J\&]T� � � �� = � � � � � � � = � Q . �Then � � � � � � =

�� is a tubular family of type ��J�.CJ��E� and � � = � � � � � � � � =

�Q - � � � � � =

�. Moreover the set

of dimension vectors ��[J\&]T� � � " � [ � � � � � � =�

� " coincides with the set of roots � of � � � � �� =

��

such that �� = �� = � � � � � � � = � Q . �

see [45, 5.2]. For . [ � � we may now define the tubular family

� � Q�� = � for . Q��T�� 4= �� for R � . � �T�� for . Q RN�� = � �

�� = � � for . �BR �

Clearly we have � � = �� Q � . We shall denote by �� the tube of the family � � indexed byI [ � . Thus, for all points I except the three exceptional points I = ��I 4��I�6 , �� is a homogeneoustube, and �� ! � ��

�H � �� have respective ranks �J�.C and � .

Lemma 15.2. For . [ � � we have � � Q � � � � " � [ � � and � is non-projective " .Proof. For . [ �*RN� � " this may be verified directly by comparing the dimension vectors of theobjects in � � (resp. � � ) which we describe in 20.6 (resp. 20.7) with the classes in � � (resp. � � )which we can construct explicitly, see Theorem 15.5. For R � . � � the result is clear from thecase � Q7R of Ringel’s result which we explained above. For . � R it follows from the same resultfor � QD0 � after applying the isometry

�� from Lemma 15.1 and (14). �


Corollary 15.3. The restriction� ��0�1 � � � �*R�" of the map � �� 0��0�1 � �E� � � ��[I\&] � �D� 61

� � � , is well-defined and surjective. Moreover, we have� � = � R ��Q � ��=P� � 4 � � 6 � � 8 � � � "E�� = � � � � ��Q � � � � � � � � � � 4 � � � � � � 8 "E� � � [ � � � �

� � = � � �J� ��Q � � �J� � � �J� � ��=P� � �J� � � 6 � � �/� � � � "E� � � [ � � � �If � [ � � :�� *� � � � � � " the set

� � = �� contains precisely one element.

We leave the proof to the reader.

15.9. Since�

is tubular of tubular type ��J�.CJ��E� , we have that � � Q � (see [34, 35]). Of coursethis can also be verified directly by a simple calculation.

Define the rank ( � �&[2� of a vector [ [ � = � as the minimal @�� such that � 2 �&[2��Q [ . Hence,if [ is an imaginary root ( � �&[ ��Q � , and if [ is a real root ( � �&[2��Q �J�.C or � .

The quasi-length � �&[2� of [ is defined as the greatest common divisor of the entries in theimaginary root

� �&[2��Q �� +�-, = �

��&[2� �

In other words, writing� �&[ ��Q�� we have � �&[2��Q�� .8 � . Set also

G � ��&[ ��Q �� &[2� � �&[2� �

Then [ [ � � if and only if G � ��&[2��Q�� where . Q 8C6 � and � � ��.8 ��Q � .These definitions are motivated by the fact that for a non-projective (indecomposable)

�0 -module

� which has quasi-length � in a tube of rank � we have

(15) � �� +��Q� if ��6 � if � "�� ( � �� +��Q

� if �� if � "��

(Of course our ( � should not be confused with the rank function on � � �� +�� ).15.10. For . [ �� , � [ � ��J�.CJ��" and � [ � � let

� � ��)��Q ��[ [ � � "3( � �&[2��Q �.�� &[2��Q�� "so that

� � Q � =�� = � = � 4 � 6 � � � �� )� �

Lemma 15.4. (a) If [ [ � � �� then � � ��)��Q � � 1 �&[2��"!3�Q �� " ;(b) If � � � and � divides � , then � � ��)��Q� ;(c) If � � � and � does not divide � , then � � ��)� consists of � real roots;(d) If � Q � , then � � � �*��Q � � � � � " where 8C6 ��Q . and � � ��.8 ��Q�� .

Proof. Let [ [ � � ��)� . Clearly, ( � � �/�&[2��Q ( � �&[2� , � � �/�&[2��Q � �&[ � , and � � ��&[ ��Q�� &[2� .Moreover, by Equation (13) we have �/�&[ � [ � � . Hence the � -orbit of [ is contained in � � �� .

Conversely, if � � ��)� is nonempty, then it consists of a single � -orbit. This follows from (10)and Lemma 15.2 (remember that � � is a tubular family of type ��J�.CJ��E� ). Note that all the inde-composable modules lying on some homogeneous tube of �� and having the same quasi-length �have the same class � � � � in � � � � � , where 8C6 ��Q . and � � �� .8 � Q�� . Moreover for � Q �J�.CJ��the element � � � � is also the class of � modules with quasi-length � lying on the tube of rank � of� � . It follows that there is no positive root [ with ( � �&[ ��Q � � � and � �&[2� divisible by � . �


15.11. Let � [ � �J�.CJ��" . For R A � ��@ A � 0 � and � � ��@ � �Q � RN��R � set

� � � � 2 � ��)��Q�� )�where . Q7@/6�� and �>Q�� @ � . By Lemma 15.4, � � � � 2 � ��)� has cardinality � . In Section 20.8we list all elements of the sets � � � � 2 � ��)� , that is, a distinguished set of

� � C � C �� C � Q � � Rreal elements, say � = � �� 4 8 � , of � � .

We constructed this set in the following way. Note first, that it coincides with the set

�� [ ��\"�R A ��*�� A � " �Now consider the algebra

� �which is obtained by restricting

�to the full subquiver of

� � � �4=

�

which is obtained by removing the vertices � = and� = . Notice that if � is a real root, then � 0 � = ! � � 0�(8 ! �/� is a real root with support in

� �. This is tilted of type � , thus the associated quadratic

form � � has 240 roots � � = � �� 4 8 � . Find them for example as the orbits of the dimension vectorsof the � indecomposable projective

� �-modules under the corresponding Coxeter transformation

� � . Since� �

is tilted of type � , the transformation � � has order C�R (the Coxeter number of � ).Next, define integers � �� .8 �� .8 � �� by

�� J� �2Q �� with R A � � �� A � �� 2Q � 8 �� 8 � �� with R AB8 � �� A � �

for �A��WA � � R . Finally set � � Q � �� 0 � �� 0 8 �� /� �We can now give the following explicit construction of all real roots in � � .

Theorem 15.5 (Construction of � � ��)� ). Let . [ � � , � [ � �J�.CJ��" and � [ � � not divisible by � .Write . Q 8C6 � with � [ � , 8 [ � and � � ��.8 ��Q�� (if . Q�� set � Q7R and 8 Q�� ). Write

� Q � � � � � � � � 8 Q �-8 � � 8 � �with � � � � � � �.8 � �.8 � � [ � such that R A � � � �.8 � � A�� 0 � . Then

� � �� Q � � � � � � � � � " � [ � � � � � � � � � � ��)� " �Proof. Let � be a real root of rank � , � an imaginary root and set [ Q � � � . Then [ is a real rootand �/�&[2��Q �� , hence [ has also rank � . Moreover, if �#Q � � � � � � and � [ � � � � � � � � � � ��)� then� �&[2��Q � � � � � � � � � � � � � � � � Q � � � � , therefore [ belongs to � � ��)� . Thus, since � � ��)� and � � � � � � � � � ��)�both have cardinality � we see that the first subset is the translate of the second one by � � � � � � . �

Thus, the 240 positive real roots listed in Section 20.8 yield a complete description of the infiniteset of all positive real roots of � . Note that the classes of these 240 roots in � = � 6 ( � � � �E� form a

finite root system of type � , and we recover that � is an elliptic root system of type � = �4=�� .

15.12. The set � �� of Schur roots is defined as

� �� Q ��[ [ � � " � � � �� [ � � �&[�� A ��" �It is easy to see that the set of imaginary Schur roots consists of the

�� with � � �� .8;��Q � .

The Schur roots can also be characterized in terms of rank and quasi-length, namely

� �� Q ��[ [ ��\" � �&[ � A\( � �&[ � " �This comes from the identity

� � � �� [ � � �&[��J� ��Q � � �&[ �( � �&[ � �


Thus the set of real Schur roots is equal to the union of all subsets � � ��)� for . [ � � , ��Q �J�.CJ��and �/A � �� . Using 15.10, this implies that �� contains exactly

� � � � � � C � � � � � � Q C��roots, one imaginary and the others real. Note that all the 240 roots listed in Section 20.8 are Schurroots.

The Schur roots of � � are related to the Schur roots of�0 (see 10.1) in the following way.

Suppose that � [ � is non-projective and has quasi-length � in a tube of rank � . Since all tubesin� �O are standard, the endomorphism ring of � is non-trivial if and only if either � � � , or � Q �

and � has a non-trivial endomorphism that factors over a projective module in the same tube.

Corollary 15.6. A module � [ � has trivial endomorphism ring if and only if � � � Q R , or� � � [ � �� and [I\&]B� �D� does not belong to the following list:

�/� � � � � � � � � 1with � Q ��.CJ� � and 3GQ �J� � .

Proof. This follows from (15) together with our description of � � and � � . �16. CASE 1�� : PARAMETRIZATION OF THE INDECOMPOSABLE IRREDUCIBLE COMPONENTS

We shall now explain how � � parametrizes (i) the indecomposable 0 -modules and (ii) the setof dimension vectors of indecomposable

�0 -modules modulo the Galois group action. From (ii)

we shall deduce the main result of this section, namely the parametrization of the indecomposableirreducible components of 0 -modules by � �� .

16.1. By 14.6, the indecomposable 0 -modules are in one-to-one correspondence with the�0 -

modules of the class�

. This class decomposes into tubular families �� as shown in 15.8. Theprojective modules � = and � � appear at the mouth of the non-homogeneous tube I� � � ! , the mod-ule � 6 at the mouth of the non-homogeneous tube I� � � H , and the modules �54 and � 8 at the mouthof the non-homogeneous tube � � � ! . As a result we obtain the following parametrization of theindecomposable 0 -modules by � � .

Proposition 16.1. Let . [ � � , � [ � ��J�.CJ��" and � [ � � . Then the following hold:

(a) If � does not divide � then there exists a one-to-one correspondence between � � ��)� and theset of indecomposable 0 -modules of quasi-length � in the non-homogeneous tube � � � with � 1 Q � . This correspondence maps [ [ � � ��)� to � Q � � � � with � � �VQ [ .

(b) If �GQ � then � � � �*� Q � � � � � " where 8C6 �\Q . and � � ��.8 ��Q�� . There is an infiniteset of indecomposable 0 -modules � Q�� ,� with � � � Q � � � � , parametrized by theweighted projective line � . More precisely, for each ordinary point I [ � there is anindecomposable module of quasi-length � in the tube � � � , and for each exceptional pointI21 , there are � 1 indecomposable modules of quasi-length � on � � � .

(c) The only indecomposable 0 -modules not appearing in the above lists are the five inde-composable projective modules.

16.2. Recall from Corollary 15.3 that we have a canonical map� ��1 � � � �*R�" . We are going

to define a ‘right inverse’ � of�. Define � �� 0 1 � � � �0�� by

� �� Q�� 0 � G � �*RN� ��4 H " � 4 0 � G�� *RN� �(8 ! " � 8 if � [ � � �� if � [ � � � �� 0 � G � �*RN� � = ! " ��=W0 � G�� *RN� � 6�� " � 6 0 � G�� *RN� � � � ! " � � if � [ � � �� 0 � 4�H � 4 0 � 8 ! � 8 if � [ � � � �

Proposition 16.2. With the above definition of � we have:


(a) � induces a well-defined and injective map �*�� 0 1� �Q � �0 � 6 � ;

(b) The only elements of�

not in the image of � are

� �/� � � � � � � � � � 1with � Q ��.CJ� � and 3�Q �J� � ;

(c) The map � restricts to a map � � � �� 0 1�� Q

� �� . The only elements of

�� not

in the image of � � are ��=P� �� .Proof. First we have to show that � �� is the dimension vector of an indecomposable

�0 -module.

This is clear for � [ � � � . For � [ � � � we notice that � 61�� 0 I 4�H � 4 0 I 8 ! � 8 gives an isometry� � � � � 0 1 � � � � � � = � �

�� , so that � � � � � � consists of dimension vectors of objects in - � � = � �

�.

The remaining two cases are treated directly. It is easy to calculate the map�

(Corollary 15.3)explicitly with (9). It follows that

� � Q �� and the rest of (a) follows since�

is a transversal forthe action of � on G�� 0�� , see Proposition 14.1. Now we obtain � 8 � from the description of thefibres of

�in Corollary 15.3, and (c) follows from Corollary 15.6. �

16.3. Collecting the results of Theorem 10.1, Theorem 10.3 and Proposition 16.2 we can nowstate the following parametrization of the indecomposable irreducible components of varieties of0 -modules and of the corresponding multisegments. Let

� 1 � �BA 3 A � � be the irreduciblecomponents containing the five indecomposable projective 0 -modules. Let �21 denote the corre-sponding multisegments, namely

� 1 Q� � 1 � =+�-, = � �� 3G0 � � � �#3 Q ��

Theorem 16.3. (a) The map [ 61 � � 5�� is a one-to-one correspondence from the set � �� of Schur roots of the Ringel form � 0/�P0��in � = � to the set G�� V�+')(�(*� 0��0 � � = � �� " .

(b) The map [ 61 � � � �&[2��is a one-to-one correspondence from �� to G�� V� � � 0 �� = � �� " .

Note that the descriptions of �� *� � and�

are completely explicit, so that we get a veryconcrete parametrization of the factors arising in the canonical factorization of the elements of � . .

Example 3. (i) Let [ Q � = �4 6 =6 6= 4 = � [ � � � . Then �J�&[2��Q [ and� � � �&[2��WQ � �� .C � � � � �J�.C � � �4CJ� � � � �4CJ� � � � � � � � � � � � � � � �

(ii) Let [ Q � � = �� = �� [ � � . Then �J�&[2��Q� � �= = �4 =� = =� = � and

� � � �&[2��Q � �� J�� J�.C � � �4CJ� � � � � � � � � �17. CASE 12� : THE COMPONENT GRAPH

Recall that the component graph �� 0�� has for vertices the indecomposable irreducible compo-nents of the varieties of 0 -modules, and two vertices

5 = and5 4 are connected by an edge if and

only if5 = U 5 4 is an irreducible component, or equivalently �;LJM =O � 5 = � 5 4 ��Q R . There are edges

from the irreducible components� � � ��A � A � � to every other vertex. The following theorem

describes all remaining edges. In agreement with Theorem 16.3 (a), we shall label the verticesother than

� �by the elements of � �� .


In order to state the theorem we introduce the following definition. We call a pair of Schur roots�&[�� critical if the following three conditions hold:

� ��[�� " F � � ��E� for some � [ � � and � [ � � ,� �&[�� Q7R Q �� [ � ,� � �&[ � � � �� .

Theorem 17.1. Two Schur roots [ and � are connected by an edge in �� 0�� if and only if thefollowing two conditions hold:

(i) �&[�� R and ��[ � � R ,(ii) �&[�� is not critical, or �&[�� R where � Q � G � � 3 � � " �&[�� 1 �� Q7R�" .

Proof. This follows from an adaptation of [24, Theorem 1.3, Lemma 6.4]. �Thus, the edges of the component graph of 0 are completely determined by the bilinear form

� 0/�P0�� and the Coxeter matrix � . Moreover there is an edge between 9 and � if and only if thereis an edge between ��&[ � and �/�� .

18. PROOF OF THEOREM 10.3

It was already proved in [24] that�

and � are well-defined and bijective. The map � is bijectiveby definition. It remains to explicitly construct the map

� Q � � = � � . It is enough to proveTheorem 10.3 for 0\Q 0 � , since 0 2 ( @ Q �J�.CJ� � ) are full convex subalgebras of 0 � .

We will use the following result from [5, Theorem 1, 2]: Let Y be a tame quasi-tilted basicalgebra, and let [ be a dimension vector of an indecomposable Y -module. Then �� &YG��[2� hasat most two irreducible components, and �� &Y ��[2� is irreducible if and only if [ is not of one ofthe following forms:

(a) [ Q � �� where � and � are connected positive vectors with disjoint support, �� Q R ,� � � � � � Q � and � A � for all entries � of � ,

(b) [ Q � � � � where � and � � are connected positive vectors with �� Q �� Q R ,�� Q � and �� Q7R .

The algebras� Q � � � �4= � and

� . Q � � � = � ��

are both tubular algebras, in particular they are tamequasi-tilted algebras. All connected positive vectors � for

� � � � � � =�

such that �� = � QDR areof the form � � � �� 8 � � � � = � where � RN��R � �Q �� .8 � [ � � � . We have �� = � � � � � � � = � Q �and �� = � �� = � Q 0 � . It follows that the case � 8 � above cannot occur for Y Q � � � � � � =

�,� QD0 ��R .

For Y7Q �we are in case �� precisely when [ [ � � = �+@ � � �� E�+@ � "�@ � � " where

� = �+@ ��Q � = �2 4 2 26 2 6 22 4 2 2 � � � 4E�+@ �2Q � � =2 4 2 26 2 6 22 4 2 2 � �and

� 6 �+@ ��Q � 2 22 4 2 22 2= � � � � � 8 �+@ ��Q � 2 22 4 2 22 2� = � � � � � �+@ ��Q � 2 22 4 2 22 2� � = � �For Y Q � . we are in case �� precisely when [ [ � �� = �+@ �� . � �� *�� E�+@ �� . "�@ � � " where

� = �+@ � . Q � 2 4 2 26 2 6 22 4 2 2� = � � � 4E�+@ � . Q � 2 4 2 26 2 6 22 4 2 2= � � �and

� 6 �+@ � . Q � � � =2 22 4 2 22 2 � � � 8 �+@ � . Q � � = �2 22 4 2 22 2 � � � � �+@ � . Q � = � �2 22 4 2 22 2 � �Let � be an indecomposable

�0 -module, and let [J\&] � � � be its dimension vector. If

[J\&]T� � � [ � � � �+@ � � �� +@ �� . " � Q ��J��@�� "E�


then � � �D� [ � � , and if

[I\&] � �D� [ � � � �+@ � � �� +@ �� . " � Q CJ� � � � ��@ � � "E�then � � �D� [ �T� . One easily checks that �

� �+@ � is a Schur root if and only if @ Q � . Set �� Q �

� � �*�and � .� Q �� *�� . , � A��A �

.Assume that [ [ � � �0 � is a Schur root whose support lies (up to shift) in

�or� . , and

assume [ �[ � � � � � .� " � A � A � " . By the result mentioned above this implies that � �� 0>��[2� isirreducible. Thus

5� Q 5 �� &[2� and� �&[2��Q �� 5 ��Q �� &[2� �

Next, assume [ [ � � � � � .� " ��A ��A � " . Thus �� 0 ��[2� has exactly two irreducible compo-nents. Furthermore, we know that 5

� Q � � � ��for some indecomposable

�0 -module � � .

For any vertex � 1 of� let � � be the corresponding simple

�0 -module, and let

5 � be the irre-ducible component consisting of the single point corresponding to � � . Then

�;LJM =�O � 5�� 5 � ��Q'�;LNM =�O � 5 � � 5�� P��QTRfor � Q �J� � and all 3 [ � , and

�;LJM =�O � 5�� 5 � ��Q'�;LNM =�O � 5 � E� 5�� Q7Rfor ��Q ��.CJ� � and all 3 [ � . Thus [14, Theorem 1.2] implies that

5�� U 5 � (resp.

5�� U 5 � )are irreducible components provided ��Q �J� � (resp. ��Q ��.CJ� � ). Exactly one of these irreduciblecomponents lies in �� 0 ��[ � , we denote this component by

5 � � � �&[2� . Thus �� 0>��[2� containsexactly one indecomposable irreducible component, namely

5� , and exactly one decomposable

irreducible component, namely5 � � � �&[2� .

If5 � U 5 � � is an irreducible component, then�� 5 � U 5 � � ��Q �� 5 � � � �� 5 � � � �

Thus, if [ �[ � � .6 � � � " , then5 � � � �&[2��Q 5 �� &[ � . This yields

5� Q 5 �� &[ � and� �&[2��Q �� 5 ��Q �� &[2� �

For [ [ � � .6 � � �!" it is not difficult to construct the module � � explicitly. We get� �� .6 �2Q � � �� J�� J� � � � �4CJ�.C � � � � � � �and � �� QD� �� J� � � � �4CJ�.C � � � � � � � � � � � � � � �

The only Schur roots in � � �0�� whose support is (up to shift) not contained in�

or� . are

� = Q� � == 4 =6 4= 4 =� = � � � 4 Q

� = �= 4 =4 6= 4 == � � � � 4 �� 8(these are all in � � ), and

� = Q� = � �= =� 4 == == � � � � � 4 Q

� � � == == 4 �= =� � = � ��=P� � 6 � � �(these are all in �L� ). Here ��=*� �� are (up to shift) the dimension vectors of the indecompos-able projective

�0 -modules as displayed in 15.3.


For each [ [ � ��=*� �� = � � 4 � � = � � 4 " , there exists an indecomposable�0 -module � � such

that5�,Q � � � �� . For [ [ � ��= � �� = � � 4 " it is easy to construct � � explicitly. In these

cases, it follows that5� Q 5 �� &[2� , thus� �&[2��Q �� 5 ��Q �� &[2� �

For any element � [ �0 and any�0 -module � , let � � � � � 1 � be the linear map defined by

the�0 -module action of � on � , i.e. � � � � � � Q � � . If � is a submodule of a module � , then

we have( � � � � � � �\(

� � � � � �for all � . Now let

[?Q � = Q� � == 4 =6 4= 4 =� = � �

In Section 20.6 we can see that � � contains two indecomposable submodules �� !

and �� !

withdimension vectors

� �= Q� � �= 4 =6 4= 4 =� = � and

� � �= Q� � =� = =4 == = ��

respectively. We get 5� �!Q � � � � �

!� and

5� � �!Q � � � � � �

!� �

Since the support of� �= and

� � �= lies in� . and

�, respectively, we get� � � �= ��Q �� 5 � �

!�2Q �� = � and

� � � � �= ��Q �� 5 � � �!�2Q �� = � �

This enables us to compute the ranks of the maps V � �� !

and CV � �� !

for all paths � in� of the

form ' � 1 ' � � = � 1 �� ' � 1 , � A � A �9A �. Then we can use the above rank inequality, and get the

rank of CV � � �!

for any path � . It turns out that� � � = ��Q �� 5 � ! �2Q �� = � �The case [ Q � 4 is done in a similar way, and we get again� � � 4*��Q �� 5 � H �2Q �� 4*� �

19. CONCLUDING REMARKS

19.1. By Proposition 3.3, the preprojective algebra 0 Q � �� is tame if and only if the quiver is of Dynkin type 1�� or �58 . Using the same methods as in this paper it is possible to obtaina complete analogue of Theorem 1.3 for type � 8 . In this case, 0 has a Galois covering which isisomorphic to the repetitive algebra of a tubular algebra of type � CJ�.CJ�.CE� , and the corresponding

root system is an elliptic root system of type � = �4=�� . We plan to give a detailed analysis of this casein a forthcoming publication.

19.2. It is shown in [3] that for all Dynkin types, the algebra �� has the structure of an (upper)cluster algebra, and that it has finite type as a cluster algebra if and only � is of Lie type 1 2 �+@?A � � .In that case one can associate to �� a root system � called its cluster type, which controls thecombinatorics of the cluster variables and of the cluster sets. More precisely, the cluster variablesare parametrized by the set � � � = of almost positive roots of � , and the pairs of cluster variableswhich can occur simultaneously in a cluster set can be explicitly described in terms of � anda piecewise linear Coxeter transformation acting on � � � = . The Cartan matrix Y of � can beobtained by a certain symmetrization procedure from the principal part < �� of the exchangematrix of �� at certain vertices � of its exchange graph (see [20]).

For � of Lie type 1 4 � 176N� 158 , the algebra � �4�� has cluster type 1 = � 176N� � � respectively [3].As mentioned in 9.10, these root systems also occur in our setting in the following way. For


1

2

34

10 5 6 7 8 9

FIGURE 2. The Dynkin diagram of � =��4=��

@SQ �J�.CJ� � , we have �� 0 2 � �Q� � �� 2V�� where � 2 is a quiver of type 1 = � 176N� � �

respectively.

19.3. At the moment, there is no notion of cluster type for the algebras �� which are not offinite type as cluster algebras. The results of this paper strongly suggest that if such a cluster typeexists for � of Lie type 1�� (resp. � 8 ), then it should be the elliptic type � = �4=�� (resp. � =��4=�� ) in thenotation of Saito, or the tubular type ��J�.CJ��E� (resp. � CJ�.CJ�.CE� ) in the language of Ringel. Rememberin particular that � �� 0W�P� �Q

� � �� +� �� where � is a weighted projective line of type ��J�.CJ��E� .19.4. Here is another remark supporting that guess. For � of Lie type 1 � one can find a vertex� of the exchange graph of �� at which the principal part of the exchange matrix is (up tosimultaneous permutations of rows and columns)

< ��Q

��

R R � R R R R R R RR R R � R R R R R 0 �0 � R R � R R R R R 0 �R 0 � 0 � R 0 � R R R R �R R R � R 0 � R R R 0 �R R R R � R 0 � R R RR R R R R � R 0 � R RR R R R R R � R 0 � RR R R R R R R � R RR � � 0�� R R R R R

��We suggest to take as symmetric counterpart of < �� the matrix

Y Q

��

� R 0 � R R R R R R RR � R 0 � R R R R R 0 �0 � R � 0 � R R R R R 0 �R 0 � 0 � � 0 � R R R R �R R R 0 � � 0 � R R R 0 �R R R R 0 � � 0 � R R RR R R R R 0 � � 0 � R RR R R R R R 0 � � 0 � RR R R R R R R 0 � � RR 0 � 0 � � 0 � R R R R �

��

Note that Y has two positive entries off the diagonal, so it differs from the matrix obtained from< �� by the symmetrizing procedure of [20]. It turns out that Y is exactly the Cartan matrix of the

root system � = �4=�� , that is, the matrix of scalar products of a basis of simple roots in the sense ofSaito [50]. It can be visualized with the help of the Dynkin type diagram of Figure 2 in which an


ordinary edge between � and 3 means that � � 1Q �1 � Q 0 � and the dashed line between�

and � Rmeans that � 8 �4= � Q � = � � 8>Q�� (see [50]).

There is a similar fact for type � 8 and � = �4=�� .

20. PICTURES AND TABLES

20.1. The Auslander-Reiten quivers of 0 4 , 0 6 , 0 8 .

�� ==��7

77

== �� =�

��777

�= ��

CC��

�= =�

CC��

The Auslander-Reiten quiver of 0 4

� ��= �=� =��8

88

� === �� = �=� �

��888

� �� =�� 8

88

� �== �� 8

88

BB��

� �� =� ��8

88

BB�� = ==� ��8

88

//

BB�� == ==� �//� �== =��

��888

BB�� =� �� ==� �

BB�� = ��

BB�� =� =�� 8

88

BB��

� �� === �

BB�� = �=� =�� The Auslander-Reiten quiver of 0 6

42C

HR

ISTO

FG

EISS,B

ER

NA

RD

LE

CL

ER

C,A

ND

JAN

SCH

RO

ER

� ��

� ��

� �

��===

� ��

� ��

� �

��===

� ��

� ��

� �

��===

� ��

� ��

� �

@@��

��===

� ��

� ��

� �

��===

� ��

� ��

� �

��===

� ��

� ��

� �

��===

@@��

� ��

� �

��===

� ��

� ��

� �

� ��

� ��

� �

@@��

��===

� ��

� ��

� �

@@��

��===

� ��

� ��

� �

@@��

��===

� ��

� ��

� �

@@��

��===

� ��

� ��

� �

@@��

��===

� ��

� ��

� �@@��

��===

� ��

� ��

� �

��===

@@��

� ��

� �

��===

@@��

� ��

� �

��===

@@��

� ��

� �

��===

@@��

� ��

� �

��===

@@��

� ��

� �

��===

@@��

� ��

� �

� ��

� ��

� �

//

� ��

� ��

� �

//

��===

@@��

� ��

� �

//

��...

....

....

..

� ��

� ��

� �

//

��===

@@��

� ��

� �

//

� ��

� ��

� �

//

��===

@@��

� ��

� �

//

� ��

� ��

� �

//

��===

@@��

� ��

� �//

��...

....

....

..

� ��

� ��

� �

//

��===

@@��

� ��

� �

//

� ��

� ��

� �

//

��===

@@��

� ��

� �

� ��

� ��

� �

@@��

� ��

� �

@@��

� ��

� �

@@��

� ��

� �

@@��

� ��

� �

@@��

� ��

� �

@@��

� ��

� �

� ��

� ��

� �

GG��

� ��

� �

GG��

The Auslander-Reiten quiver of

��


20.2. The indecomposable multisegments in � � � � .

� = Q � �� 4 = Q � ��.C � � � � � � �� 4 Q � �J�� 4 4 Q � �� 4CJ�.C � � � � � � �� 6 Q �4CJ�.C � � 4 6 Q � �� J�� 4CJ� � �� 8 Q � � � � � � 4 8 Q � �� J� � �� Q � �� 4�� Q � �� 4CJ� � �� Q � �� J�� 4 � Q � �� J�.C � � � � � � �� Q � �J�.C � � 4 � Q � �� J�.C � � � � � � �� Q � �J�� 4CJ�.C � � 4 � Q � �� J� � �� Q �4CJ� � � � 4�� Q � ��.C � � �4CJ� � �� = � Q �4CJ�.C � � � � � � � � 6 � Q � �� J�.C � � �4CJ� � �� =�= Q � ��.C � � 6 = Q � �� J�.C � � �4CJ�.C � � � � � � �� = 4 Q � �� J�� 4CJ�.C � � 6 4 Q � �� J� � � � �4CJ�.C �� = 6 Q � �� 4CJ�.C � � 6 6 Q � ��.C � � � �J�� 4CJ� � �� = 8 Q � �� J�.C � � 6 8 Q � �� J�� J�.C � � �4CJ� � �� = � Q � �� J�.C � � 6�� Q � �� .C � � � �J�� 4CJ� � �� = � Q � �J� � � � 6 � Q � �� J� � � � �4CJ�.C � � � � � � �� = � Q � �J�� 4CJ�.C � � � � � � � � 6 � Q � �� = � Q � �J�.C � � � � � � � � 6 � Q � ��.C � � � �J� � �� = � Q � �J�� 4CJ� � � � 6�� Q � �� J�.C � � �4CJ� � �� 4 � Q � �J�.C � � �4CJ� � � � 8 � Q � �� J�� 4CJ�.C � � � � � � �

� 6 �$$JJ

J� 8 �

$$JJJ� 8

$$JJJ

� =�= ::ttt

$$JJJ

� = �$$JJ

J� =

$$JJJ

� = �$$JJ

J

::ttt � = 4$$JJ

J� 8� 4 =

::ttt

$$JJJ

� � ::ttt

$$JJJ

� 4 8::ttt

$$JJJ

� 4 4::ttt

$$JJJ

� � ::ttt

$$JJJ

� 4 6::ttt

$$JJJ� 6��

$$JJJ

::ttt � = �$$JJ

J

::ttt � = 8$$JJ

J

::ttt � 6 �$$JJ

J

::ttt � = 6$$JJ

J

::ttt � = �$$JJ

J

::ttt � 6�� 4 // � 6 8 //

$$JJJ

::ttt � 6 � //

��777

7777

� 4 � //

$$JJJ

::ttt � 4 � // � 6 = //

$$JJJ

::ttt � 6 // � 6 4 //

$$JJJ

::ttt � 4 � //

��777

7777

� 4�� //

$$JJJ

::ttt � 4�� // � 6 6 //

$$JJJ

::ttt � 4� 4 �::ttt � � ::ttt � = � ::ttt � = � ::ttt �

�

::ttt � � ::ttt � 4 �� 6��

CC�� 6 �

CC��

The Auslander-Reiten quiver of 0 8 in terms of multisegments


20.3. Extensions between indecomposable 0 6 -modules.

��8

88�

��:::

��6

66& 4

��>>>

�

��:::

��8

88

& =??��

AAA

�

BB��

��===

�

BB��

��;;;

� �??��

AAA

A �BB��

��;;;

�

BB��

��===

��

AA��

@@�� BB��

>>}}}} �@@��

AA��

20.4. Extensions between indecomposable 0 8 -modules.

& = A

AA�

��===

��=

==�

��===

��=

==�

��;;;

��

��>>>

AA��

@@@

@@�� @

@@

@@�� @

@@

@@��

@@@

@@�� <

<<

BB��

�""D

DDD

<<zzzz � @

@@

??~~~ �

@@@

??~~~ � @

@@

??~~~ �

@@@

??~~~ ��>

>>

@@�� // � //

��>>>

@@�� // � //

@@@

??~~~ � // � // @

@@

??~~~ � // � //

@@@

??~~~ � // � //

@@@

??~~~ � // � //

��<<<

AA��

<<zzzz �??~~~ �

??~~~ �??~~~ �

??~~~ �@@��

�""D

DDD�

@@@

�

@@@

� @

@@�

@@@

��>

>>�

��;

;;

@@��

��===

??~~~ �

��===

??~~~ �

��===

??~~~ �

��===

??~~~ ��9

99

AA��

& 4 A

AA

>>}}}�

��===

@@��

��===

@@��

��===

@@��

��===

@@��

��;;;

AA�� // � //

��>>>

AA�� // � // @

@@

@@�� // � // @

@@

@@�� // � // @

@@

@@�� // � //

@@@

@@�� // � //

��<<<

BB��

<<zzzz �??~~~ �

??~~~ �??~~~ �

??~~~ �@@��

�""D

DDD�

@@@

� @

@@�

@@@

� @

@@�

��>>>

��

��>>>

@@�� @

@@

??~~~ �

@@@

??~~~ �

@@@

??~~~ � @

@@

??~~~ ��<

<<

AA��

� A

AAA

<<zzzz ��=

==

??~~~ �

��===

??~~~ �

��===

??~~~ �

��===

??~~~ ��;

;;

@@�� & 6 // � //

��;;;

AA�� // � //

��===

@@�� // � //

��===

@@�� // � //

��===

@@�� // � //

��===

@@�� // � //

��999

BB��

>>}}}} �@@��

@@�� @@��

@@�� AA��

�""D

DDD �

@@@

� @

@@�

@@@

� @

@@�

��>>>

��

��>>>

@@�� @

@@

??~~~ �

@@@

??~~~ �

@@@

??~~~ � @

@@

??~~~ ��<

<<

AA��

�""D

DDD

<<zzzz � @

@@

??~~~ �

@@@

??~~~ �

@@@

??~~~ �

@@@

??~~~ ��>

>>

@@�� // � //

��;;;

@@�� // � //

��===

??~~~ � // � //

��===

??~~~ � // � //

��===

??~~~ � // � //

��===

??~~~ � // � //

��999

AA�� & 8

>>}}}�

@@�� @@��

@@�� @@��

AA��

AAA

��;

;;�

��===

��=

==�

��===

�

��===

�& �

AAA

>>}}}�

��===

@@��

��===

@@�� =

==

@@��

��===

@@��

��;;;

AA��

�""D

DDD

>>}}} ��>

>>

AA��

@@@

@@��

@@@

@@��

@@@

@@��

@@@

@@�� // � //

""DDDD

<<zzzz � // � //

@@@

??~~~ � // � // @

@@

??~~~ � // � // @

@@

??~~~ � // � //

@@@

??~~~ � // � //

��>>>

@@��

<<zzzz �@@��

??~~~ �??~~~ �

??~~~ �??~~~ �

�""D

DDD�

��>>>

� @

@@�

@@@

�

@@@

� @

@@�

�""D

DDD

<<zzzz � @

@@

??~~~ �

@@@

??~~~ �

@@@

??~~~ �

@@@

??~~~ ��>

>>

@@��

� A

AA

<<zzzz ��;

;;

@@��

��===

??~~~ �

��===

??~~~ �

��===

??~~~ �

��===

??~~~ �� // & � //

AAA

>>}}} � // � //

��===

@@�� // � //

��===

@@�� // � //

��===

@@�� // � //

��===

@@�� // � //

��;;;

AA��

>>}}} �AA��

@@��

@@��

@@��

@@��


20.5. The graph �� .

(1,3) (1,4) (1,5) (1,6) (1,9) (1,10) (1,11) (1,12) (1,13) (1,14)(1,15) (1,21) (1,22) (1,23) (1,24) (1,25) (1,26) (1,27) (1,28) (1,29)(1,30) (1,31) (1,32) (1,35) (1,36) (2,4) (2,5) (2,6) (2,7) (2,8)(2,11) (2,12) (2,15) (2,16) (2,17) (2,18) (2,19) (2,20) (2,21) (2,23)(2,27) (2,28) (2,33) (2,34) (2,35) (3,7) (3,8) (3,9) (3,10) (3,11)(3,12) (3,13) (3,14) (3,15) (3,16) (3,17) (3,20) (3,22) (3,24) (3,29)(3,30) (3,31) (3,32) (3,36) (4,5) (4,6) (4,9) (4,10) (4,16) (4,17)(4,18) (4,19) (4,20) (4,21) (4,22) (4,23) (4,24) (4,25) (4,26) (4,27)(4,28) (4,29) (4,30) (4,33) (4,34) (4,35) (4,36) (5,6) (5,8) (5,11)(5,12) (5,13) (5,15) (5,17) (5,19) (5,21) (5,22) (5,23) (5,25) (5,27)(5,28) (5,29) (5,33) (5,35) (6,7) (6,11) (6,12) (6,14) (6,15) (6,16)(6,18) (6,21) (6,23) (6,24) (6,26) (6,27) (6,28) (6,30) (6,34) (6,35)(7,8) (7,10) (7,11) (7,12) (7,14) (7,15) (7,16) (7,17) (7,18) (7,20)(7,26) (7,27) (7,30) (7,31) (7,34) (8,9) (8,11) (8,12) (8,13) (8,15)(8,16) (8,17) (8,19) (8,20) (8,25) (8,28) (8,29) (8,32) (8,33) (9,10)(9,12) (9,13) (9,16) (9,17) (9,19) (9,20) (9,22) (9,23) (9,24) (9,25)(9,28) (9,29) (9,30) (9,32) (9,36) (10,11) (10,14) (10,16) (10,17) (10,18)(10,20) (10,21) (10,22) (10,24) (10,26) (10,27) (10,29) (10,30) (10,31) (10,36)(11,12) (11,13) (11,14) (11,15) (11,17) (11,18) (11,20) (11,21) (11,22) (11,26)(11,27) (11,29) (11,30) (11,31) (11,33) (11,34) (11,35) (12,13) (12,14) (12,15)(12,16) (12,19) (12,20) (12,23) (12,24) (12,25) (12,28) (12,29) (12,30) (12,32)(12,33) (12,34) (12,35) (13,15) (13,17) (13,22) (13,25) (13,28) (13,29) (13,32)(14,15) (14,16) (14,24) (14,26) (14,27) (14,30) (14,31) (15,16) (15,17) (15,22)(15,24) (15,27) (15,28) (15,31) (15,32) (15,36) (16,17) (16,18) (16,19) (16,20)(16,23) (16,24) (16,26) (16,27) (16,28) (16,30) (16,31) (16,32) (16,34) (16,36)(17,18) (17,19) (17,20) (17,21) (17,22) (17,25) (17,27) (17,28) (17,29) (17,31)(17,32) (17,33) (17,36) (18,20) (18,21) (18,26) (18,27) (18,30) (18,34) (19,20)(19,23) (19,25) (19,28) (19,29) (19,33) (20,21) (20,23) (20,29) (20,30) (20,33)(20,34) (20,35) (21,22) (21,26) (21,27) (21,29) (21,30) (21,33) (21,34) (21,35)(22,25) (22,27) (22,28) (22,29) (22,31) (22,32) (22,36) (23,24) (23,25) (23,28)(23,29) (23,30) (23,33) (23,34) (23,35) (24,26) (24,27) (24,28) (24,30) (24,31)(24,32) (24,36) (25,28) (25,29) (26,27) (26,30) (27,28) (27,31) (27,36) (28,32)(28,36) (29,30) (29,33) (29,35) (30,34) (30,35) (31,36) (32,36) (33,35) (34,35)


20.6. The non-homogeneous tubes in � � .= 4 =6 6= 4 =

��99

= 4 =6 6= 4 =��9

9

= 4 =6 6= 4 =��9

9

= 4 =6 6= 4 == 4 =6 6= 4 =

��99

= 4 =6 6= 4 =��9

9

= = =4 4= = =��9

9

BB�� 4 �4 4= = =��9

9

BB�� = = =4 4� 4 ��9

9

BB��

= = �4 == = �

BB�� = == 4� = =BB�� = = �4 == = �

= � == =� = �

BB�� = �= == � =BB�� = �= =� = �

BB�� = � == =� = �

� �= 4 =6 6= 4 =� ��2

22

� == 6 48 84 6 == ��2

22

� �= 4 =6 6= 4 =� ��2

22

� �= 4 =6 6= 4 =� ��2

22

= �4 6 =8 8= 6 4� =��2

22

� �= 4 =6 6= 4 =� ��2

22

� �= 4 =6 4= 4 =� ��2

22

FF�� == 4 =6 6= 4 =� ��2

22

FF�� = 4 =6 6= 4 == ��2

22

FF�� = 4 =4 6= 4 =� ��2

22

FF�� = �= 4 =6 6= 4 =� ��2

22

FF�� = 4 =6 6= 4 =� =��2

22

FF�� = 4 =6 4= 4 =� �� == 4 =6 4= 4 =� �

��222

FF�� = = �4 4� = =� ��2

22

FF�� = 4 =4 6= 4 == ��2

22

FF�� = �= 4 =4 6= 4 =� ��2

22

FF�� = =4 4= = �� 2

22

FF�� = 4 =6 4= 4 =� =��2

22

FF��

� == 4 =6 4= 4 =� =

FF��

��222

� �= = �4 =� = =� �

FF��

��222

� �= = �= 4� = =� �

FF��

��222

= �= 4 =4 6= 4 == �

FF��

��222

� �� = == 4= = ��

FF��

��222

� �� = =4 == = ��

FF��

��222

� == 4 =6 4= 4 =� =� �= = �4 =� = =� =

��222

FF�� = = �= =� = =� ��2

22

FF�� = �= = �= 4� = =� ��2

22

FF�� = == 4= = �= ��2

22

FF�� = == == = �� 2

22

FF�� =� = =4 == = �� 2

22

FF��

� �� = ��

FF�� = = �= =� = =� =

FF��

��222

= �= = �= =� = =� �

FF�� =� � ��

FF�� = == == = �= �

FF��

��222

� =� = == == = ��

FF�� = �� = �= = �= =� = =� =

FF�� =� = == == = �= �

FF��


20.7. The non-homogeneous tubes in � � .

=4 44 6 44 4=��9

9

�= == 4 == =��9

9

�= == 4 == =��9

9

=4 44 6 44 4== == 4 == =��9

9

= == 4 == =��9

9

�= == 4 == ==��9

9

BB�� = == = == =��9

9

BB�� == == 4 == =��9

9

BB��

= �= = �= �

BB�� =� = =� =BB�� = �= = �= �

�� = ��

BB�� = == = == ==

BB��

��99

== == = == =�

BB�� = �� == == = == ==

BB��

� �= == 4 == =� ��2

22

� == 4= 6 =4 == ��2

22

� �= == 4 == =� ��2

22

� �= == 4 == =� ��2

22

= �4 == 6 == 4� =��2

22

� �= == 4 == =� ��2

22

� �= == 4 �= =� ��2

22

FF�� == == 4 == =� ��2

22

FF�� = == 4 == == ��2

22

FF�� = =� 4 == =� ��2

22

FF�� = �= == 4 == =� ��2

22

FF�� = == 4 == =� =��2

22

FF�� = == 4 �= =� �� == == 4 �= =� �

��222

FF�� = �= = =� =� ��2

22

FF�� = =� 4 == == ��2

22

FF�� = �= =� 4 == =� ��2

22

FF�� == = == �� 2

22

FF�� = == 4 �= =� =��2

22

FF��

� == == 4 �= =� =��2

22

FF�� = �= = �� =� ��2

22

FF�� = �� = =� =� ��2

22

FF�� = �= =� 4 == == ��2

22

FF�� =� = == �� 2

22

FF�� == = �= �� 2

22

FF�� == == 4 �= =� =� �= �= = �� =� =

��222

FF�� = �� = �� =� ��2

22

FF�� = �= �� = =� =� ��2

22

FF�� =� = == �= ��2

22

FF�� =� = �= �� 2

22

FF�� =� == = �= �� 2

22

FF��

� �� = � ��

FF�� = �� = �� =� =

FF��

��222

= �= �� = �� =� �

FF�� =� ��

FF�� =� = �= �= �

FF��

��222

� =� =� = �= ��

FF�� = � �� = �= �� = �� =� =

FF�� =� =� = �= �= �

FF��


20.8. The sets � � � � 2 � ��)� .The sets � � � � 2 � ��E� , � � ��@Z� [ � 4

��

��

��

��

��

The sets � � � � 2 � � CE� , � � ��@Z� [ � 6�� !�"��

��

�� #��"�� $��

��

�� #��%�� $��

�� &��&�&��

�� !�%��

��

�� #��#'�� (��&��&�&��

�� &��&�&��

�� !��#'�� (��

�� &��

��

�� )*��

�� +#'��

�� ,)*��

��


The sets � � � � 2 � ��E� , � � ��@Z� [ � �� )*��

��

�� )*��

��

�� # �� )*��

�� )*��

��

�� )*��

��

�� )*��

��

�� &��

�� )*�� &��

�� ) ��

�� &�&��

�� ) �� &��

�� &�&�� ,) ��

�� &��

�� %�� $��&�� &�&��

�� &�� &��

�� &�&�� &��

�� %��

�� &��

��

�� &��

�� %�� $��

��

��

�� #'�� &��

�� &��&�&�� &�&��

�� &�� &��&�&��

�� &�&��

�� #��%��

��

��

�� #'�� $��

�� &�� &��

�� &��

�� &��

�� &��&�&�� &��& ��

�� &�� &��&�&��

�� &��&��

�� &��

�� #'��

��

��

�� $��

�� &��&�&�� &��

�� &��&�&��

�� &��

�� &��&�&�� &��

��&�� &��&�&��

�� &��&�&�� &��& ��

�� &��&�&��

�� !�%��

��

��

��


�� #��#'��

��

��

�� &��

��

�� &��

��

�� &��&�� &��&�&��

�� &�� &��&��

�� &��&�&�� &��

�� '�� & ��

�� &��&��

�� &��&�&��

��&�� &��& ��

�� &��&�&��

�� '�� &��&�&�� &��&�&��

�� &��&�� &��&�&��

�� &��&�&�� &��&��

�� &��&��

��

�� &��&��

��

�� #�� (��

��

��

�� '�� &��&��

�� &��&��

�� &��&�� &��&��

��

�� !��#'�� (��

��

��

�� #��

��

��

��

�� #�� '�� &��

�� &��

��

�� !� � ��

��

��

��

�� !��

��

��

�� !� �'��

��

��

��

��

�� ) ��

�� )*��

�� +#'�� )*��

��

�� )*��

��


�� )*��

��

�� ,) ��

�� (��

�� ,)*��

�� ) ��

��

�� '�� ,)*�

��

�� ) ��

��

REFERENCES

[1] M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras. Corrected reprint of the 1995 original.Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge (1997), xiv+425pp.

[2] R. Bautista, On algebras of strongly unbounded representation type. Comment. Math. Helv. 60 (1985), no. 3,392–399.

[3] A. Berenstein, S. Fomin, A. Zelevinsky, Cluster algebras III: upper bounds and double Bruhat cells. Duke Math.J. 126 (2005), no. 1, 1–52.

[4] A. Berenstein, A. Zelevinsky, String bases for quantum groups of type� � . I.M. Gelfand Seminar, 51–89, Adv.

Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI (1993).[5] G. Bobinski, A. Skowronski, Geometry of modules over tame quasi-tilted algebras. Colloq. Math. 79 (1999), no.

1, 85–118.[6] K. Bongartz, Algebras and quadratic forms. J. London Math. Soc. 28 no. 3 (1983), 461–469.[7] K. Bongartz, A geometric version of the Morita equivalence. J. Algebra 139 (1991), no. 1, 159–171.[8] P. Caldero, Adapted algebras for the Berenstein Zelevinsky conjecture. Transform. Groups 8 (2003), no. 1, 37–

50.[9] P. Caldero, A multiplicative property of quantum flag minors. Representation Theory 7 (2003), 164–176.

[10] P. Caldero, R. Marsh, A multiplicative property of quantum flag minors II. J. London Math. Soc. 69 no. 3 (2004),608-622.

[11] F. Chapoton, S. Fomin, A. Zelevinsky, Polytopal realizations of generalized associahedra. Canad. Math. Bull. 45(2002), 537–566.

[12] W. Crawley-Boevey, On the exceptional fibres of Kleinian singularities. Amer. J. Math. 122 (2000), 1027–1037.[13] W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001),

257–293.[14] W. Crawley-Boevey, J. Schroer, Irreducible components of varieties of modules. J. Reine Angew. Math. 553

(2002), 201–220.[15] V. Dlab, C.M. Ringel, The module theoretical approach to quasi-hereditary algebras. In: Representations of

algebras and related topics (Kyoto, 1990), 200–224, Cambridge Univ. Press, Cambridge (1992).[16] P. Dowbor, A. Skowronski, On Galois coverings of tame algebras. Arch. Math. 44 (1985), 522–529.[17] P. Dowbor, A. Skowronski, Galois coverings of representation-infinite algebras. Comment. Math. Helv. 62 (1987),

311–337.[18] S. Fomin, A. Zelevinsky, Cluster algebras I: Foundations. J. Amer. Math. Soc. 15 (2002), 497–529.[19] S. Fomin, A. Zelevinsky, � -systems and generalized associahedra. Annals of Math. 158 (2003), no. 3, 977–1018.[20] S. Fomin, A. Zelevinsky, Cluster algebras II: Finite type classification. Invent. Math. 154 (2003), 63–121.[21] P. Gabriel, Auslander-Reiten sequences and representation-finite algebras. In: Representation Theory I (Carleton,

1979), 1-71. Lecture Notes in Math. 831, Springer-Verlag, Berlin (1980).[22] P. Gabriel, The universal cover of a representation-finite algebra. In: Representations of algebras (Puebla, 1980),

68–105, Lecture Notes in Math. 903, Springer-Verlag, Berlin (1981).[23] W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional

algebras. In: Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 265–297, LectureNotes in Math. 1273, Springer-Verlag, Berlin (1987).

[24] C. Geiß, J Schroer, Varieties of modules over tubular algebras. Colloq. Math. 95 (2003), 163–183.[25] C. Geiss, J. Schroer, Extension-orthogonal components of nilpotent varieties. Trans. Amer. Math. Soc. (to appear)[26] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras. London Mathe-

matical Society Lecture Note Series 119. Cambridge University Press, Cambridge (1988), x+208pp.[27] D. Happel, C.M. Ringel, The derived category of a tubular algebra. In: Representation theory, I (Ottawa, Ont.,

1984), 156–180, Lecture Notes in Math. 1177, Springer-Verlag, Berlin (1986).


[28] D. Happel, D. Vossieck, Minimal algebras of infinite representation type with preprojective component.Manuscripta math. 42 (1983), 221–243.

[29] M. Kashiwara, On crystal bases of the � -analogue of universal enveloping algebras. Duke Math. J. 63 (1991),465–516.

[30] M. Kashiwara, Y. Saito, Geometric construction of crystal bases. Duke Math. J. 89 (1997), 9–36.[31] B. Leclerc, Imaginary vectors in the dual canonical basis of �� . Transformation Groups 8 (2003), 95–104.[32] B. Leclerc, Dual canonical bases, quantum shuffles and � -characters. Math. Z. 246 (2004), 691–732.[33] B. Leclerc, M. Nazarov, J.-Y. Thibon, Induced representations of affine Hecke algebras and canonical bases of

quantum groups. In ‘Studies in memory of Issai Schur’, Progress in Mathematics 210, Birkhauser (2003).[34] H. Lenzing, A K-theoretic study of canonical algebras. 433–454, CMS Conf. Proc. 18 (1996).[35] H. Lenzing, Coxeter transformations associated with finite dimensional algebras. In: Computational methods for

representations of groups and algebras (Essen, 1997), Progress in Math. 173, Birkhauser 1999.[36] H. Lenzing, H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular

algebra. In: Representations of algebras (Ottawa, ON, 1992) 313–337, CMS Conf. Proc. 14, Providence RI(1993).

[37] G. Lusztig, Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. 3 (1990), 447–498.[38] G. Lusztig, Quivers, perverse sheaves and quantized enveloping algebras. J. Amer. Math. Soc. 4 (1991), 365–421.[39] G. Lusztig, Affine quivers and canonical bases. Publ. math. I.H.E.S. 76 (1994), 365–416.[40] G. Lusztig, Constructible functions on the Steinberg variety. Adv. Math. 130 (1997), 365–421.[41] G. Lusztig, Semicanonical bases arising from enveloping algebras. Adv. Math. 151 (2000), 129–139.[42] R. Marsh, M. Reineke, Personal communication.[43] M. Reineke, Multiplicative properties of dual canonical bases of quantum groups. J. Algebra 211 (1999), 134–

149.[44] I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS 44 (1997), 546–556.[45] C. M. Ringel, Tame algebras and integral quadratic forms. Lecture Notes in Math. 1099, Springer-Verlag (1984),

xiii+376pp.[46] C. M. Ringel, The preprojective algebra of a quiver. In: Algebras and modules II (Geiranger, 1966), 467–480,

CMS Conf. Proc. 24, AMS 1998.[47] C. M. Ringel, The multisegment duality and the preprojective algebras of type

�. Algebra Montpellier Announce-

ments 1.1 (1999) (6 pages).[48] C. M. Ringel, Representation theory of finite-dimensional algebras, In: Representations of algebras, Proceedings

of the Durham symposium 1985, 9–79, LMS Lecture Note Series 116, LMS 1986.[49] J. Schroer, Module theoretic interpretation of quantum minors. In preparation.[50] K. Saito, Extended affine root systems I. Publ. Res. Inst. Math. Sci. 21 (1985), 75–179.[51] A. Zelevinsky, Personal communication.[52] A. Zelevinsky, The multisegment duality. Documenta Mathematica, Extra Volume ICM Berlin 1998 III, 409–417.

CHRISTOF GEISS

INSTITUTO DE MATEMATICAS, UNAMCIUDAD UNIVERSITARIA

04510 MEXICO D.F.MEXICO

E-mail address: [email protected]

BERNARD LECLERC

LABORATOIRE LMNOUNIVERSITE DE CAEN

F-14032 CAEN CEDEX

FRANCE


JAN SCHROER

DEPARTMENT OF PURE MATHEMATICS

UNIVERSITY OF LEEDS

LEEDS LS2 9JTENGLAND


semicanonical bases and preprojective algebras · 2006-12-11 · semicanonical bases and...

Documents