UPPSALA DISSERTATIONS IN MATHEMATICS

56

On the Clebsch-Gordan problemfor quiver representations

Martin Herschend

Department of MathematicsUppsala University

UPPSALA 2008

Dissertation at Uppsala University to be publicly examined in Häggsalen, ÅngströmLaboratory, Thursday, May 22, 2008 at 13:15 for the degree of Doctor ofPhilosophy. The examination will be conducted in English.

AbstractHerschend, M. 2008. On the Clebsch-Gordan problem for quiver representations.Uppsala Dissertations in Mathematics 56. v+34 pp. Uppsala.ISBN 978-91-506-2002-3

On the category of representations of a given quiver we define a tensor product point-wise and arrow-wise. The corresponding Clebsch-Gordan problem of how the tensorproduct of indecomposable representations decomposes into a direct sum of indecom-posable representations is the topic of this thesis.

The choice of tensor product is motivated by an investigation of possible waysto modify the classical tensor product from group representation theory to the caseof quiver representations. It turns out that all of them yield tensor products whichessentially are the same as the point-wise tensor product.

We solve the Clebsch-Gordan problem for all Dynkin quivers of type A, D and E6,and provide explicit descriptions of their respective representation rings. Furthermore,we investigate how the tensor product interacts with Galois coverings. The results ob-tained are used to solve the Clebsch-Gordan problem for all extended Dynkin quiversof type An and the double loop quiver with relations βα = αβ = αn = βn = 0.

Keywords: quiver, quiver representation, tensor product, Clebsch-Gordan problem,representation ring, bialgebra, Galois covering.

Martin Herschend, Department of Mathematics, Uppsala University,Box 480, SE-751 06 Uppsala, Sweden

Copyright c© Martin Herschend, 2008

ISSN 1401-2049ISBN 978-91-506-2002-3urn:nbn:se:uu:diva-8663 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-8663)

List of Papers

This thesis is based on the following papers, which are referred to in thetext by their Roman numerals.

I Herschend, M. (2005) Solution to the Clebsch-Gordan problem forrepresentations of quivers of type An. J. Algebra Appl., 4(5):481–488.

II Darpö, E., Dieterich, E., and Herschend, M. (2005) In which di-mensions does a division algebra over a given ground field exist?Enseign. Math. (2), 51(3-4):255–263.

III Herschend, M. (2007) Galois coverings and the Clebsch-Gordanproblem for quiver representations. Colloq. Math., 109(2):193–215.

IV Herschend, M. (2008) Tensor products on quiver representations.J. Pure Appl. Algebra, 212(2):452–469.

V Herschend, M. (2008) On the representation ring of a quiver. Al-gebr. Represent. Theory, to appear.

VI Herschend, M. (2008) On the representation rings of quivers ofexceptional Dynkin type. Bull. Sci. Math., to appear.

Reprints were made with permission from the publishers.

iii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Clebsch-Gordan problem . . . . . . . . . . . . . . . . . . . . . . . 11.2 Representation theory of linear categories . . . . . . . . . . . . . . 41.3 Representation ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Summary of paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Summary of paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Summary of paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Summary of paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Summary of paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6 Summary of paper VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

v

1. Introduction

1.1 The Clebsch-Gordan problemLet V andW be representations of a group G over an algebraically closedfield k of characteristic zero. Then their tensor product V ⊗W is alsoa representation of G via the action g(v ⊗ w) = (gv)⊗ (gw). Recall thefamous Krull-Schmidt Theorem.

Theorem 1.1. Let V be a representation of the group G of finite length.Then V decomposes uniquely (up to isomorphism and order) into a directsum of irreducible representations

V →n⊕

i=1

Vi.

Theorem 1.1 provides an explicit description of all representations ofG as soon as unique representatives of the isomorphism classes of ir-reducible representations are found, i.e. the problem of classifying allirreducible representations is solved.

Now let V → ⊕ni=1 Vi and W → ⊕m

j=1Wi be Krull-Schmidt de-compositions. Since the tensor product commutes with direct sums weobtain

V ⊗W →n⊕

i=1

m⊕j=1

Vi ⊗Wj .

To find the decomposition of V ⊗ W into irreducibles it is enough todecompose Vi ⊗Wj into irreducibles for all i and j.

Thus, for a full description of the tensor product it suffices to find theKrull-Schmidt decomposition of V ⊗W for all irreducible representationsV and W . The problem of finding these decompositions is called theClebsch-Gordan problem.

Once the classification problem has been solved, one can attempt tosolve the Clebsch-Gordan problem by trying to find, for each pair ofobjects in the classifying list, a formula for the decomposition of theirtensor product into a direct sum of irreducibles from the list. The for-mulae appearing in such a solution are called Clebsch-Gordan formulae.

The Clebsch-Gordan problem can be posed for any Krull-Schmidt cat-egory equipped with a tensor product that respects direct sums. Aclassical example is provided by the generalization of the group case

1

to modules over Hopf algebras [12]. In the present thesis we considerrepresentations of quivers.

A quiver Q is a quadruple (Q0, Q1, t, h), where Q0 is the set of verticesand Q1 the set of arrows. The maps t, h : Q1 → Q0 map each arrowα ∈ Q1 to its tail tα and head hα respectively. Thus a quiver is anoriented graph, possibly with loops and multiple arrows. We write x α→ yto state that tα = x and hα = y.

A representation V of a quiver Q consists of a collection of vectorspaces V (x) over a field k and k-linear maps

V (α) : V (tα)→ V (hα),

where x ∈ Q0 and α ∈ Q1.We define the tensor product of quiver representations point-wise and

arrow-wise. Let V and W be representations of Q. Their tensor productis the representation V ⊗W of Q defined by

(V ⊗W )(x) = V (x)⊗W (x),

for each x ∈ Q0 and

(V ⊗W )(α) = V (α)⊗W (α),

for each α ∈ Q1. This definition is similar to the definition of the directsum V ⊕W in so far as

(V ⊕W )(x) = V (x)⊕W (x),

(V ⊕W )(α) = V (α)⊕W (α).

We say that a representation U of Q is indecomposable if it only decom-poses trivially, i.e. if U → V ⊕W implies V = 0 or W = 0 but notboth. In what follows we will only consider point-wise finite dimensionalrepresentations, i.e. those V that satisfy dimV (x) <∞ for all x ∈ Q0.

Since the tensor product of quiver representations is defined point-wiseit inherits many properties from the tensor product of vector spaces,such as associativity, commutativity and distributivity with respect tothe direct sum. Because of the last property it is meaningful to pose theClebsch-Gordan problem.

With each representation V of Q we associate its dimension vectordimV : Q0 → Z, x �→ dimV (x). It is worth noting that

dim(U ⊗ V )(x) = dim(U(x)⊗ V (x)) = dimU(x) dimV (x),

for all x ∈ Q0 and dim(U ⊕ V ) = dimU + dimV , for all representationsU, V of Q.

2

For quiver representations the Clebsch-Gordan problem was first con-sidered in the case of the loop quiver, where it was solved over an al-gebraically closed ground field of characteristic 0 by Huppert [13], andindependently by Martsinkovky and Vlassov [17].

We restate the solution here in a particular case which is important tothe Clebsch-Gordan problem in many other situations. For each λ ∈ k

and n ∈ N \ {0} let Jλ(n) be the Jordan block of size n and eigenvalueλ:

Jλ(n) =

⎡⎢⎢⎢⎢⎣λ 1

. . . . . .

λ 1λ

⎤⎥⎥⎥⎥⎦

Theorem 1.2. Let k be a field of characteristic zero. For all λ, μ ∈k \ {0} and l,m ∈ N \ {0} the following formula holds:

Jλ(l)⊗ Jμ(m) →min(l,m)−1⊕

k=0

Jλμ(l +m− 2k − 1)

The Jordan blocks classify all indecomposable linear operators only ifthe ground field is algebraically closed. The formula in Theorem 1.2 onlyholds in characteristic zero. The case of positive characteristic is studiedin [14].

In what follows, whenever assumptions on the ground field are made,the only purpose is to ensure that Theorem 1.2 holds. If the Clebsch-Gordan problem for the loop quiver is solved for some other field k, thenthat solution can easily be incorporated in any of the solutions to thecases considered below to obtain a solution to that case for the field k.

For quivers with more than one vertex the Clebsch-Gordan problemwas first studied by Dieterich. In his study of lattices over curve singu-larities he provided a partial solution to the Clebsch-Gordan problem forthe four factor space quiver D4 [5].

Historically the term Clebsch-Gordan problem comes from invarianttheory. In [4] Clebsch and Gordan investigated binary algebraic forms,i.e. homogeneous polynomials f(x1, x2). The space Sm of all binaryalgebraic forms of degree m ∈ N can be viewed as a representation ofSL2 via the action(

a b

c d

)f(x1, x2) = f(ax1 + cx2, bx1 + dx2).

This representation is irreducible.The tensor product Sm⊗Sn of two such representations is as a vector

space identified with the space of all polynomials which are homogeneous

3

of degree m in the variables x1, x2 and homogeneous of degree n in thevariables y1, y2. Let f(x1, x2, y1, y2) be such a polynomial. In [4] it isshown that the following formula holds for all m ≥ n:

f =n∑

k=0

α(n)k (x1y2 − y1x2)kΔn−kDn−kΩkf.

Here Δ, D and Ω are certain differential operators which are invariantunder the action of SL2 and the α(n)

k are numerical coefficients. More-over, the polynomial Dn−kΩkf only depends on the variables x1, x2 andis homogeneous of degree m+ n− 2k.

In the representation theoretic setting this formula gives a decompo-sition of Sm⊗Sn into a direct sum of irreducible representations of SL2

(cf. [19]). The importance of this formula in quantum mechanics wasalready observed in [22].

It is interesting to note that this original formula of Clebsch and Gor-dan is very similar to the one in Theorem 1.2, which in turn is reproducedin many of the solutions to tame cases below.

1.2 Representation theory of linear categoriesQuivers were introduced by Gabriel in [7]. Their representation theorycan be viewed as a special case of the representation theory of linearcategories. From [9] we collect those basic notions from that theorywhich are necessary prerequisites for what follows.

Let k be a field. A category A is called linear if the set A(a, b) ofmorphisms from a to b is a vector space over k for all a, b ∈ A and thecomposition maps are k-bilinear. A functor between linear categories iscalled linear if its maps between morphism spaces are linear.

Let A be a small linear category. An A-module is a linear functor

m : A → Modk,

where Modk denotes the category of all vector spaces over k. It is calledpoint-wise finite dimensional if dimm(a) <∞ for each object a ∈ A.

A morphism of A-modules m and n is a natural transformation φ :m→ n. The category of all A-modules is denoted by ModA and the fullsubcategory of all point-wise finite dimensional A-modules by modA.

Let Q be a quiver. A path of length d > 0 from x ∈ Q0 to y ∈ Q0 is asequence of arrows αd . . . α1 positioned as below.

xα1 �� hα1

α2 �� · · · αd−1 �� tαdαd �� y

Moreover, there is for each vertex x ∈ Q0 a path of length 0, which wedenote by ex.

4

With each quiver Q we associate its path category Q. The object classof Q is the set of vertices Q0 and the morphism sets Q(x, y) consist ofall paths from x to y. Composition of paths is given by concatenation.

We also consider the linearized path category kQ. It has the sameobjects as Q, and for all x, y ∈ Q0 the morphism space kQ(x, y) is thevector space having Q(x, y) as basis. The composition maps in kQ arethe bilinear extensions of the composition maps in Q.

Each kQ-module m determines a representation M of Q defined byM(x) = m(x) for each vertex x ∈ Q0 and M(α) = m(α) for each arrowα ∈ Q1. This gives rise to an equivalence between ModkQ and thecategory of all representations of Q. It respects notions such as point-wise finite dimensionality and direct sums. We also define the tensorproduct for kQ-modules as for representations of Q.

In case Q is finite, i.e. both Q0 and Q1 are finite, we consider the pathalgebra of Q. As a vector space it is

A =⊕

x,y∈Q0

kQ(x, y).

The product of two paths is given by their composition if they are com-posable in kQ. Otherwise it is zero. This provides a third viewpoint onquiver representations as mod kQ is equivalent to the category of finitedimensional A-modules.

By considering bialgebra structures on the path algebra we obtain analternative approach to define tensor products of quiver representations.How this relates to the point-wise tensor product defined above is dis-cussed in paper IV. The question when the path algebra is a Hopf algebrahas been studied in [3], however without considering the correspondingClebsch-Gordan problem.

Sometimes the point-wise tensor product can be extended to quiverswith relations. An ideal I in a linear category A is a family of sub-spaces I(a, b) ⊂ A(a, b) such that A(b, y)I(a, b)A(x, a) ⊂ I(x, y) for alla, b, x, y ∈ A. The quotient category A/I has the same objects as A andthe morphism spaces are A/I(a, b) = A(a, b)/I(a, b). The compositionmaps in A/I are induced from the composition maps in A.

If A is small, then the category modA/I is identified with the fullsubcategory of modA that consists of all modules m satisfying m(α) = 0for all α ∈ I.

Let Q be a quiver. We say that an ideal I in kQ is semimonomial if itis generated by elements of the form γ or γ−μ where γ and μ are paths inQ. In paper III it is shown that the point-wise tensor product of kQ/I-modules is well-defined in case I is semimonomial. A characterizationof semimonomial ideals is given by the following proposition.

5

Proposition 1.3. Let Q be a quiver and I an ideal in kQ. Then I issemimonomial if and only if the set {γ + I(x, y)|γ ∈ Q(x, y)} \ {0} is abasis of kQ/I(x, y) for all x, y ∈ Q0.

Proof. Set P =⋃

x,y∈Q0(Q(x, y) \ I(x, y)). Define the equivalence rela-

tion ∼ on P by γ ∼ μ if and only γ − μ ∈ I(x, y) for some x, y ∈ Q0.Let I ′ ⊂ kQ be the ideal generated by the family of subsets

{γ − μ | γ, μ ∈ Q(x, y) ∪ {0}} ∩ I(x, y) ⊂ kQ(x, y),

where x, y ∈ Q0. Observe that I ′ ⊂ I, and that I ′ = I if and only if Iis semimonomial.

Further set

I(x, y) =⎧⎨⎩

∑γ∈Q(x,y)

aγγ

∣∣∣∣∣∣ aγ ∈ k,∑γ∈A

aγ = 0 for all A ∈ P/∼

⎫⎬⎭ .

We claim that this family of vector spaces defines an ideal I ⊂ kQ. Let∑aγγ ∈ I(x, y) and μ ∈ Q(y, z). Then

μ(∑

aγγ)=∑

aγμγ.

Let A ∈ P/∼. Since γ ∼ γ′ implies μγ ∼ μγ′ we have for each B ∈ P/∼that μγ ∈ A and γ ∈ B holds if and only if μB ⊂ A and γ ∈ B. Hence∑

μγ∈A

aγ =∑

μB⊂A

∑γ∈B

aγ = 0

and thus kQ(y, z)I(x, y) ⊂ I(x, z). The fact that I(y, z)kQ(x, y) ⊂I(x, z) is shown similarly. We conclude that I is an ideal and I ′ ⊂ I.

We proceed to show that I ⊂ I ′ by induction. Let∑n

i=1 aiγi ∈ I(x, y)with ai = 0. If n = 0, then

∑ni=1 aiγi = 0 ∈ I ′(x, y). If n > 0, then

there is i0 such that γi0 ∼ γ1. We obtainn∑

i=1

aiγi = a1(γ1 − γi0) +n∑

i=2

a′iγi.

Since a1(γ1 − γi0) ∈ I ′(x, y) ⊂ I(x, y) we get∑n

i=2 a′iγi ∈ I(x, y) and

by induction∑n

i=2 a′iγi ∈ I ′(x, y). Hence

∑ni=1 aiγi ∈ I ′(x, y). In other

words I ′ = I.Set B(x, y) = {γ + I(x, y)|γ ∈ Q(x, y)} \ {0} for all x, y ∈ Q0. Then

B(x, y) spans kQ/I(x, y). Moreover, B(x, y) is linearly independent ifand only if

∑γ∈Q(x,y) aγγ ∈ I(x, y) implies

∑γ∈A aγ = 0 for all A ∈

P/ ∼. This condition is equivalent to the inclusion I(x, y) ⊂ I(x, y). Inother words B(x, y) is a basis in kQ/I(x, y) for all x, y ∈ Q0 if and onlyif I is semimonomial.

6

To solve the Clebsch-Gordan problem we need a classification of theindecomposable modules. The representation finite quivers, i.e. thosefor which there are only finitely many isomorphism classes of indecom-posables, are exactly those of Dynkin type, as was shown by Gabriel[7].

To describe the indecomposables in this case we consider the Tits formof a quiver Q. It is the integral quadratic form

qQ : ZQ0 → Z,

defined byqQ(d) =

∑x∈Q0

d(x)2 −∑

α∈Q1

d(tα)d(hα).

A vector d ∈ ZQ0 is called a root of qQ if qQ(d) = 1 and positive if

d(x) ≥ 0 for all x ∈ Q0.In the Dynkin case the positive roots are the dimension vectors of the

indecomposables. Here is the precise statement cited from [9, p. 63].

Theorem 1.4. The number of isoclasses of indecomposable representa-tions of Q is finite if and only if qQ is positive definite. If this is thecase, the map [m] �→ dimm provides a bijection between the set of theseisoclasses and the set of positive roots of qQ.

Thus for a Dynkin quiver Q the solution to the Clebsch-Gordan prob-lem can be described as follows. Each pair (m,n) of indecomposablemodules corresponds to a pair (d, e) of positive roots of qQ. The tensorproductm⊗n has the dimension vector de defined by (de)(x) = d(x)e(x).It decomposes into a direct sum of indecomposables

m⊗ n →l⊕

i=1

ui.

Each summand ui corresponds to a positive root di = dimui, and wehave

de =l∑

i=1

di.

This condition can be used to reduce the number of possible decompo-sitions of m ⊗ n. The Clebsch-Gordan problem for Dynkin quivers isconsidered in papers V and VI.

Another fact that can be used when investigating possible decompo-sitions is that the tensor product of linear maps is multiplicative withrespect to the rank. How this concept extends to quiver representationsis studied by Kinser in [15].

For the extended Dynkin quivers the indecomposable representationsare also classified but there are infinitely many and they must therefore

7

be parametrized in some way. When considering the Clebsch-Gordanproblem it is vital that the parametrization of the indecomposables iscompatible with the tensor product in order to obtain digestible formu-lae.

An example of how different the solutions may look depending onthe chosen classification can be found in paper I where two differentclassifications are considered for the Kronecker quiver • ⇔ •.

Quivers that are neither of Dynkin nor extended Dynkin type arecalled wild and the classification problem is to date unsolved for all ofthem.

1.3 Representation ringThe presentation of the solution to the Clebsch-Gordan problem as alist of formulae is often hard to digest, in part as it may depend on theorientation of the quiver. It is therefore difficult to get an overview ofthe solutions even in comparatively simple cases such as for quivers oftype D.

This motivates the introduction of the representation ring which en-codes the solution to the Clebsch-Gordan problem in its multiplicativestructure.

Let Q be a quiver and S(Q) the set of isoclasses of its representations.We define the operations addition and multiplication on S(Q) by

[m] + [n] = [m⊕ n]

and

[m] · [n] = [m⊗ n].

This endows S(Q) with the structure of a commutative semiring. Therepresentation ring R(Q) = K(S(Q)) is the Grothendieck ring (in thesense of [16]) associated with S(Q). For more details see paper V.

As an abelian group, the ring R(Q) is freely generated by the isoclassesof indecomposable modules. The structure constants for the multiplica-tion in this Z-basis are the coefficients in the Clebsch-Gordan formulae.In the Dynkin case there are only finitely many isoclasses of indecom-posables. Hence R(Q) is a free Z-module of finite rank. This yields thefollowing proposition from paper V.

Proposition 1.5. If Q is a Dynkin quiver, then the representation ringR(Q) is a commutative Z-order.

8

1.4 Discussion of resultsIn this section we present an overview of the results from the variouspapers contained in the present thesis. The Clebsch-Gordan problem forquiver representations is considered in two different situations: the tamecases An and the double loop quiver with relations αβ = βα = αn =βn = 0, and the representation finite case of Dynkin quivers.

For the tame cases the main tools are Theorem 2.7 and Corollary 2.9.These results describe the interaction between Galois coverings and thepoint-wise tensor product. They reduce parts of the Clebsch-Gordanproblem for the base quiver to the Clebsch-Gordan problem for the cov-ering quiver. They also show that a classification given in terms ofcoverings often is well-suited for solving the Clebsch-Gordan problem.

In the Dynkin case a complete solution has been found for all quiversof type A, D and E6. These solutions are built up successively in thesense that the solution in case A yields part of the solution in case D,and the remaining part is accomplished by direct matrix calculations.These results are then used to achieve a partial solution in case E6, andagain the remaining part is accomplished by certain matrix calculations.

In paper VI we outline how the same techniques could be used in thecase E7, and subsequently E8. However, the amount of work needed todo the parts to be handled by matrix calculations increases rapidly, notleast due to the fact that the number of orientations doubles in eachstep. It therefore seems that computer calculations will be necessary tofollow this program through and obtain a solution to the Clebsch-Gordanproblem for all Dynkin quivers. However, it remains to find a suitablealgorithm.

The uniformity of the known descriptions of the representation ringsof Dynkin quivers is striking. For each quiver Q of type A, D or E6 therepresentation ring turns out to be of the form

R(Q) →n∏

r=1

Rkr ,

where Rk = Z[T1, . . . , Tk]/(TiTj | 1 ≤ i, j ≤ k) for all k ∈ N. It shouldbe stressed that the increasing sequence of natural numbers (k1, . . . , kn)and even the isomorphism involved depend heavily on the orientation ofQ.

The identity element in each factor of∏n

r=1Rkr corresponds to anidempotent in R(Q) and each variable Ti to a nilpotent element. Theseidempotents and nilpotents form a multiplicative Z-basis of R(Q). Thusthe sum of the number of nilpotents of this type and the number ofidempotents of this type is the Z-rank of R(Q), which coincides withthe number of indecomposables. By Theorem 1.4, this number does notdepend on the orientation of Q.

9

If one fixes the underlying graph of Q but varies its orientation, thenumber of idempotents and nilpotents changes. Inspecting the resultsfor D and E6 one can make the following qualitative statement: the morearrows that are equally oriented with respect to the branching point ofQ, the larger the number of nilpotents. Moreover, the orientation ofarrows closer to the branching point plays a more important role thanthat of arrows further away.

The only paper in the present thesis which is not concerned with theClebsch-Gordan problem is paper II, which deals with finite dimensionaldivision algebras. It is a testimony of my minor involvement in the workdone by Darpö and Dieterich in this field and is included for the sake ofcompleteness.

10

2. Summary of papers

2.1 Summary of paper IPaper I contains the solution to the Clebsch-Gordan problem for quiversof type An, i.e. with the underlying graph

a0

α0

����

����

a1 α1· · ·

αn−1an

αn

��������

This result extends a solution to the special case of the Kronecker quiver• ⇔ • contained in the authors masters thesis [11].

The classification of indecomposable modules is given in termsof strings and bands. Let s ∈ {0, . . . , n} and l ∈ N. The stringmodule S(s, l) is defined as follows. The basis vectors in the vectorspaces S(s, l)(a0), . . . , S(s, l)(an) are denoted e0, . . . , el. For eachν ∈ {0, . . . , n}, the vector space S(s, l)(aν) has as basis those ei forwhich s + i ≡ ν mod n. The linear map S(s, l)(αν) maps ei to ei+1 orei−1 depending on the orientation.

Let λ ∈ k \ {0} and m ∈ N \ {0}. The band module Bλ(m) is definedas follows. All vector spaces Bλ(m)(aν) are equal to k

m. The linearmap Bλ(m)(αν) is the identity if ν = n and Bλ(m)(αn) is given by theJordan block of eigenvalue λ and size m.

The strings and bands classify all indecomposable kQ-modules [9],and the solution to the Clebsch-Gordan problem with respect to thisclassification is given by the following theorem.

Theorem 2.1. For all s, r ∈ {0, . . . n} with r ≤ s, scalars λ, μ ∈ k \ {0}and l,m ∈ N the following formulae hold:1.

S(s, l)⊗ S(r,m) →[m+r−s

n+1 ]⊕k=0

S(s,min(l,m+ r − s− k(n+ 1)))

⊕[ l+s−r

n+1 ]⊕k=1

S(r,min(m, l + s− r − k(n+ 1)))

2.S(s, l)⊗Bμ(m) → mS(s, l), where m = 0

11

3.

Bλ(l)⊗Bμ(m) →min(l,m)−1⊕

k=0

Bλμ(l +m− 2k − 1), where l,m = 0

As discussed in section 1.2, the complexity of the Clebsch-Gordanproblem can vary depending on the choice of cross-section of indecom-posable modules in the classification. In paper I we see an example ofthis phenomenon. Here the classification is given in terms of strings andbands. The solution to the Clebsch-Gordan problem consists of the threeformulae in Theorem 2.1.

For the Kronecker quiver the solution is also given with respect tothe classification obtained by Auslander-Reiten theory. Here we need allseven formulae of Corollary 1 in paper I to obtain a complete solution,even though it is a special case of the solution in Theorem 2.1.

Thus it appears that the classification given by strings and bands iswell suited when dealing with tensor products.

The three formulae in Theorem 2.1 are proved individually. The firsttwo, i.e. those involving strings, are proved by providing explicit basesfor each of the summands in the decomposition, much inspired by themethods used in [11]. The last case is reduced to the same problem forthe loop quiver which is solved in Theorem 1.2.

A consequence of Theorem 2.1 is that a string module tensored withany module decomposes into a direct sum of string modules. This is aspecial case of a more general phenomenon coming from the interactionof the tensor product with Galois coverings studied in paper III.

2.2 Summary of paper IIPaper II differs from all other papers contained in the present thesisin so far as it does not deal with representation theory. It concernsfinite dimensional division algebras. In this section algebras are neitherassumed to be associative nor unital. A non-zero algebra A is called adivision algebra if the linear maps Lx, Rx : A→ A defined by Lx(y) = xyand Rx(y) = yx are bijective for all x ∈ A \ {0}.

Denote by N (k) the set of all natural numbers n such that there existsa division algebra over the field k with dimension n. The main result isthe following theorem.

Theorem 2.2. Let k be a field. Then

N (k) =

⎧⎪⎨⎪⎩{1} if k is algebraically closed{1, 2, 4, 8} if k is real closedunbounded if k is non-closed

.

12

Here non-closed means neither algebraically nor real closed. In case k isalgebraically closed the statement follows from the following proposition.

Proposition 2.3. (Gabriel 1994) If k is an algebraically closed fieldand A is a k-algebra with 1 < dimA <∞, then A has zero divisors.

The proof of Proposition 2.3 relies on the fact that every k-linearoperator has an eigenvalue in case k is algebraically closed. In case kis non-closed a classical result by Artin and Schreier dealing with irre-ducible polynomials is used. Denote by M(k), the set of all degrees ofirreducible polynomials in k[X].

Proposition 2.4. (Artin, Schreier 1927) For every field k which isnot algebraically closed, the following statements are equivalent.

1. [k : k] is finite.

2. M(k) is bounded.

3. k is real closed.

Hence, if k is non-closed then M(k) is unbounded. Since each irre-ducible polynomial gives rise to a field extension whose degree equalsthe degree of the polynomial, we obtainM(k) ⊂ N (k) and thus N (k) isalso unbounded. In case k = R, there is the following famous result onN (k).Proposition 2.5. (Hopf 1940; Bott, Milnor, Kervaire 1958) Everyfinite dimensional real division algebra has dimension 1, 2, 4 or 8.

In view of the classical real division algebras R, C, H, O it followsthat N (R) = {1, 2, 4, 8}. In paper II we proceed to extend this resultfrom R to arbitrary real closed fields using model theory. Recall that twofields are called elementarily equivalent in the language of rings if theysatisfy the same first order sentences in this language. Now consider thefollowing fact.

Proposition 2.6. (Tarski 1931) Any two real closed fields are elemen-tarily equivalent.

For each natural number n we formulate the statement n ∈ N (k) asa first order sentence in the language of rings. Thus for each real closedfield k

N (k) = N (R) = {1, 2, 4, 8}where the first equality is by Proposition 2.6 and the second by Propo-sition 2.5. This completes the proof of all three statements of Theorem2.2.

13

2.3 Summary of paper IIIPaper III investigates the relationship between Galois coverings and thepoint-wise tensor product. The results obtained are used to solve theClebsch-Gordan problem for two tame cases, in both of which the clas-sification is given via coverings.

Covering theory was introduced into representation theory by Gabriel[8], [1]. Originally it served to compute indecomposable modules in therepresentation finite case. We briefly outline the basic notions of Galoiscoverings. For more details and definitions see section 3 in paper III. Forthe original definitions we refer to [8].

If G is a group acting freely and locally bounded on a small linearcategory Γ, then one obtains the quotient category Λ = Γ/G and alinear functor

P : Γ→ Λ,

sending objects x ∈ Γ to their orbits P (x) = Gx. It is called a Galoiscovering.

To illustrate this we present one of the examples considered in paperIII. Let Q be a quiver of type A

∞∞, i.e. with underlying graph

· · · α−1 a0α0 a1

α1 · · ·such that the orientation is periodic with period n+1. Then G = Z actson Q by translating arrows and vertices n+1 steps. This action extendsto Γ = kQ, and the quotient category Λ is isomorphic to kQ′, where Q′

is a quiver of type An.We return now to the general setting. The push-down functor

P∗ : modΓ→ modΛ

is the left adjoint of the pull-up functor

P ∗ : modΛ→ modΓ.

We say that a Λ-module n is of the first kind if n → P∗m for some Γ-module m, and denote by mod1 Λ the full subcategory of modΛ formedby all modules of the first kind. In the example above, the indecompos-able modules of the first kind are precisely those which are isomorphicto some string module.

It is shown in [8] that if Γ is locally representation finite then so isΛ, and in that case mod1 Λ = modΛ. However, as seen in the exampleabove this is in general not the case if Λ is representation infinite.

Dowbor and Skowronski investigate in [6] the category mod2 Λ of mod-ules containing no direct summand of the first kind. The indecomposablemodules among these are exactly those which are isomorphic to someband module, in the case of An.

14

Let Q be a quiver and G a group acting freely on Q. Let I be aG-invariant semimonomial ideal of kQ. Set Γ = kQ/I. The quotientcategory Λ = Γ/G can be interpreted as the linearized path category ofa quiver Q′, modulo a semimonomial ideal. The interaction between thecovering functor P∗ and the tensor product is described in the followingtheorem from paper III.

Theorem 2.7. For all m ∈ modΓ and n ∈ modΛ there is an isomor-phism

(P∗m)⊗ n → P∗(m⊗ (P ∗n)).

From Theorem 2.7, it follows that the tensor product of a module ofthe first kind and any other module will be a module of the first kind.This can be rephrased in terms of the representation ring.

Corollary 2.8. Let I be an ideal of the representation ring R(Γ), gener-ated as an abelian group by some subset X ⊂ S(Γ). Let φ : R(Γ)→ R(Λ)be the group morphism induced by P∗. Then φ(I) is an ideal of R(Λ). Inparticular, the abelian group generated by the modules of the first kind isan ideal of R(Λ).

Proof. Since φ is a group morphism, φ(I) is a subgroup of R(Λ). Let

ψ : R(Λ)→ R(Γ)

be the group morphism induced by P ∗. If a ∈ S(Λ) and b ∈ X, then itfollows from Theorem 2.7 that φ(b)a = φ(bψ(a)) ∈ φ(I). Since R(Λ) isgenerated by S(Λ) and I is generated by X as abelian groups it followsthat φ(I) is an ideal.

The tensor product of two modules of the first kind can be computedmore explicitly. Let n ∈ modΓ and g ∈ G. We denote by gn thetranslated module n ◦ g−1

Corollary 2.9. For all m,n ∈ mod(Γ) there is an isomorphism

(P∗m)⊗ (P∗n) →⊕g∈G

P∗(m⊗ gn)).

To solve the Clebsch-Gordan problem using these results one has toconsider m ⊗ n in three different cases, assuming firstly that m,n ∈mod1 Λ, secondly that m ∈ mod1 Λ and n ∈ mod2 Λ, and thirdly thatm,n ∈ mod2 Λ.

The first case is completely reduced to the Clebsch-Gordan problem forΓ-modules by Corollary 2.9. In the second case one needs to decomposeP ∗n for all n ∈ mod2 Λ and then apply Theorem 2.7. However, theseresults do not help in the last case which has to be dealt with in someother way.

15

As mentioned earlier, in case Q′ is of type An, the covering quiver Q isof type A

∞∞. The Clebsch-Gordan problem for Q is essentially the sameas for quivers of type A, and thus the first two cases can be handledquite easily in the way just described. Luckily the last case can reducedto the loop quiver.

We also apply Theorem 2.7 and Corollary 2.9 in the case of the doubleloop quiver

a�

�

with relations βα = αβ = αn = βn = 0. The classification of its inde-composables was first achieved by Gelfand and Ponomarev in their in-vestigation of indecomposable representations of the Lorentz group [10].Its Auslander-Reiten quiver is computed in [2].

The covering quiver in this case is a mesh quiver with suitable relations.The indecomposable representations of the covering quiver are classifiedby the characteristic representations satisfying the relations.

All subquivers corresponding to these characteristic representationsare of type A, A∞ or A

∞∞. We call these subquivers lines. The inde-composable modules of the first kind are all of the form P∗(χL) for somefinite line L. The modules of the second kind, correspond in a similarway to infinite lines which are G-periodic. This terminology is borrowedfrom [6].

The program outlined above is carried out in this case. The resultsobtained relate the Clebsch-Gordan problem to the combinatorics of linesin the covering quiver. This inspires the author to carry through the lastcase by suitable chosen bases, thus completing the solution.

These results can of course be applied to other situations. However,they are useful only when the Clebsch-Gordan problem for the coveringquiver is easier to solve than for the base quiver.

2.4 Summary of paper IVThe point-wise tensor product is a naive modification of the tensor prod-uct coming from group representation theory in the sense that the arrowsact diagonally, i.e. as grouplike elements. What other possible tensorproducts are there that have this property? How does the point-wise ten-sor product relate to more classical situations in which a tensor productappears? The purpose of paper IV is to answer these questions.

The tensor product in the representation theory of groups is a specialcase of the one which arises in the representation theory of Hopf algebras[12], which in turn can be generalized to the setting of bialgebras. Theset-up is as follows.

16

A comultiplication on an associative algebra A is a linear map

Δ : A→ A⊗A.

IfM,N are A-modules, thenM⊗N is an A⊗A-module. The elementsa ∈ A act on M ⊗N via Δ by

a(m⊗ n) = Δ(a)(m⊗ n),

for all m ∈ M , n ∈ N and a ∈ A. If Δ is an algebra morphism thisaction provides an A-module structure on M ⊗ N . We denote this A-module by M ⊗Δ N . Hence we obtain a tensor product on the categoryof A-modules, whenever an algebra morphism Δ : A→ A⊗A is given.

Paper IV discusses three modifications of this set-up to the frameworkof quiver representations.

Let Q be a quiver and A the path algebra of Q with coefficients in k.The first approach is to consider A-modules instead of modules over thelinearized path category kQ. Define the comultiplication

Δ : A→ A⊗A,

by Δ(γ) = γ ⊗ γ for each path γ in Q. The problem with this set-up isthat Δ is not an algebra morphism in general. It is multiplicative andlinear, but it does not respect the identity element.

To salvage the situation we introduce so-called premodules. A pre-module over A is a vector space M together with a bilinear map

A×M →M,

which satisfies(ab)m = a(bm),

for all a, b ∈ A and m ∈ M . In this terminology an A-module is apremodule where 1A acts as the identity morphism onM . If all elementsact as the zero morphism, the premodule is called trivial. We cite thefollowing proposition from paper IV.

Proposition 2.10. Every premodule M over a k-algebra A has twouniquely determined subpremodules M1 and M0 such that M1 is an A-module, M0 is trivial and

M =M1 ⊕M0.

With the action of A defined by Δ the vector space M ⊗ N is apremodule and the A-module M ⊗N := (M ⊗N)1 corresponds exactlyto the point-wise tensor product of M and N considered as kQ-modules.Hence we recover the point-wise tensor product with this approach.

17

The second approach investigates possible modifications of Δ to makeit an algebra morphism. Since we are only interested in tensor productsfor which the arrows act diagonally we impose the condition Δ(α) =α⊗ α.

We say that Δ is partitioning if there exists a partition Q0 × Q0 =⋃k∈Q0

Ek such that

Δ(ek) =∑

(i,j)∈Ek

ei ⊗ ej .

Essentially this means that Δ respects the quiver structures underlyingA and A⊗A respectively.

Theorem 2.11. LetΔ : A→ A⊗A

be a partitioning algebra morphism satisfying

Δ(α) = α⊗ αfor each arrow α in Q. Let M and N be A-modules. Then there is anisomorphism of A-modules

M ⊗Δ N →M ⊗N ⊕⊕k∈Q0

dkSk

where Sk is the simple module corresponding to the vertex k and

dk =∑i,j

dimM(i) dimN(j)

where (i, j) ∈ Ek and i = j.

According to Theorem 2.11, the only difference in the decompositionofM ⊗ΔN andM ⊗N is the direct summand

⊕k∈Q0

dkSk. So from thepoint of view of the Clebsch-Gordan problem the tensor products ⊗Δ

and ⊗ are equivalent.The last approach is to replace A with the linearized path category

kQ. The tensor product of linear categories is defined as follows.Let A and B be small linear categories. The linear category A⊗ B is

defined by

Ob(A⊗ B) = ObA×ObB,(A⊗ B)((a, b), (a′, b′)) = A(a, a′)⊗k B(b, b′)

where the composition is defined by (α′ ⊗ β′)(α⊗ β) = α′α⊗ β′β.Now define the linear functor

Δ : kQ→ kQ⊗ kQ

18

by Δ(x) = (x, x) for all x ∈ Q0 and Δ(γ) = γ ⊗ γ for all paths γ in Q.Let m,n be kQ-modules. We define the kQ⊗ kQ-module m⊗ n by

(m⊗ n)(x, y) = m(x)⊗ n(y)for each x, y ∈ Q0 and

(m⊗ n)(α⊗ β) = m(α)⊗ n(β)for each α, β ∈ Q1.

It turns out that m ⊗ n → (m ⊗ n)Δ. Thus the point-wise tensorproduct appears naturally when the bialgebra set-up is modified to thecategorical setting. In this set-up it is also straightforward to show thatmod kQ is a monoidal category under the point-wise tensor product.

2.5 Summary of paper VThe main result of paper V is the solution to the Clebsch-Gordan prob-lem for quivers of type A and D. For quivers of type D the result dependsheavily on the orientation and thus quite many formulas are required togive the complete solution. To obtain a better overview of the solutionthe representation ring is introduced. For quivers of type A and D it isdescribed explicitly in Theorem 3 and Theorem 5 respectively.

The main tools are characteristic representations and matrix calcu-lations. Let Q be a quiver and P any subquiver. The characteristicrepresentation χP associated with P is defined by

χP (x) =

{k if x ∈ P0

0 if x ∈ P0

for each vertex x ∈ Q0 and

χP (α) =

{1k if α ∈ P1

0 if α ∈ P1

for each arrow α ∈ Q1.The terminology is motivated by the observation that characteristic

representations behave with respect to the tensor product as charac-teristic functions behave with respect to multiplication. Indeed, for allsubquivers P 1, P 2 of Q, the formula

χP 1 ⊗ χP 2 → χP 1∩P 2 (2.1)

holds. For each kQ-module m, we let S(m) be the support of m, i.e. thesubquiver of Q consisting of all points and arrows at which m does notvanish. We have

χS(m) ⊗m → m.

19

A characteristic representation χP is indecomposable if and only ifP is connected. Hence formula (2.1) provides a partial solution to theClebsch-Gordan problem. If Q is of type A, then the indecomposablecharacteristic representations classify all indecomposables and thus for-mula (2.1) provides the complete solution.

The elements [χP ] ∈ R(Q) are idempotents. This motivates the rele-vance of the following ring theoretic proposition from paper V.

Proposition 2.12. Let R be a commutative ring and S a finite set ofidempotents, closed under multiplication and containing 0. The relation≤ on S, defined by e ≤ f if and only if ef = e, is a partial order. Themap S → R, e �→ e′ is defined recursively by e′ = e−∑f<e f

′. Then thefollowing statements hold.1.

e =∑f≤e

f ′

2. For all e, f ∈ S the equality

e′f ′ =

{e′ if e′ = f ′

0 if e = f

holds. In particular the set

S′ = {e′ | e ∈ S}

consists of pairwise orthogonal idempotents.

The definition of the elements e′ in Proposition 2.12 is recursive, butit can be made explicit using the Möbius inversion formula. We brieflyintroduce this concept following Rota [18].

Let P = (P,≤) be a finite partially ordered set. The incidence algebraI(P ) consists of all functions f : P × P → Z such that f(x, y) = 0 ifx ≤ y. It is equipped with the product

(f ∗ g)(x, y) =∑

x≤z≤y

f(x, z)g(z, y).

The zeta function ζ ∈ I(P ) is defined by

ζ(x, y) =

{1 if x ≤ y

0 otherwise.

It is invertible in I(P ), and its inverse μ is called the Möbius function ofP .

20

Theorem 2.13. (Möbius inversion formula) Let P be a finite par-tially ordered set and G an abelian group. Let h, g : P → G. If

g(x) =∑y≤x

h(y),

thenh(x) =

∑y≤x

μ(y, x)g(y).

Now let P be the set of idempotents S ⊂ R considered in Proposition2.12 and G = Z

S . Define the homomorphism φ : G → R by φ(de) = e′,where de : S → Z is defined by de(f) = δef . Further define h, g : S → G,by h(e) = de and g(e) =

∑f≤e df . Observe that for each e ∈ S the

equalitiesφ(g(e)) =

∑f≤e

φ(df ) =∑f≤e

f ′ = e

and φ(h(e)) = φ(de) = e′ hold.Since g(e) =

∑f≤e h(f), Theorem 2.13 applies and

h(e) =∑f≤e

μ(f, e)g(f).

Applying φ we obtaine′ =

∑f≤e

μ(f, e)f.

Proposition 2.12 and its relation to the Möbius inversion formula issimilar to a technique used in the theory of semigroup algebras (cf. [20],[21]).

Using Proposition 2.12 in conjunction with the properties of charac-teristic representations discussed above the following results are shownto be valid.

Theorem 2.14. Let Q be a quiver and {Qi}ni=1 a finite set of subquivers

of Q, closed under intersections and containing Q. Set ei = [χQi ], R =R(Q) and Ri = R(Qi). Then the following statements hold true.1. The rings eiR and Ri are isomorphic.2.

1 =n∑

i=1

e′i

is a decomposition of 1 into orthogonal idempotents.3.

R =n⊕

i=1

e′iR

is a decomposition of R into ideals.

21

4. The map

R→n∏

i=1

e′iR, r �→ (e′ir)ni=1

is a canonical isomorphism of rings.5. The map

e′iR→eiR∑

ej<eiejR

, x �→ x+∑

ej<ei

ejR

is an isomorphism of rings.

Corollary 2.15. Let Q be a finite connected quiver without cycles and{Qi}n

i=1 be the set of all connected subquivers of Q. Set ei = [χQi ] andR = R(Q). Then

R =n∏

i=1

e′iR.

Furthermore the following statements hold.1. As an abelian group the ring e′iR is generated by {e′i[m] | m ∈indkQ, S(m) = Qi}.

2. If Q is a Dynkin quiver, then e′iR is freely generated by {e′i[m] | m ∈indkQ, S(m) = Qi} and all e′i[m] are distinct as [m] ranges throughall isoclasses of indecomposable kQ-modules with support Qi.

3. If Qi is of type A, then

e′iR = e′iZ.

In case Q is a Dynkin quiver, Corollary 2.15 applies and thus providesa factorization of the representation ring, with one factor for each con-nected subquiver of Q. Moreover, a Z-basis is provided for each of thefactors and in case the subquiver is of type A, the corresponding factoris isomorphic to Z.

In particular, if Q is of type An, then

R → Zn(n+1)

2

since there are n(n+1)2 connected subquivers of Q and they are all of type

A.Formula 2.1 provides a partial solution to the Clebsch-Gordan problem

for quivers of type D. The cases which are not solved by the use ofcharacteristic representations are handled by matrix calculations. Theresults are found in Propositions 5,6 and 7 in paper V, which solve theClebsch-Gordan problem for all quivers of type D.

Paper V also provides an explicit description of the representation ringR(Q) for all quivers Q of type D. Corollary 2.15 provides a partial de-scription of R(Q). It remains to describe those factors of R(Q) which

22

correspond the connected subquivers of type D. Let P be such a sub-quiver and let G be the Z-basis of [χP ]′R provided by Corollary 2.15.Using the Clebsch-Gordan formulae mentioned above we find that for allm,n ∈ G their product satisfies mn ∈ {m,n, 0}. This is used to provethat [χP ]′R is isomorphic to a direct product of rings of the form Rk,where the precise factorization depends on the orientation.

2.6 Summary of paper VIPaper VI deals with the representation rings of quivers of exceptionalDynkin type. For all quivers of type E6, the representation ring is de-scribed explicitly. An outline is also given as to how similar descriptionspossibly can be found for quivers of type E7 and E8.

The methods rely heavily on the results on quivers of type A andD in paper V. The following proposition from paper VI is essentially asummary of results from paper V.

Proposition 2.16. Let Q be a Dynkin quiver, R its representation ringand {Qi}i∈I the set of all connected subquivers of Q. Set ei = [χQi ].Then the following statements hold.1.

R =∏i∈I

e′iR

2. For each i ∈ I, the set {e′i[m] | m ∈ indkQ, S(m) = Qi} is aZ-basis of e′iR.

3. Multiplication is given by

e′i[m]e′i[n] =

r∑k=1

e′i[uk],

where

m⊗ n =r⊕

k=1

uk ⊕s⊕

l=1

vl

is the decomposition of m⊗n into indecomposables such that S(uk) =Qi and S(vl) = Qi.

4. The ring e′iR is determined up to isomorphism by the quiver Qi andis in this sense independent of the quiver Q.

23

Let Q be of type E6, i.e. having the underlying graph

a1

α1

����

����

a2α2

x cγ

b1

β1

�������

b2.β2

Proposition 2.16 reduces the description of representation ring R = R(Q)to the description each of its factors e′iR. If Qi is a proper connectedsubquiver of Q, then it is of type A or D and e′iR is described explicitlyin paper V.

It remains to describe e′R, where e = [χQ]. Proposition 2.16 pro-vides such a description up to the knowledge of certain Clebsch-Gordanformulae. The ring e′R has the Z-basis

G = {e′[m] | m ∈ indkQ, S(m) = Q}.

Let m be an indecomposable kQ-module with full support. Then dimmis a strictly positive root of the Tits form qQ. It is a fact that all suchroots have entry 1 at the vertex a2.

Hence if n is another such module the dimension of m⊗n at a2 is also1. In particular m⊗n has at most one omnipresent direct summand. Letu be this summand if it exists and otherwise set u = 0. By Proposition2.16, we have

e′[m]e′[n] = e′[u].

In other words, the Z-basis G for e′R is multiplicative. To describe e′Rit remains to find the module u for each m and n.

Incidentally the same argument goes through for E7 but fails for E8

since in that case there is a positive root with all entries greater or equalto 2. It is as yet unknown if the corresponding basis is multiplicative forE8 or not.

Paper VI makes further use of the results from paper V to determinea multiplication table for the elements in G. Methods are developedthat induce indecomposable representations of Q from representationsof smaller quivers. The modules obtained are exactly those for whichthere is an arrow x

α→ y in Q such that the dimensions at x and y areequal.

If two modules m and n have a common such arrow, then the prob-lem of decomposing m ⊗ n can be reduced to a similar problem for thesmaller quiver, which must be of type A or D. Together with resultsfrom paper V, this procedure yields a partial multiplication table for themultiplicative basis G. It is found as Table 1 in paper VI.

24

Filling in the remaining gap by use of matrix calculations much in thespirit of paper V, it turns out that the basis G allows a partition intodisjoint sets E and N such that E consists only of idempotents and forall m,n ∈ N the product satisfies mn = 0. By Proposition 2.12, the set

E′ = {f ′ | f ∈ E ∪ {0}}

consists of pairwise orthogonal idempotents which sum up to the identityelement. Hence we obtain the factorization

e′R →∏f∈E

f ′R.

Moreover, f ′R is generated by the set f ′N ∪ {f ′}. It follows thatf ′R → Rk, where k is the rank of the Z-module generated by f ′N .

25

Summary in Swedish

Clebsch-GordanproblemetIdén bakom representationsteori är att försöka förstå abstrakta algebra-iska strukturer med hjälp av linjär algebra, som är ett väl utrett ämne.Representationsteorin härstammar från grupprepresentationsteorin därelementen i en given grupp representeras av linjära operatorer på ettvektorrum.

Varje representation delas upp i en direkt summa av ouppdelbararepresentationer på ett unikt sätt. Detta kallas för representationensKrull-Schmidtuppdelning som kan jämföras med heltalens primtalsfak-torisering. På samma sätt som studiet av primtal utgör en viktig del avtalteorin är problemet att klassificera alla ouppdelbara representationercentralt inom representationsteorin.

Givet två representationer V och W av en grupp G kan man bil-da deras tensorprodukt V ⊗W , vilken återigen är en representation avG. Om V =

⊕mi=1 Ui och W =

⊕nj=1 U

′j är representationernas Krull-

Schmidtuppdelningar så är

V ⊗W =m⊕

i=1

n⊕j=1

Ui ⊗ U ′j

För att hitta Krull-Schmidtuppdelningen av V ⊗W räcker det med attkänna till uppdelningen av U ⊗U ′ för alla ouppdelbara representationerU och U ′. Att hitta dessa kallas för Clebsch-Gordanproblemet. Clebschoch Gordan stötte på ett liknande problem inom invariantteorin [4] vil-ket senare kunde tolkas som lösningen till Clebsch-Gordanproblemet förgruppen SL2. Problemet har tillämpningar i kvantmekanik där grupp-representationsteori används flitigt.

Om klassifikationsproblemet är löst kan man lösa Clebsch-Gordan-problemet på följande vis: för varje par av element ur den klassificerandelistan anges en formel för uppdelning av deras tensorprodukt, i en direktsumma av ouppdelbara representationer från listan.

På senare tid har representationsteorin spritt sig från att bara behand-la grupper till ett flertal typer av algebraiska objekt, som till exempel ko-ger, vilka introducerades av Gabriel [7]. Ett koger Q består av en mängdpunkter Q0 och en mängd pilar Q1, varje pil α ∈ Q1 har en startpunktx och en slutpunkt y vilket betecknas x α→ y.

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En representation V av Q består av en samling vektorrum V (x) ochlinjära avbildningar

V (α) : V (x)→ V (y),

där α ∈ Q1 är sådan att x α→ y och x ∈ Q0.Låt V och W vara representationer av ett koger Q. Tensorproduk-

ten V ⊗W definieras punktvis och pilvis, det vill säga (V ⊗W )(x) =V (x)⊗W (x) för varje x ∈ Q0 och (V ⊗W )(α) = V (α)⊗W (α) för varjeα ∈ Q1. Precis som för grupprepresentationer har varje kogerrepresen-tation en Krull-Schmidtuppdelning och vi kan därför studera Clebsch-Gordanproblemet.

Koger kan delas in i tre klasser. Den första är Dynkinkogren vil-ka Gabriel visade är precis de för vilka det finns ändligt många iso-morfiklasser av ouppdelbara representationer. Lösningen till Clebsch-Gordanproblemet kan i detta fall presenteras som en ändlig lista medformler. Nästa klass är de utökade Dynkinkogren. De har oändligt mångaisomorfiklasser av ouppdelbara representationer, men klassifikationspro-blemet är löst. Övriga koger kallas vilda och klassifikationsproblemet ärännu olöst för samtliga av dem.

Lösningen till Clebsch-Gordanproblemet framställd som en lista avformler blir ofta svåröverskådlig. För att få mer kvalitativ beskrivningav lösningen introduceras representationsringen, vilken har lösningen tillClebsch-Gordanproblemet kodad i sin multiplikativa struktur.

I denna avhandling löses Celbsch-Gordanproblemet för alla Dynkin-koger av typ A, D och E6 genom att explicita beskrivningar av repre-sentationsringen ges. Lösningarna byggs upp successivt så att lösningentill fallet A används för att lösa delar av fallet D. Den resterande delenlöses sedan med hjälp av matrisberäkningar. Alla dessa resultat användssedan för att lösa delar av fallet E6, men återigen så måste en betydligdel hanteras med hjälp av matrisberäkningar.

Det finns även en skiss på hur dessa metoder kan användas för att lösade resterande fallen E7 och E8. Det bör dock poängteras att arbetsbördansom krävs för att genomföra de delar som tidigare hanterades med hjälpav matrisberäkningar ökar kraftigt i varje steg.

I avhandlingen behandlas även två tama fall, närmare bestämt utö-kade Dynkinkoger av typ An och det dubbelöglekogret med relationernaαβ = βα = αn = βn = 0. I båda dessa fall finns det oändlig mångaisomorfiklasser av ouppdelbara representationer. För att lösa Clebsch-Gordanproblemet måste dessa därför parametriseras på något sätt. Detär avgörande att parametriseringen är kompatibel med tensorproduktenför att en överskådlig lösning ska kunna uppnås.

I artikel I löses Clebsch-Gordanproblemet för koger av typ An. Häranvänds den klassifikation av ouppdelbara representationer som ges avsträngar och band. Den visar sig vara bättre anpassad för Clebsch-

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Gordanproblemet än till exempel klassifikationen som ges av Auslander-Reitenteori.

Strängar och band är ett specialfall av en klassifikation som ges avövertäckningsteori. I artikel III undersöks sambandet mellan övertäck-ningar och den punktvisa tensorprodukten. Resultaten används för attåterskapa lösningen till Clebsch-Gordanproblemet för koger av typ An.De använd även för att lösa Clebsch-Gordanproblemet för dubbelögle-kogret med relationerna αβ = βα = αn = βn = 0.

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Acknowledgements

First of all I would like to thank my supervisor Ernst Dieterich for hisunending encouragement and good advice. I am grateful to the GraduateSchool in Mathematics and Computing (FMB) for the financial support Ihave received. Many thanks goes out to my colleagues at the Departmentof Mathematics in Uppsala, in particular Erik Darpö and Erik Melin forthe interesting discussions about mathematics. I am indebted to LisaHellman for her support and understanding. Finally I would like tothank my family and friends.

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