Dimension Vectors of Indecomposable Objects for Nilpotent Operators of Degree 6 with One Invariant Subspace

Formulas for the dimension vectors of all objects M in the category S(6~)$\mathcal {S}(\tilde {6})$ of nilpotent operators with nilpotency degree bounded by 6, acting on finite dimensional vector spaces with invariant subspaces in a graded sense, are given (Theorem 2.3). For this purpose we realize a tubular algebra Λ, controlling the category S(6~)$\mathcal {S}(\tilde {6})$, as an endomorphism algebra of a suitable tilting bundle over a weighted projective line of type (2,3,6) (Theorem 3.6). Using this description and a concept of mono-epi type, the interval multiplicity vector of an object in S(6~)$\mathcal {S}(\tilde {6})$ is introduced and determined (Theorem 2.8). This is a much finer invariant than the usual dimension vector.


Introduction
In 1934 G. Birkhoff investigated the combinatorial structure of p-groups and their subgroups. More precisely he studied the category S(A) which objects are pairs (Z, Z ), where Z is a finitely generated module over Z p n and Z is a submodule of Z. For n ≤ 5 he was able to solve the problem completely, in contrast to the case n = 6, which is still waiting for a proper treatment.
The problem becomes even more interesting if we replace Z p n by other uniserial rings, in particular by the algebras A = k[x]/(x n ), where k is a field. Then the submodule category S(A), which will be further called S(n), is just the category of finite-dimensional vector spaces over k endowed with nilpotent operators of nilpotency degree bounded by n and invariant subspaces.
The categories S(n) were studied quite intensively recently. D. Simson determined the representation type of S(n), dependently on n [25]. In fact, he solved the problem of the representation type not only for submodule categories, but more generally, for the categories of chains of submodules of arbitrary lengths. His solution, to some extend, is similar to that in the original situation. More precisely, if n ≤ 5 (respectively, n ≥ 7) the submodule category S(n) is representation finite (respectively, wild) and for n = 6 it is tame (see also [22]). The further essential progress concerning the categories S(n) was done by C. M. Ringel and M. Schmidmeier in [23] (see also [21,22,24]). They described in detail the structure of S(n) in the most interesting case n = 6, showing that the Auslander-Reiten quiver of S (6) consists only of P 1 (k)-families of tubes.
A basic role in understanding the structure of the S(n)'s, in [23] is played by the categories S(ñ), which are the Z-graded variants of the S(n)'s. It occurs that the natural covering functors F n : S(ñ) → S(n), defined by forgetting the grading, have nice properties, in particular, they are dense provided n ≤ 6. An elegant description of the structure of S(6) is based just on this fact and refers to the shape of a fundamental domain D in S(6) (more precisely in its Auslander-Reiten quiver) with respect to the action of the group Z given by shift of the grading. D is a disjoint union of P 1 (k)-families of tubes each of them of tubular type (2,3,6). It is proved that the functor F 6 yields a bijection between the sets of isoclasses of all objects from D and of isoclasses of all indecomposable objects from S (6).
Using completely different methods in [10] it was shown that the stable category of S(ñ) is equivalent to the stable category of vector bundles over a weighted projective line of weight type (2, 3, n) in the sense of [7]. Here the notion of a stable category of vector bundles means that in the category of vector bundles we factor out the ideal of all morphisms which factor through finite direct sums of line bundles. The investigation of such categories and their connections to categories of nilpotent operators with chains of invariant submodules was continued in [11] and [12].
The main aim of this paper is to determine the dimension vectors of the indecomposable objects in S (6), with a particular focus on the exceptional ones. We consider these objects as representations over a certain algebra R given by a quiver with relations (for the precise definition of R see Section 2.1). In fact, R is the universal cover of the 2 × 2 triangular matrix algebra T 2 (k[x]/(x 6 )) and it admits a natural action of the group Z such that the induced action on the category mod(R) of finite dimensional right R-modules by restriction to S (6) coincides with the Z-action mentioned before. Notice that in contrast to objects belonging to homogeneous tubes (all of them clearly admit non-trivial selfextensions), the general problem of determining the dimension vectors was not solved in [23]. Recall that an object X in an abelian k-category is called exceptional if End(X) is a skew field (in case k is algebraically closed this means that End(X) = k, if End(X) is finite dimensional) and Ext n (X, X) = 0, for all n ≥ 1. By general facts the exceptional objects are uniquely determined by their dimension vectors, so their knowledge is useful. The description of these dimension vectors, given in Theorem 2.3, is expressed in terms of certain matrices which are basically obtained from the tubular mutations in the sense of [14,16,17]. These functors were introduced there in order to classify the indecomposable sheaves over a tubular weighted projective line.
A crucial point in the proof of our result is played by an explicit realization of a certain tubular algebra which was introduced in [23]. This algebra is a subcategory of R, if both, and R, are regarded as k-categories; hence, all modules over can be treated as modules over R. Moreover, has the property that all indecomposable objects of the fundamental domain D for S (6), with the exception of those in a certain rank-6 tube, under the canonical restriction belong to the category mod( ) of finite dimensional right -modules. We find a concrete tilting object T in the category coh(X) of coherent sheaves over a weighted projective line X of type (2,3,6) such that its endomorphism algebra End(T ) is isomorphic to . The tilting sheaf T is given as a direct sum of some vector bundles of rank smaller than or equal to 6 and the proof of the realization above is mainly K-theoretical. Then using the equivalence of the bounded derived categories D b coh(X) D b mod( ) we can adopt the techniques of tubular mutations and telescopic functors in a similar way as in the case of canonical algebras of tubular type [3,4,17,18]. We stress the fact that the proposed method of finding a tilting object in the category coh(X) is quite general and can be used for tilting realizations for other concealed-canonical algebras. We intend to study further tubular cases related to problems of chains of submodules in a forthcoming paper.
As a byproduct of our tilting theory approach we obtain that for all indecomposable objects in S (6), except of the members of some very special tubes, the linear structure maps are monomorphisms or epimorphisms. This is an important result which is used in further considerations.
In this paper we will also study the interval multiplicities for the indecomposablemodules and objects of S (6). Recall that an indecomposable module over a linear quiver is given by an interval of the vertex set where the chosen vector spaces for all points of this interval are k and the chosen linear maps for the arrows are identities. We use this for a convenient description of the finite dimensional modules over the algebra and the related algebra R. Note that R treated as a k-category contains two disjoint copies of a categoryÃ, whose ordinary quiver is the infinite equioriented linear quiver with vertex set Z, such that each object of R belongs to some of them. Applying these ideas we will in 2.7 define for a module M in mod(R) the interval multiplicity vector m(M), which is a much finer invariant than the dimension vector of M and it is useful in order to describe the possibly simplest matrix representation for M.
The second main result of this paper is the determination of the interval multiplicities for indecomposable objects in S(6) (Theorem 2.8). In order to do so we introduce the concept of abstract mono-epi types and representations belonging to them. We study these notions for uniserial path algebras and their quotients, and finally for the algebra . Moreover, we give also a description of so called strict types in terms of the sections in the Auslander-Reiten quivers of the considered algebras and recognize the interval decompositions in case of the two distinguished types arising by means of our tilting procedure.
The results above will be used in a forthcoming joint paper with M. Schmidmeier [5] in order to show that all exceptional objects in S (6) can be exhibited by matrices having as coefficients only 0 and 1, a result which is related to work of P. Gabriel on Dynkin quivers [6], C. M. Ringel on path algebras of finite quivers without oriented cycles [20] and also our work on canonical algebras [13,18] and [4].
The paper is organized as follows. In Section 2 we introduce the notions and notations, which are necessary to present our main results, Theorem 2.3 and Theorem 2.8. There also some facts following from these theorems are formulated (Corollaries 2.5 and 2.9). Section 3 is devoted to the proof of Theorem 3.6 on the realization of the tubular algebra as an endomorphism algebra of a concrete tilting sheaf over a weighted projective line X. In particular, we show that is isomorphic to the endomorphism algebra of a tilting bundle with the prescribed rank vector, if these two algebras have the same Cartan matrices (Proposition 3.9). In Section 4, the comparison result (Theorem 4.1) and its consequence (Proposition 4.6), which yields a description of the quasi-simple exceptional objects from D in terms of sheaves, are formulated. We also prove Theorem 2.3, applying the previous result, and perform a detail discussion of the mono-epi property for indecomposable -modules with suitable slopes (Propositions 4.10 and 4.12, Corollary 4.14). Section 5 contains a systematic survey of basic facts concerning the notion of mono-epi types for the case of uniserial path algebras and their factors (Lemma 5.3, Proposition 5.4), which ends up with the classification result (Theorem 5.7). They are adopted to the context ofÃ (Proposition 5.9) and next applied in the situation of some special types (Proposition 5.11). The considerations of this part of the paper are completed by the proof of Theorem 2.8. Section 6 is devoted to some examples, illustrating how to apply our two results in practice for selected values of the index.
We would like to thank very much Markus Schmidmeier for pointing out, during joint discussions on the "0,1-property" for objects of S (6) in May 2015 in Toruń, that some elements of our project he has developed independently earlier with Claus M. Ringel (e.g. some elements of Corollary 4.14), and for convincing us about the importance of the interval decompositions in our considerations. Unfortunately, up to our knowledge, the precise formulations of the results by Ringel and Schmidmeier (and their proofs) are not available in any form, which make them possible to quote.
In the paper we use standard definitions and notation, which are well known and commonly used. For example, we denote by N (respectively, by N m ) the set of all natural numbers with 0 (respectively, greater than or equal to m). We use the similar notation in case of the sets Z and Q of all integers and rational numbers, respectively. For any n ∈ N 1 (respectively, m, n ∈ N 1 such that m ≤ n) we set [n] := {1, . . . , n} (respectively, [m, n] := {m, . . . , n}). If A is a commutative ring then by A m we always mean the free A-module of rank m, consisting of the column vectors of the respective size. For a matrix P ∈ M m×n (A), by P j we denote the vector in A m being the j th column of P , where j ∈ [m]; moreover, we set P i,j : For basic information concerning modules and representation theory of algebras (respectively, derived categories of module categories) we refer to [1] (respectively [8]). The most important specialized notions, for the benefit of the reader will be briefly recalled in the next consecutive sections. All fields used in the paper for simplicity are assumed to be algebraically closed.
The authors would like to thank the referee for the comments and suggestions, which considerably contributed to improvements in the presentation of the results.

The main results
We start by fixing the notations which are necessary to formulate our results.

2.1
Let (Q,˜ ) be an infinite bounded quiver, whereQ is given below It is clear that the cate-goryS := S(6) in a natural way can be regarded as a full subcategory of the categorỹ H := mod(R) = rep k (Q,˜ ) of finite dimensional right modules over the locally bounded category R associated to (Q,˜ ) (equivalently, all finite dimensional representations of (Q,˜ )), consisting of all these M = (M v , M α ) v∈Q 0 ,α∈Q 1 for which M ν j is a monomorphism for every j ∈ Z. The category R is equipped with a natural action of the group Z such that the induced action (z, M) → z M onH (andS) is given by "shifting of M by z to the right", which means that ( z M) n = M n−z and ( z M) n = M (n−z) , for any n ∈ Z. (By some technical reasons, in the definition ofQ we use here another notation as in [23]; e.g. the indexing of vertices ofQ is such that s(α n ) = n + 1 and h(α n ) = n).
Recall that the description of the structure ofS from [23], in particular, of the indecomposable objects and the Auslander-Reiten components, is fully given (up to the Z-shift) by a fundamental domain D being a disjoint union of P 1 (k)families T γ of tubes inS, for γ ∈ Q 0 = {q ∈ Q : q ≥ 0}, each of tubular type (2,3,6). (Note that this fundamental domains differs slightly from the original one). All the families but T 0 consist of regular tubes, in T 0 only the rank-6 tube is not regular. The shifts m D, for various m ∈ Z, are pairwise disjoint and the Auslander-Reiten quiver of the categoryS is equal to the union m∈Z m D. If necessary, we treat further D also as a corresponding class of indecomposable objects inS which is closed under isomorphisms. We fix the following notation. For any γ ∈ Q 0 and i ∈ [3] such that (γ , i) = (0, 3), we denote by M i,0 (γ ), . . . , M i,p i −1 (γ ) fixed representatives of the consecutive (in the sense of Auslander-Reiten translation τ − ) isoclasses of quasi-simple objects from the mouth in the exceptional tube of rank p i in the family T γ , where p 1 = 2, p 2 = 3 and p 3 = 6. (In fact, the indexing of quasi-simples in the tubes is provided in some canonical coherent way, which is independent on γ , see 4.7). Moreover, for any l ∈ N 1 , we denote by M i,s,l (γ ) a fixed object from the tube of rank p i in T γ , determined uniquely up to isomorphism, which is of quasi-length l and has quasi-socle isomorphic to M i,s (γ ), where s ∈ Z p i (clearly, we can assume M i,s,1 (γ ) = M i,s (γ )). We give explicit formulas determining directly the dimension vectors dim k M i,s (γ ) (respectively, dim k M i,s,l (γ )).

2.2
LetṠ,Ṙ, U ∈ M 10 (Z) be the following triple of invertible (over Z) matrices given beloẇ In fact, these matrices correspond to some important operations on the Grothendieck group level, their actual meaning will be explained in Lemma 3.11 and Corollary 4.2. To any γ = γ ∞ γ 0 ∈ Q 0 , with γ ∞ ∈ N, γ 0 ∈ N 1 being coprime, we associate the matrix γ ∈ M 10 (Z), defined as follows: where c(q) = [c 1 ; c 2 , . . . , c m ] is the continued fraction presentation of the rational number q ∈ Q, for q = q(γ ) : The proof of the theorem needs longer preparations and will be given in Section 4.7.

2.4
To formulate the precise description of the dimension vectors for all indecomposable objects inS we denote by κ i,s,1 (γ ) the expressions on the right hand side of the formula ( * * ), for the respective triples (γ , i, s). Moreover, for any γ ∈ Q 0 we set additionally, for any s ∈ Z 6 and 2 ≤ r ≤ 5, with (γ , s) = (0, 3), we denote by κ 3,s,r (γ ) the expression in the rth row and sth column of the table below (by technical reasons it is splitted into two parts) The following formula can be easily obtained from Theorem 2.3.

Corollary 2.5
For any (γ, i, s) as above and l ∈ N 1 , the dimension vector of the indecomposable object M i,s,l (γ ) inS ⊆H is given by the expression where κ i,s,l (γ ) := quo p i (l) · ξ( 10 γ ) + κ i,s,rem p i (l) (γ ) . γ ), for any indecomposable object M of quasilength l from homogeneous tubes in the family T γ .
(In the formulation above quo p i (l) and rem p i (l) denote the integer quotient and the remainder of l modulo p i , respectively). Remark 2.6 (a) The dimension vectors of the indecomposables from the tube encoded by the pair (γ , i) = (0, 3) are known (see [23, (2.3)]. In fact, they can be easily recovered from those for the mouth objects, which are exhibited in [23, (1.3)]. Due to non-stability of the tube the formulas have a slightly irregular shape, so we do not present them here.
(b) There is also an alternative simple description of the dimension vectors of the indecomposables from homogeneous tubes given in terms of the vectors h γ (cf. Section 4.4).

2.7
As a byproduct of the results above we are able to determine the interval multiplicity of indecomposable objects fromS.

Corollary 2.9
For any pair γ ∈ Q 0 and l ∈ N 1 , all the objects M i,s,lp i (γ ), where i ∈ [3] and s ∈ Z p i , provided γ ∈ Q 0 \ {0, 1}, i ∈ [2] and s ∈ Z p i , provided γ = 1, and respectively, i = 1 and s ∈ Z 2 , provided γ = 0, have the same interval multiplicity vectors. Moreover, these vectors are equal to the vector m(M), for any indecomposable object M of quasi-length l from a homogeneous tube in the family T γ . Remark 2.10 The interval multiplicity vectors for the indecomposables from the exceptional regular tubes, encoded by the pairs (γ , i) = (1, 3), (0, 2), do not have such a homogeneous and nice shape. Nevertheless, they can be reconstructed from those for quasisimple objects in these tubes, with some extra effort (the details will be given in [5]). The quasi-lenght 1 case, however not "completely regular", is easy to handle by the shape of the dimension vectors (see Section 6, the lists L 1 and L ∞ ). The tube encoded by (0, 3) needs a separate treatment, which uses the description of its members given in [23].

3.1
A fundamental role in the understanding of the structure and the description of indecomposable objects of the categoryS = S (6) in [23] is played by the algebra := kQ/ of the bounded quiver (Q, ), where Q is the quiver and is the two-sided admissible ideal α 6 . . . α 1 , β 1 γ 1 − γ 2 α 3 , β 2 γ 2 − γ 3 α 4 in the path algebra kQ in Q (we follow the definition of kQ from [1, II.1], in [23] is denoted by ). We will always identify the category mod( ) of right finite dimensional -modules with the category rep k (Q, ) of finite dimensional representations of (Q, ). The following fact [23,1] shows the role of in studying of the categoryS.

3.3
Recall that is a tubular algebra, in the sense of [19], of type (2, 3, 6) (see [23]). The structure of the Auslander-Reiten quiver of mod( ) looks as follows: • P is a preprojective component which coincides with the preprojective component of mod( 0 ), • T = (T (γ ) ) γ ∈ Q 0 consists of separating P 1 (k)-families of tubes of type (2, 3, 6), all but T (0) and T (∞) consisting only of stable ones (i.e. not containing a projective or injective module), • Q is a preinjective component which coincides with the preinjective component of mod( ∞ ).
The tubular families in play a basic role in the description of D. More precisely, for every γ ∈ Q + := Q 0 \ { 0}, the P 1 (k)-family T γ inS is just equal to the imageT (γ ) of the family T (γ ) via the functor(−). Similarly, the same holds for T 0 and T (0) , which consist of all regular tubes in T 0 and T (0) , respectively. (In both cases we delete the unique non-regular tube, i.e. this of rank 6). Notice also that the rank-6 tubes in T 0 and T (0) (and in T (∞) ) are also strongly related (see [23,1] for the details).

3.4
In the proof of our main result we need not only the fact that is a tubular algebra of type (2, 3, 6), but also a realization of as the endomorphism algebra of a concrete tilting bundle T = 10 i=1 T i in coh(X), where X is a weighted projective line of tubular weighed type (2, 3, 6) (see Theorem 3.6, cf. [15,Proposition 3.6]). This allows us to treat -modules (in consequence some objects ofS) as objects of D b coh(X), and to apply techniques developed for studying coherent sheaves to our problem.
In the construction of this realization we use rather detailed information on the structure of the category of graded sheaves over weighted projectives lines, introduced by Geigle and Lenzing [7] for a better understanding of the module theory over canonical algebras of Ringel [19]. Their idea is based on associating to a canonical algebra C = C(p, λ) defined by the sequences p = (p 1 , . . . , p t ) ∈ N t and λ = (λ 1 , . . . , λ t ) ∈ (P 1 (k)) t consisting of pairwise different points λ i , the so-called weighted projective line X = X(p, λ). X is defined as the graded projective spectrum X := Proj L (S), where S is the commutative k-algebra admitting a natural grading S = x∈L S x by the rank-one abelian ordered group L = L(p) on generators x 1 , x 2 ,. . . , x t with relations p 1 x 1 = p 2 x 2 = · · · = p t x t =: c, such that the degree of each X i is just x i . Roughly speaking X(p, λ) consists of exceptional points x 1 , . . . , x t , corresponding to λ 1 , . . . , λ t , of multiplicities p(x 1 ) = p 1 , . . . , p(x t ) = p t , and the remaining points corresponding to x = x which are ordinary and have multiplicity p(x) = 1. (One can identify points x ∈ X with the corresponding λ ∈ P 1 (k), as above). Then one considers the abelian hereditary category coh(X) consisting of all L-graded coherent sheaves over X. The sheaf Consequently, T O allows to define in a standard way a triangle equivalence D b coh(X) D b mod(C) (see [7] for all details).
The category coh(X) admits Serre duality [7]. As a consequence coh(X) has almost split sequences, the Auslander-Reiten translation τ X in coh(X) is given by shift with the dualizing Similarly as in the case of a smooth projective curve, for coherent sheaves F over X we have concepts of rank and degree, defined as integers rk(F ) ∈ N and deg(F ) ∈ Z (see [7, 1.8 and 2.8]). The rank and degree functions induce Zlinear forms rk, deg : K 0 (X) → Z, where K 0 (X) is the Grothendieck group of the category coh(X), which is a free abelian group with a standard natural Z-basis Section 3.5 for their matrices in case (2, 3, 6)). Moreover, it is equipped with the Z-bilinear form eulf : K 0 (X) 2 → Z, called the Euler form, given by the formula Recall from [7] that each indecomposable element in coh(X) is a locally free sheaf, called a vector bundle, or a sheaf of finite length. Denote by vect(X) (respectively coh 0 (X)) the category of vector bundles (respectively finite length sheaves) on X and, for any q ∈ Q = Q∪{∞}, by C q the full subcategory of coh(X) formed by all sheaves whose indecomposable summands F satisfy the equality μ(F ) = q. Then coh(X) = vect(X)∨coh 0 (X), coh 0 (X) = C ∞ and vect(X) = q∈Q C q . It is also known that and that for each x ∈ X the category of L-graded finite length modules mod L 0 O X,x over the stalk O X,x is a uniserial category having p(x) simples. Moreover, all C q , for q ∈ Q, are abelian categories, additionally uniserial (with a length function = q called quasilength), enjoying the same tubular structure of the Auslander-Reiten quiver as coh 0 (X), if p is of tubular type (e.g. for p = (2, 3, 6)). Recall also that then, due to semistability arguments, for any indecomposable F, G in coh(X) we have always Moreover, we have the formulas where μ(F ) = d r is an irreducible fraction presentation of the slope of F and ρ(F ) denotes a τ X -order of F. (In particular, we always have ρ(F ) = 6 and (F ) = 1, if rk(F ) = 1 or deg(F ) = 1, for p = (2, 3, 6), see e.g. [14]).

3.5
From now on, if not restated, we assume that p = (2, 3, 6). Then the Auslander-Reiten quiver of each category C q consists of P 1 (k)-families T q (X) = {T q λ (X)} λ∈X of pairwise orthogonal standard stable tubes, which are almost all homogeneous except of three of them , being of the ranks 2, 3 and 6, respectively. We will consider B O as the ordered basis of the shape For any i = 1, 2, 3 and j ∈ Z p i we set All S i,j 's are simple torsion sheaves. For each i = 1, 2, 3, the set {S i,j } j ∈Z p i forms a full list of consecutive with respect to τ −1 X quasi-simple objects in the exceptional tubes in C ∞ . The families {S i,j } yield another ordered basis (note that the definition does not depend on the choice of i).

Theorem 3.6
The algebra admits a realization ∼ = End X (T ), where T = 10 i=1 T i is the unique, up to isomorphism, tilting sheaf over a weighted projective line X of the tubular type (2,3,6), having the following properties: (a) the sequences rk(T ) = (rk(T i )) i∈ [10] , deg(T ) = (deg(T i )) i∈ [10] and μ(T ) = (μ(T i )) i∈ [10] of ranks, degrees and slopes of indecomposable direct summands , are equal to the consecutive columns of the matrix Remark 3.7 The shape of the sequences rk(T ) and deg(T ) (consequently of μ(T )), for the tilting sheaf T realizing as E := End X (T ), under some natural "normalizing conditions" for T is canonically determined. One can show that if T is a tilting bundle, then the equality [rk(T l )] l∈ [10] = h 1 (:= [1, 3, 5, 6, 5, 3, 1; 4, 3, 1]) always holds, once we assume that coh 0 (X) =< T ∞ (X)> corresponds via the triangle equivalence :  [10] = h ∞ . In fact, these two equalities follow from some formula which holds for tilting sheaves over all weighted projective lines of tubular type, and can be treated as a tool for finding rk(T ) and deg(T ) of potential "tilted realization" of a given tubular algebra in general situations (this will be discussed in a subsequent publication).
The proof of Theorem 3.6, we present below (see Section 3.12), has a K-theoretical character and uses the technique of the so-called telescoping functors [14]. We start by a rather general fact which says how to determine, in some nice situations, the Cartan matrix of the algebra End X ( i∈ [10] T i ) via the matrix of the respective values of the Euler form.
Proof To prove the assertion it suffices to construct a surjective algebra homomorphism ψ : kQ → E such that ψ( ) = 0. The equality C = C E implies in particular that dim k = dim k E; hence, the homomorphism ψ : kQ/ → E induced by ψ is then an isomorphism.
Denote by the set consisting of all 35 pairs (i, j ) ∈ [10] 2 such that i = j and c E i,j = c i,j = 0 (hence equal to 1). For any (i, j ) ∈ we fix a nonzero map θ j,i ∈ Hom X (T i , T j ) (clearly, Hom X (T i , T j ) = kθ j,i ). From the shape of C E it follows that we have the following configuration of morphisms in coh(X) ..,n . It occurs that the remaining maps θ j,i are equal, up to nonzero scalars, to compositions of these from the diagram above. Let be the following 11 element set: , (3,8), (4,9), (5, 10), (8,9), (9, 10)}.
We show that for each pair (i, j ) ∈ and each sequence i for each pair (i, j ) ∈ such a sequence always exists.

3.10
Before we pass to the proof Theorem 3.6 we briefly collect some necessary notions and facts. We notice first that by the very definition of B S , the matrix V (of the respective base change) with columns being the coordinate vectors of the consecutive members of B S with respect to B O , and the matrix G B S (= V tr · G B O · V ) of the Euler form with respect to B S , have respectively the following forms: Moreover, the matrices rk B S and rk B S of the rank and degree homomorphisms rk, deg : K 0 (X) → Z with respect to the basis B S (and the standard basis {1} of Z) are given as follows: In our considerations a crucial role is played by a pair of autoequivalences R and S of the derived category D b coh(X), defined as tubular mutations (see [16]). They are nicely controlled on the level of the Grothendieck group. We have the following commutative diagram: where π : D b coh(X) −→ K 0 (D b coh(X)) ∼ = K 0 (X) denotes the canonical passage to the Grothendieck class and , ς are the Z-linear automorphisms of K 0 (X) induced by R, S, respectively ( and ς preserve the Euler form eulf : K 0 (X) 2 → Z). The matrices of the maps , ς can be easily computed from the very definition of the functors (see [3,7,14,16] for the details). In fact, we have the following identification showing the actual meaning of the two first members of the triple (Ṙ,Ṡ, U ) of matrices, defined in Section 2.2.

Lemma 3.11
The matrices of the Z-linear transformations , ς : K 0 (X) → K 0 (X) in the basis B S coincide withṘ andṠ, respectively.

3.12
Restricted to coh(X), R in contrast to S, is no longer an autoequivalence. Nevertheless, some special restrictions of R and S yield isomorphisms for all 0 ≤ q ≤ ∞ of categories. As a result, for any q ∈ Q there exists a unique canonical sequence of powers of R and S such that their composition called telescoping functor and given by the formula where q = [c 1 ; . . . , c m ] is the continued fraction presentation of q ∈ Q, yields an isomorphism C ∞ ∼ = C q . (We set additionally ∞,∞ := Id). For any q ∈ Q, we denote by φ q,∞ the automorphism of K 0 (X) controlling q,∞ , which is defined by the formula ( * * ) with the R and S replaced by and ς , respectively. It is clear that we have and that the matrix of φ q,∞ in the basis B S is equal to the matrix˙ q,∞ ∈ M 10 (Z), wherė q,∞ is defined by the formula ( * * ) now with the functors R and S replaced by the matriceṡ R andṠ, respectively. (We set additionally φ ∞,∞ := id K 0 (X) and˙ ∞,∞ := I 10 ).
Next we prove that T = i∈ [10] T i is a tilting sheaf. Observe that the familyT of quasi simple sheaves satisfies the condition (b) from Lemma 3.8 (one has to consider only the pairs (i, j ) = (4, 8), (6,9), (7,10)). Next we have to determine the matrix GT : [10] consisting of the values of the Euler form on all the pairs (T i , T j ). A direct calculation via the formula GT = ( B S ) tr · G B S · B S yields the equality GT = C . Then a case by case verification of positivity of the respective coefficients of C shows that also the condition (a) from Lemma 3.8 holds forT . Consequently, T is a tilting sheaf and C E = C , where E = End X (T ).
To complete the proof note that ∼ = E, by Proposition 3.9. Notice that the uniqueness, up to isomorphism, of T follows from the fact that exceptional sheaves are determined uniquely by their Grothendieck classes [14,17]. In this way our proof is finished.

-Modules Versus Coherent Sheaves, Applications
From now on we assume that T is precisely as in the Theorem 3.6 and the isomorphism ∼ = End X (T ) is given by the induced homomorphism ψ, defined in the proof of Proposition 3.9. Moreover, all basic vectors θ j,i ∈ Hom X (T i , T j ), for (i, j ) ∈ , are chosen in a compatible way, i.e. θ j,i , for (i, j ) ∈ , correspond to images of the arrows from Q 1 via ψ and satisfy the relations defining , whereas the remaining θ j,i , for (i, j ) ∈ \ , are given as the respective compositions of the previous ones.
Under the setting above we have the following comparison result. [10] and B E := {[S i ]} i∈ [10] , where {S i } i∈ [10] is a full list of standard simple -modules, form bases of K 0 (X) = K 0 ( ). where rk B E and rk B E denote matrices of the forms rk, deg : K 0 (X) → Z with respect to the basis B E , respectively.

Theorem 4.1 (a) The full right derived functor
Proof Follows immediately from Theorem 3.6 and the shape of the respective matrices (see also the general result concerning the tilting procedure [7]).
As a straightforward consequence we obtain also an explanation of the actual role of the third matrix from the triple (Ṙ,Ṡ, U ), defined in Section 2.2.

4.3
Due to Theorem 4.1 we can identify -modules with the corresponding sheaves (or their copies shifted by the translation functor [1] in the derived category). Note that we have in our disposal the notion of rank and degree for -modules, hence also of the slope (see Section 4.1(b) for the explicit formulas). We explain below details of this identification. Now let coh + (X) (respectively coh − (X)) be the full subcategory of vect(X) formed by all vector bundles whose indecomposable summands F satisfy the condition Ext 1 X (T , F) = 0 (respectively Hom X (T , F) = 0). Further, we denote by mod + ( ) (respectively mod 0 ( ), mod − ( )) the full subcategories of mod( ) formed by all -modules whose indecomposable summands have positive rank (respectively zero rank, negative rank). Finally, coh ≥ (X) (respectively, mod ≥ ( )) denotes the additive closure of coh + (X) ∪ coh 0 (X) (respectively, mod + ( ) ∪ mod 0 ( )).
Then under the equivalence : D b coh(X) → D b mod( ) -coh + (X) corresponds to mod + ( ) by means of F → Hom X (T , F), -coh 0 (X) corresponds to mod 0 ( ) by means of F → Hom X (T , F), -coh − (X) [1] corresponds to mod − ( ) by means of F[1] → Ext 1 X (T , F). The structure of the Auslander-Reiten quiver of mod( ) and the shape of its components, for the algebra of tubular type can be now derived alternatively from the description of the Auslander-Reiten quiver of coh(X) by applying tilting theory [14,Theorem 5.7]. In particular, one can precisely reconstruct the one-parameter families of tubes forming T from those for coh(X) and present T in a form T = (T q ) q∈ Q , slightly different as originally. Namely, consists of the components described as follows: • for each q ∈ Q with q > 1 or q < 0 a family (T q λ ) λ∈X = (T q λ ( )) λ∈X of stable tubes, being the images of (T q λ (X)) λ∈X (in the second case in fact of (T q λ (X) [1]) λ∈X in coh(X) [1]), whose ranks equal the weights of λ; • a family (T 1 λ ) λ∈X = (T 1 λ ( )) λ∈X of tubes such that T 1 λ , for λ ∈ X \ {λ 3 }, are stable tubes, being the images of (T 1 λ (X)) λ∈X\{λ 3 } , whose ranks equal the weights of λ, and a tube T 1 λ 3 is obtained from the stable tube T 1 λ 3 (X) of rank 6 by deletion of 2 corays ending at the vertex τ X T 7 and τ X T 10 , respectively (T 1 λ 3 contains 2 projective modules P (7) and P (10)); • a family (T 0 λ ) λ∈X = (T 0 λ ( )) λ∈X of tubes such that T 0 λ , for λ ∈ X \ {λ 3 }, are stable tubes, being the images of (T 0 λ (X)) λ∈X\{λ 3 } (in fact of (T 0 λ (X) [1]) λ∈X ), whose ranks equal the weights of λ, and a tube T 0 λ 3 is obtained from the stable tube T 0 λ 3 (X) of rank 6 by deletion of the ray starting at the vertex T 1 , (in fact T 1 [1]; T 0 λ 3 contains the injective -module I (1)); • the preprojective and preinjective components P and Q are formed by the images of sheaves F from the tube families T q (X), for 0 ≤ μ(F ) ≤ 1, such that Ext 1 X (T , F) = 0 and Hom X (T , F) = 0, respectively.
Notice that the objects from the preprojective component P and the tubes (T q λ ) λ∈X with 1 ≤ q < ∞ (respectively, from the preinjective component Q and the tubes (T q λ ) λ∈X with q ≤ 0) form the category mod + ( ) (respectively, mod − ( )). Moreover, the objects from (T ∞ λ ) λ∈X form the subcategory mod 0 ( ).

4.4
Now we briefly compare the description of tube families in given above with the original one (see Section 3.3).
An indecomposable -module M lies in a tube T q λ , for λ ∈ X, of the Auslander-Reiten quiver of mod( ) if and only if the equality μ(M) = q holds (can be immediately verified once we know dim k M). On other hand for a fixed γ = γ ∞ γ 0 ∈ Q 0 the dimension vectors of all -modules from the tubes in the family T (γ ) have index γ (see [19] for the precise definition). Recall that for each homogeneous tube they form the set N 1 · h γ with h γ : 2, 3, 3, 2, 1, 0; 2, 1, 0], h ∞ = [0, 1, 2, 3, 3, 2, 1; 2, 2, 1] ∈ Z 10 are the standard generators of the radical spaces for the tame concealed algebras 0 and ∞ , respectively. (Note that gcd{(h γ ) i : i ∈ Q 0 } = 1, since we always assume that γ ∞ , γ 0 ∈ N are coprime). Therefore, the number q ∈ Q, for γ ∈ Q 0 , such that the full subcategories of mod( ) formed by indecomposables from T (γ ) and T q coincide, is determined by the equality q = μ(h γ ). Consequently, we infer that We have the following.

Lemma 4.5
The mapping γ → q(γ ) and its inverse q → γ (q), given for q = a b , with a ∈ Z and b ∈ N coprime, by the formula γ (q) = a−b a , yield a bijection Q , we denote by F q,i,j := τ −j X F q,i,0 , j ∈ Z p i , the "consecutive" quasi-simple sheaves in the exceptional tube of the rank p i in the family T q (X), where p 1 = 2, p 2 = 3, p 3 = 6 (in the moment the choice of F q,i,0 is not important, cf. Section 4.7).

4.7
From now on we assume that for any q ∈ Q and any pair (i, j ), with i ∈ [3] and j ∈ Z p i , the sheaves F q,i,j are precisely equal to the images of the simple torsion sheaves S i,j via the telescoping functor q,∞ , i.e. F q,i,j := q,∞ (S i,j ). (Clearly, F ∞,i,j = S i,j ). Moreover, for each pair (γ , i) ∈ Q 0 × [3], (γ , i) = (0, 3), the indexing of the quasi-simple objects M i,s (γ ), for s ∈ Z p i , in the mouth of the corresponding tube in the categoryS is such that σ γ,i is the identity permutation.

4.8
The comparison theorem yields also a possibility to determine the mono-epi property (and eventually the mono-epi type) for indecomposable -modules and R-modules being objects ofS. We start by formulating the basic definition.
Let (Q, ) be for a moment an arbitrary bounded quiver and A = kQ/ the associated k-algebra (respectively, k-category). As usually we identify right A-modules with representations of (Q, ). We denote by = (Q, ) the set of all oriented paths δ in Q of positive length, which do not belong to . We say that M = (M v , M γ ) v∈Q 0 ,γ ∈Q 1 ∈ mod(A) is a path mono-epi representation (mono-epi epresentation, in short), if for any δ =

Proposition 4.10 Let F be an indecomposable sheaf in coh
• The map M (i,j ) is a monomorphism for every (i, j ) ∈ e and an epimorphism for every (i, j ) ∈ <1 m , additionally it is a monomorphism for every (i, j ) ∈ ∞ m and an epimorphism for every 1 then M (i,j ) is a monomorphism for every (i, j ) ∈ e ∪ ∞ m and an epimorphism for every (i, j ) ∈ <1 m ; moreover, for each pair (i, j ) ∈ 1 m , the map M (i,j ) is an isomorphism for all F lying in any tube from the family T 1 except of the one distinguished exceptional tube T 1 λ(i,j ) , λ(i, j ) ∈ {λ 1 , λ 2 , λ 3 }, and it is a monomorphism (respectively, an epimorphism) for all but precisely one quasi-simple objects F in T 1 λ(i,j ) , which one we will denote by F (respectively, F ). • If μ(F ) = ∞ then M (i,j ) is a monomorphism for every (i, j ) ∈ e and an epimorphism for every (i, j ) ∈ <1 m ∪ 1 m ; moreover, for each pair (i, j ) ∈ ∞ m , the map j ) is an isomorphism for all F lying in any tube from the family T ∞ except of the one distinguished exceptional tube T ∞ λ(i,j ) , λ(i, j ) ∈ {λ 1 , λ 2 , λ 3 }, and it is a monomorphism (respectively, an epimorphism) for all but precisely one quasi-simple objects F in T ∞ λ(i,j ) , which one we will denote by F (respectively, F ).
Proof Fix an indecomposable F in coh ≥ (X). Clearly, M (i,j ) is always a monomorphism, if rk(T i ) > rk(T j ). Now assume that rk(T i ) ≤ rk(T j ), equivalently, θ j,i is a monomorphisms.
We have a short exact sequence where Q i,j := Coker θ j,i . The cokernel Q i,j is an exceptional sheaf as a result of a mutation of the exceptional pair (T i , T j ) (see [17, 4.2]), so it is indecomposable. Hence, Q i,j lies in exceptional tube in coh(X) with slope (The slopes of all Q i,j , for (i, j ) ∈ m , can be easily computed by ( * * ) and Theorem 3.6(a), see Table 1 below). Moreover, each Q i,j belongs to coh ≥ (X), since the functor Ext 1 X (T , −) is right exact.
Consider the long exact sequence for the functor Hom X (−, F), induced by ( * ). Due to the equality Ext 1 X (T , F) = 0 its beginning looks as follows: j ) is a monomorphism if and only if Hom X (Q i,j , F) = 0. Notice that this is always the case if μ(F ) < μ(Q i,j ), or if μ(F ) = μ(Q i,j ) and F belongs to another tube than Q i,j . Moreover, in the tube of coh(X) containing Q i,j there exists precisely one quasi-simple object F = F (i, j ) with Hom X (Q i,j , F ) = 0; namely, the quasi-top of Q i,j . Note that it belongs to coh ≥ (X), since all epimorphic images of Q i,j have this property.
Similarly, M (i,j ) is an epimorphism if and only if Ext 1 X (Q i,j , F) = 0, or equivalently due to Serre duality if Hom X (F , τ X (Q i,j )) = 0. In particular, this is always the case if μ(F ) > μ(Q i,j ), or if μ(F ) = μ(Q i,j ) and F belongs to another tube than Q i,j . Moreover, in the tube of coh(X) containing Q i,j there exists precisely one quasi-simple object F = F (i, j ) with Hom X (F , τ X (Q i,j )) = 0; namely, the quasi-socle of τ X (Q i,j ). We claim that if μ(Q i,j ) ≥ 1 then F belongs to coh + (X). Notice that eventual problems can appear only for the pairs (i, j ) ∈ m with μ(Q i,j ) = 1, for those additionally ρ(Q i,j ) = 6. Nevertheless our claim holds true also in this case, since by the shape of the dimension vectors of the -modules S from the mouths of the tubes T 1 λ 3 and T 1 λ 2 (can be immediately determined by the formulas ( * ) and ( * * ) in Section 4.7) one easily indicates for each of the five pairs (i, j ) ∈ 1 m the (unique) module S = S (i, j ) such that the structure map

S (i,j )
: S j → S i in S is not an epimorphism (see L 1 from Section 6 and Table 1, cf. also Remark 4.11). Now, the assertions follow from the considerations above by a simple analysis of Table 1 containing the slopes of all the cokernels Q i,j , for (i, j ) ∈ m . (b) Using the list L 1 , one can also easily indicate for each of the five pairs (i, j ) ∈ 1 m (the unique) S from the union of the mouths of the tubes T 1 λ 3 and T 1 λ 2 , such that the structure map S (i,j ) : S j → S i in S is not a monomorphism. The same can be done for each of the four pairs (i, j ) ∈ ∞ m , in an epimorphism as well as a monomorphism version for the map S (i,j ) , where now S belongs to the mouth of the tube T ∞ λ 3 (use L ∞ from Section 6). (c) For any (i, j ) ∈ 1 m we have λ(i, j ) = λ 2 or λ(i, j ) = λ 3 (i.e. ρ(Q i,j ) = 3 or ρ(Q i,j ) = 6, respectively). The sequence 6,6,3,3,6 describes the ranks of the tubes T 1 λ(i,j ) (containing Q i,j ) for the consecutive five pairs (i, j ) ∈ 1 m , ordered as in the definition of the set 1 m (apply (b)). For any (i, j ) ∈ ∞ m we have λ(i, j ) = λ 3 , since ρ(Q i,j ) = 6 (see (a)).

Proposition 4.12
Let F be an indecomposable sheaf in coh (X) such that μ(F ) ≤ 0.
• The map M (i,j ) is an epimorphism for every (i, j ) ∈ m and a monomorphisms for every (i, j ) ∈ >0 e , additionally it is a monomorphism for every (i, j ) ∈ 0 e , provided μ(F ) < 0.
• If μ(F ) = 0 then M (i,j ) is an epimorphism for every (i, j ) ∈ m and a monomorphisms for every (i, j ) ∈ >0 e ; moreover, for each pair (i, j ) ∈ 0 e , the map M (i,j ) is an isomorphism for all F lying in any tube from the family T 0 except of the one distinguished exceptional tube T 0 λ(i,j ) , λ(i, j ) ∈ {λ 1 , λ 2 , λ 3 }, and it is a monomorphism (respectively, an epimorphism) for all but precisely one quasi-simple objects F in T 0 λ(i,j ) , which one we will denote by F (respectively, F ).
Proof We use arguments similar to those from the proof of Proposition 4.10. Fix an indecomposable F in coh − (X). The category coh(X) is hereditary so the functor Ext 1 X (−, F) is right exact. Consequently, M (i,j ) is an epimorphism for every (i, j ) ∈ m , since in this case θ j,i : T i → T j is a monomorphism. Now assume that (i, j ) ∈ e , equivalently, that θ j,i is an epimorphism. Then we have a short exact sequence where K i,j := Ker θ j,i . The kernel K i,j is an exceptional sheaf as a result of a mutation of the exceptional pair (T i , T j ) (see [17, 4.2]). Hence, K i,j belongs to an exceptional tube in coh(X) with slope (The slopes of all K i,j , for (i, j ) ∈ e , can be easily computed by ( * * ) and Theorem 3.6(a), see Table 2 below). Consider the long exact sequence for the functor Hom X (−, F), induced by ( * ). Due to the equality Hom X (T , F) = 0 its nontrivial part looks as follows: is an epimorphism if and only if Ext 1 X (K i,j , F) = 0, or equivalently due to Serre duality if Hom X (F , τ X (K i,j )) = 0. Notice that this is always the case if μ(F ) > μ(K i,j ), or if μ(F ) = μ(K i,j ) and F belongs to another tube than K i,j . Moreover, in the tube in coh(X) containing K i,j there exists precisely one quasi-simple object F = F (i, j ) with Hom X (F , τ X (K i,j )) = 0; namely, the quasi-socle of τ X (K i,j ).
Similarly, M (i,j ) is a monomorphism if and only if Hom X (K i,j , F) = 0. This is always the case if μ(F ) < μ(K i,j ), or if μ(F ) = μ(K i,j ) and F belongs to another tube than K i,j . Moreover, in the tube in coh(X) containing K i,j there exists precisely one quasi-simple object F = F (i, j ) with Hom X (K i,j , F ) = 0; namely, the quasi-top of K i,j .
We claim that if μ(K i,j ) ≤ 0 (in fact, μ(K i,j ) = 0, see Table 2) then both, F and F , belong to coh − (X). Notice that eventual problems can appear only for the pairs (i, j ) ∈ e with μ(K i,j ) = 0, for those additionally ρ(K i,j ) = 6. Nevertheless, our claim is valid also in this case, since by the shape of the dimension vectors of the -modules S from the mouths of the tubes T 0 λ 3 and T 0 λ 2 (can be determined by the formulas ( * ) and ( * * ) in Section 4.7, for each of the seven pairs (i, j ) ∈ 0 e , one can easily indicate among the modules S the (unique) pair, S = S (i, j ) and S = S (i, j ), such that the structure map S (i,j ) : S j → S i in S is not an epimorphism, for S = S (i, j ), respectively is not a monomorphism, for S = S (i, j ) (see L 0 from Section 6 and Table 2, cf. also Remark 4.13). Now the assertions follow from the considerations above by a simple analysis of Table 2 containing the slopes of all the kernels K i,j , for (i, j ) ∈ e .  any (i, j ) ∈ 0 e we have λ(i, j ) = λ 2 or λ(i, j ) = λ 3 (i.e. ρ(K i,j ) = 3 or ρ(K i,j ) = 6, respectively). The sequence 3,3,3,3,6,6,6 describes the ranks of the tubes T 0 λ(i,j ) (containing K i,j ) for the consecutive seven pairs (i, j ) ∈ 0 e , ordered as in the definition of the set 0 e . (c) For (i, j ) ∈ 0 e , the kernel K i,j belongs to coh − (X) if and only if (i, j ) = (2, 7), since then the equality Hom X (T , K i,j ) = 0 is equivalent to Hom X (T 1 , K i,j ) = 0.  (3,8) , M (4,9) , M (5,10) being monomorphisms. In particular, (b) If μ(M) = ∞ and M is a quasi simple module from the rank-6 tube then M is a mono-epi representation with M (3,8) , M (4,9) , M (5,10) being monomorphisms (c) If μ(M) = 1 and M is a quasi-simple from one of the two biggest tubes then M is a mono-epi representation with M (3,8) , M (4,9) , M (5,10) being monomorphisms (d) If μ(M) = 0 and M is a quasi-simple from one of the two biggest tubes then M is a mono-epi representation, with M (4,9) , M (5,10) being always monomorphisms, and M (3,8) being a monomorphism except for precisely one distinguished quasi-simple module M = M in the biggest tube, for whom M (3,8) is an epimorphism.

Remark 4.15
Applying this method one can also examine the remaining indecomposable -modules M with respect to the property of being mono-epi representation (i.e. the preprojective and preinjective ones).

5.1
Now we introduce the notion of a so-called mono-epi type which allows to understand in a more systematic and deeper way the class of mono-epi representations. In particular, we show that in the case of the categoryÃ(6) the mono-epi types determine the interval decomposition of mono-epi representations which belong to these types. We apply a precise formula for their multiplicity vectors to prove Theorem 2.8. Let again, as in Section 4.8, (Q, ) be for a moment an arbitrary bounded quiver and A = kQ/ the associated k-algebra (respectively, k-category). We fix the two element set {e, m} of symbols. A map t : → {e, m}, where = (Q, ) is as in Section 4.8 will be called a sincere abstract mono-epi type for the algebra A (an abstract mono-epi type, in short), if it satisfies the following natural condition: for any pair δ, δ ∈ , with h(δ) = s(δ ) and t (δ) = t (δ ), we have δδ ∈ and t (δδ ) = t (δ) (= t (δ )). Additionally, we say that t as above is strict, if for any pair δ, δ ∈ , with h(δ) = s(δ ) and δδ ∈ , the equality t (δ) = e always implies t (δ ) = e (equivalently, t (δ ) = m implies t (δ) = m). If Q is acyclic and t (δ) = t (δ ) for each pair of parallel paths δ, δ ∈ (i.e. such that s(δ) = s(δ ) and h(δ) = h(δ )), then to any abstract mono-epi type t we associate the relation ≺ = ≺ t ⊆ (Q 0 ) 2 consisting of all the pairs (s(δ), h(δ)), for δ ∈ with t (δ) = m, and (h(δ), s(δ)), for δ ∈ with t (δ) = e. It occurs that in good situations the reflexive closure = t of the relation ≺ yields an ordering of the vertex set Q 0 .
We say that the (mono-epi) representation M in mod(A) (respectively, the vector d = Finally, an abstract mono-epi type t is called a (real) mono-epi type, if there exists a sincere M in mod(A), which belongs to t. Observe that if Q is a full subquiver of Q, a restriction of the ideal to kQ (in the sense of path categories) and A := kQ / then the restriction t := t | : (Q , ) → {e, m} of mono-epi type t for A is always a mono-epi type for A , called the restricted type.

5.2
For any n ∈ N 1 , we denote by Q (n) the linear quiver Proof For n = 2 all the assertions are trivially satisfied, so we assume that n ≥ 3.
(a) First we briefly show that = t yields an ordering of [n]. (Note that now = {δ j,i : i < j}). Antisymmetry of holds, since i ≺ j means that t (δ j,i ) = e, if i < j, and t (δ i,j ) = m, if i > j; hence we have j ⊀ i. To check transitivity of fix i, j, l ∈ [n] such that i ≺ j and j ≺ l. Then there are six possible different <-orderings of the set {i, j, l}. If i < j < l then t (δ l,j ) = t (δ j,i ) = e, so t (δ l,i ) = e. In the case i, l < j we have t (δ j,i ) = e and t (δ j,l ) = m, and additionally δ j,i = δ j,l δ l,i , so t (δ l,i ) = e, if l > i (respectively, δ j,l = δ j,i δ i,l , so t (δ i,l ) = m, if i > l); hence i ≺ l. The remaining three cases are dual to the above onces. Notice that is linear, since any two different vertices of Q are connected by an oriented path (lying out of the ideal (0)). To finish the proof of the first part of (a), we associate to any linear ordering of [n] the map t = t : → {e, m}, given by setting t (δ j,i ) = e, if i ≺ j , and t (δ j,i ) = m, if j ≺ i, for any i < j. It is easily seen that t is an abstract mono-epi type and that the mappings t → t and → t are mutually inverse. (Of course, each linear ordering = t , interpreted as a chain of elements of [n] increasing in the sense of ≺ , can be also uniquely encoded by the sequence (σ (1), σ (2), . . . , σ (n)), where σ = σ t is a permutation of [n] such that ( * ) : i ≺ j if and only if σ −1 (i) < σ −1 (j ); in particular, for any i < j we have: t (δ j,i ) = e ⇔ σ −1 (i) < σ −1 (j ), respectively, t (δ j,i ) = m ⇔ σ −1 (i) > σ −1 (j )). If now an abstract mono-epi type t is additionally strict then either t (δ n,n−1 ) = e (respectively, t (δ 2,1 ) = m) and then is given by the sequence (1, 2, . . . , n) (respectively, (n, n − 1, . . . , 1)), or otherwise p − 1, p + 1 ≺ p, for some p ∈ [n] \ {1, n}. Then applying induction we infer that 1 ≺ 2 ≺ . . . ≺ p and n ≺ n − 1 ≺ . . . ≺ p. Conversely, let be a linear ordering of [n], for which there exists p ∈ [n] with the required property. Set t := t . We can assume that 1 < p < n, since in the opposite case, for all i < j we have t (δ j,i ) = e, if p = n (respectively, t (δ j,i ) = m, if p = 1). Then for any i < j < l, the equality t (δ l,j ) = e implies j < p, and hence t (δ j,i ) = e. Consequently, the type t is always strict.
(The cases r even and r odd correspond to the presence of the inequalities p − 1 ≺ p + 1 and p + 1 ≺ p − 1 in ([n], t ), respectively). It is not hard to see that in this situation the list of intervals if r is even, and with if r is odd, contains all, up to isomorphism, indecomposable representations in mod(A) (intervals) which belong to t. We denote the set formed by all these intervals by S(t) = S A (t) and call it the interval support of t. Simple calculation shows that always |S(t)| = n.
be a dimension vector of type t. Then the following equalities hold: We set where σ = σ t and d σ ( . But this is impossible, since for any i < j < l we have δ l,i = δ l,j δ j,i ∈ and the inequalities d i , d l ≥ d j imply d i = d j or d l = d j (cf. Section 4.8). Consequently, such p ∈ [n] always exists. If p = n (respectively, p = 1) then for a strict abstract mono-epi type t such that M belongs to t we can take t , where is given by the sequence (1, . . . , n) (respectively, (n, . . . , 1)). Assume now that 1 < p < n. Consider two linear orderings 1 of [p] and 2 of [n] \ [p − 1], which are defined by the chains 1 ≺ 1 . . . ≺ 1 p and n ≺ 2 . . . ≺ 2 p, respectively. Then any integration of these two chains into the one, yields always a linear ordering on [n] such that the associated type t is strict. It is easily seen that with a slight care, applying the arguments similar to these involving the sequences i 1 < i 3 < . . . and i 2 > i 4 > . . . , the integration process can be provided in a way respecting the property: d i ≤ d j , if i ≺ j , where i < p and j > p, or i > p and j < p. Clearly, the representation M belongs to the type t = t , for the such ordering .
We say that the ordering of the set   we need only to verify the transitivity property for . (Notice that now = {δ j,i : i < j, (i, j ) = (1, n)}). Fixing i, j, l ∈ [n] such that i ≺ j and j ≺ l, observe that if i < j < l then δ l,j , δ j,i ∈ and t (δ l,j ) = t (δ j,i ) = e, so δ l,i = δ l,j δ j,i ∈ and t (δ l,i ) = e; hence i ≺ l. The case i > j > l is dual to that above. In the remaining four cases we always have {i, l} = {1, n}, so automatically δ l,i ∈ , if i < l (respectively, δ i,l ∈ , if l < i), and comparing to the proof of Lemma 5.3, we do not need any extra arguments. Note that by the shape of the ordering is almost linear. Let now be an arbitrary almost linear ordering of [n]. We define the map t = t : → {e, m} by the same formula as in the case of A (n) . It is well defined, since for any i < j such that (i, j ) = (1, n), the elements i and j are (c) If the mono-epi representation M of A is sincere then we have d 1 , . . . , d n = 0 and δ n,1 = δ n,i δ i,1 ∈ (n) , for any i ∈ [n] ; hence, d 1 + d n ≤ d i (see Section 4.8). On the other hand the restriction M := M | Q of M to Q = (Q ) (n) is a mono-epi representation of the algebra A , so by Lemma 5.3(c) the vector dim k M belongs to some strict mono-epi type t for A . Let now t be the unique strict abstract mono-epi type for A such that t | = t . It is clear that dim k M belong to t, since t = | [n] , where = t .
If now M is not sincere with d n = 0 and d 1 ≤ d n−1 then d 1 ≤ d i , for every i ∈ [n] , since the inequality d i < d 1 (≤ d n−1 ) is impossible due to the fact that δ n−1,1 = δ n−1,i δ i,1 ∈ (see Section 4.8). Now constructing t in the same way as above we again infer that dim k M belongs to t. Note that the converse implication is obvious due to (a). The case d 1 = 0 is analogous.
Finally, suppose that d 1 , d n = 0 and d i = 0, for some i ∈ [n] . Then for any 1 < j < i (respectively i < j < n) we have d j = 0 due to the arguments as above, a contradiction. In this way the proof is complete.
From now on the formulation strict mono-epi type (if there is no other concrete specification for A) will always mean that we deal with a strict abstract mono-epi type t for the algebra A, where A = A (n) or A = A (n) , for some suitable n ∈ N.
Remark 5.5 Let i, j ∈ [n] be a pair of integers such that 1 ≤ i < j ≤ n and 2 ≤ j − i < n − 1, Q a full subqiuver of Q = Q (n) spanned on the vertex set [i, j ] ⊆ Q 0 and t an arbitrary strict mono-epi type for the path algebra A := kQ . Then there exists a strict extension t of t to A := A (n) , i.e. a strict mono-epi type t for A, with the property that for any sincere A -representation of type t its extensionM to A by zeros belongs to t. (If 1 < i < j < n, the type t is not uniquely determined, since the integration of the chains 1 ≺ . . . ≺ i − 1 and n ≺ . . . j + 1 into one chain, which has to be clearly located in = t "below" the chain given by t , can be provided in an arbitrary way). Moreover, for the algebra A = A (n) the same holds always true, provided 1 < i < j < n. For i = 1 (respectively j = n), a strict extension t of t exists if and only if j = n − 1 (respectively i = 2) and additionally t is such that 1 ≺ t n − 1 (respectively n ≺ t 2).
Below we formulate some important observations which follow directly from the proofs Sections 5.3 and 5.4.

Corollary 5.6 (a) If t is a strict mono-epi type for A then:
• I [1,n] belongs to S A (t) and for any I [i,j ] , • I [2,n−1] , I [1,n−1] , I [2,n]   It is interesting that we can also characterize strict mono-epi types in terms of sections in the Auslander-Reiten quiver A , where by a section we mean any connected full subquiver of A , which has a one element intersection with each τ A -orbit in A (see [1]). Formulating our result we will identify intervals with their isomorphism classes.

Theorem 5.7 The mapping t → S A (t) yields a bijection between the sets of all strict mono-epi types for A and all sections
in A , which is determined by the equality 0 = S A (t). In particular, for A = A (n) the sections in A correspond bijectively via the mapping → ∩ A to the sections in A , where A = (A ) (n) and A → A is a canonical embedding; for any strict mono-epi type t for A we have S A (t) ∩ ( A ) 0 = S A (t) \ {I [1,n−1] , I [2,n] Proof Since further we do not use this fact, we only outline the arguments of the proof. Assume first that A = A (n) . We say that for a pair of intervals I [i,j ] , I [i ,j ] ∈ S A the condition ( * ) holds if : either i = i + 1 and j = j or i = i and j = j − 1. Observe that is a section in A if and only if 0 consists of intervals I [i 1 ,j 1 ] , . . . , I [i n ,j n ] such that (after suitable change of indexing) each pair of consecutive intervals satisfies ( * ); in particular, we have ( * * ) : i 1 = 1 ≤ i 2 ≤ . . . ≤ i n = j n ≤ j n−1 ≤ . . . ≤ j n = n, and dim k I [i l ,j l ] = n − l + 1, for every l. Now it is clear that S A (t), for strict t, forms a section, since |S A (t)| = n and for any two consecutive elements of the list in the proof of Lemma 5.3(b) the condition ( * ) is satisfied.
Let be an arbitrary section, with 0 = {I [i l ,j l ] : l ∈ [n]} as above. We associate to the abstract mono-epi type t = t ( ), for A, by defining the permutation σ of the elements of [n]. We set σ (l) := i l , if j l+1 = j l , and σ (l) := j l , if i l+1 = i l , for l < n; and σ (n) := i n = j n . Notice that t determined by σ is strict, with p = i n = j n , since by the construction we have i l i l and j l j l , if l < l , where = t (cf. ( * * )). Now it is easy to check that S A (t ( )) = 0 , for any section , and t ( ) = t, for any strict t, where is a section such that 0 = S A (t).
Finally note that the case A = A (n) follows easily from the previous one by the description of strict mono-epi types t in Proposition 5.4(a), the formula S(t) = {I [1,n−1] , I [2,n] } ∪ S A (t ) and the fact that I [1,n−1] , I [2,n] , I [2,n−1] ∈ 0 , for any section in A . The remaining assertions are evident.

5.10
Returning back to our basic setup we assume from now on that r = 6, henceÃ =Ã(6). In the proof of Theorem 2.8 an important role is played by two pairs of special local mono-epi types forÃ, whose shape is motivated by Corollary 4.14.

Format of Data
The way we display the dimension vectors over R in the tables is consistent with that introduced in Section 2.2, the first from the left coordinates in the rows correspond to the vertices 1 and 1 inQ, respectively. As a reminder of the action of ξ , we use the "hat notation" for easier visualization of the recovery procedure of the dimension vectors dim k X q,i,s for quasi-simple exceptional -modules X q,i,s with slope q from dim k M i,s (γ )'s (see Section 4.7 for the precise definition). It is just realized by deleting the "hat coordinates" but formally one should also forget the last two zeros in the upper row of dim k M i,s (γ ). To encode multiplicity vectors we introduce the shorten notation format which is explained by the following example: m(M) = |14 2 |24|23| |16 2 |26|25|34 3 | , for M = (M s , M a , ϕ M ) in mod(R), means exactly that M s ∼ = ⊕I [1,4] 2 ⊕ I [2,4] 1 ⊕ I [2,3] 1 and M a ∼ = I [1,6] 2 ⊕ I [2,6] 1 ⊕ I [2,5] 1 ⊕ I [3,4] 3 .
Computation Aspects For the indices γ ∈ Q + , γ = 1, the entries of the tables L q(γ ) , i.e. the dimension vectors and the interval multiplicity vectors for the M i,s (γ )'s, are computed directly by the formulas from Theorems 2.3 and 2.8, respectively. The same holds in the case γ = 1, with an exception of the lower row in the second table, containing the vectors m(M 3,s (1)). Since by Corollary 4.14 all the M 3,s (1)'s are mono-epi representations of R, so even if some of them belong neither to the type (t s + , t a + ) nor to (t s − , t a − ), their interval decompositions can be immediately recovered from very simple shapes of dim k M 3,s (1)'s. In the case γ = 0, for the first table the situation is similar to this above. Now an exception is formed by the vectors m(M 2,s (0)), which can be easily determined "by hand". The second table contains the invariants for the R-modules M 3,s (0) :=X 1,3,s , where s = 1, 2, 3, 5, from the tube T 0 inS, which were not formally defined in Section 2.1. The dimension vectors dim k M 3,s (0) are computed by use of the standard formula involving the matrix 0 (Theorem 2.3 formally does not cover this case, so we apply ( * ) and ( * * ) from Section 4.7), whereas the multiplicity vectors m(M 3,s (0)) similarly as for the previous exceptions. The "stars" in empty columns indicate the lack of data for the images of (F 1,3,0 ) and (F 1,3,4 ) by the functor (ˆ) (absence by obvious reasons). The case γ = ∞, added for some symmetry, concerns the data for the R-modules M i,s (∞) :=X 0,i,s , where i ∈ [3] and s ∈ Z p i , with (i, s) = (3, 5) (not considered yet). It does not yield to much new information on invariants for the exceptional quasi-simple objects from D (the two smaller tubes shifted by −1 coincide with those for γ = 0, M 3,1 (∞) does not even belong toS). Nevertheless, it completes the description of the corresponding data for exceptional quasisimple -modules in the lacking slope q = 0. The entries of the table are computed in a very similar way as those in the case q = 1. The analogy is almost full with only difference that the matrix ∞ is not defined in Section 2.2, so using the standard formula we have do this now (cf. Section 4.7), and also that in the second table appears only one empty column (result of obvious absence of the image of (F 0,3,5 ) by (ˆ)).

Final Remark
This rather natural selection of the rationals γ provided above is not completely accidental, it is motivated by some further applications. Namely, most of the information contained in the presented tables L q(γ ) play an important role in the proof of the so-called "0-1-property" for the class of all indecomposable objects inS lying in exceptional tubes (by the very definition this means that each such object admits an Rmodule matrix presentation involving only the coefficients 0 or 1). This result is a topic of a forthcoming publication [5].