Representations of quivers and their applications.

3 lectures by

Jerzy Weyman

1. The basics of quiver representations.

The theory of quiver representations took off in early seventies after Gabriel Theoremindicated deep connections with representations of Lie algebras and root systems. Theprincipal results were obtained by Gabriel, Kac, Auslander, Reiten, Ringel, Schofield andCrawley-Boevey.

Definition 1.1. A quiver Q is a pair Q = (Q0, Q1) consisting of the finite set of vertices

Q0 and the finite set of arrows Q1. The set Q0 usually will be identified with the set

[1, n] = 1, . . . , n. The arrows will be denoted by initial letters of the alphabet. Each

arrow a has its head ha and tail ta, both in Q0;

taa−→ha

Formally both h and t are just maps from Q1 to Q0.

We fix an algebraically closed field K.

Definition 1.2. A (finite dimensional) representation V of Q is a family of finite dimen-

sional K-vector spaces V (x);x ∈ Q0 and of K-linear maps V (a) : V (ta) → V (ha).A morphism f : V → V ′ of two representations is a collection of K-linear maps f(x) :V (x) → V ′(x);x ∈ Q0, such that for each a ∈ Q1 we have f(ta)V ′(a) = V (a)f(ha). This

in fact means that the following diagrams commute

V (ta)V (a)−→ V (ha)

f(ta) ↓ ↓ f(ha)

V ′(ta)V ′(a)−→ V ′(ha)

We denote the linear space of morphisms from V to V ′ by HomQ(V, V ′). Obviouslythe representations of a quiver Q over K form a category denoted RepK(Q).

Let us consider some basic examples illustrating the meaning of representations.

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Example 1.3. The quiver Aeq2 is 1 a−→2. The category RepK(θ(1)) can be identified with

the category of linear maps V (1)V (a)−→V (2). If V (a) : V (1) → V (2) and W (a) : W (1) →

W (2) are two linear maps then their morphism is a pair of linear maps f(i) : V (i)→W (i),i = 1, 2, such that the diagram

V (1)V (a)−→ V (2)

f(1) ↓ ↓ f(2)

W (1)W (a)−→ W (2)

commutes.

Example 1.4. The one loop quiver σ(1) with σ(1)0 = 1, σ(1)1 = a with ta = ha =1. The category RepK(Q) can be identified with the category of endomorphisms of K-

vectorspaces.

Example 1.5. More generally, the m-loop quiver σ(m) has one vertex, σ(m)0 = 1 and

m arrows a1, . . . , am with tai = hai = 1.

Example 1.6. The m-subspace quiver ξ(m) has m+1 vertices, ξ(m)0 = 1, 2, . . . ,m+1and with m arrows a1, . . . , am with tai = i, hai = m+ 1.

Example 1.7. The equioriented An quiver Aeqn

Aeqn : 1→ 2→ 3→ . . .→ n− 1→ n.

We denote the arrow i→ (i+ 1) by ai.

In order to obtain first properties of categories RepK(Q) we identify them as modulecategories. We start with some preparatory definitions.

A path p in Q is a sequence of arrows p = a1, . . . , am such that hai = tai+1 (1 ≤ i ≤m− 1). We define tp = ta1, hp = ham. For each x ∈ Q0 we also include in our definitiontrivial path e(x) from x to x. For x, y ∈ Q0 we define [x, y] to be the K-vector space onthe basis of paths from x to y.

Definition 1.7. The path algebra KQ of a quiver Q is a K-algebra with a basis labeled

by paths in Q. We denote by [p] the element in KQ corresponding to the path p in Q. We

define the multiplication by the formula

[p] [q] =

[pq], if tp = hq;0, otherwise.

If p is a path with tp = x, hp = y, this includes p e(x) = p, e(y) p = p.

Notice that the trivial paths e(x) (x ∈ Q0) form an orthogonal set of idempotents inQ.

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Example 1.3.a). The algebra Kθ(1) is 3-dimensional over K. The basis consists of

[e(1)], [e(2)], [a] with the multiplication table given by formulas [e(i)]2 = [e(i)] for i = 1, 2,

[e(1)][e(2)] = [e(2)][e(1)] = 0, [a] = [e(2)][a] = [a][e(1)], [a][e(2)] = [e(1)][a] = 0.

Example 1.4.a). The algebra Kσ(1) is a polynomial ring in one variable over K, gener-

ated by [a].

Example 1.5.a). The algebra Kσ(m) is a free polynomial algebra K < [a1], . . . , [am] >on m noncommuting variables.

Proposition 1.8. The categories RepK(Q) and KQ−Mod are equivalent.

Proof. Let us define two functors giving the equivalences. The functor F : KQ−Mod→RepK(Q) is defined as follows. For a KQ-module M we define the vector spaces FM(x) :=e(x)M for x ∈ Q0. It is clear that if a ∈ Q1 then the multiplication by a sends FM(ta)ino FM(ha) because ae(ta) = e(ha)a. So the set of spaces FM(x) and of linear mapsFM(a) : FM(ta)→ FM(ha) given by the action of a define a representation FM of Q.

To define the inverse G : RepK(V ) → KQ −Mod we notice that we can take GV =⊕x∈Q0V (x) as a vector space and we can define the left action of a to be zero on all spacesV (x) with x 6= ta, and to be applying V (a) on V (ta). It is easy to check that this indeeddefines a KQ-module GV and that the functors F,G define the equivalence of categories.•

Throughout these notes we will not distinguish between the representations of Q andKQ-modules.

Definition 1.9. Representation V is indecomposable if it cannot be expressed as a proper

direct sum, i.e. from the decomposition V = V ′ ⊕ V ′′ it follows that either V ′ or V ′′ is

zero.

Corollary 1.10.(Krull-Remak-Schmidt Theorem). Every finite dimensional repre-

sentation V of a quiver Q is isomorphic to a direct sum of indecomposable representations.

This decomposition is unique up to isomorphism and permutation of factors. More pre-

cisely, if

V ∼= V1 ⊕ . . .⊕ Vm = W1 ⊕ . . .⊕Wp

with Vi, Wj nonzero and indecomposable, then m = p and there exists a permutation σ

on m letters such that for each i, 1 ≤ i ≤ m we have Vi∼= Wσ(i).

The dimension vector of a representation V is the function d(V ) defined by d(V )(x) :=dim V (x). The dimension vectors lie in the space Γ(Q) = ZQ0 of integer valued functionson Q0. The dimension vectors corespond to a sublattice Γ+(Q) in Γ(Q) of functions withnonnegative values.

We can look at a problem of classifying representations of the quiver Q of dimensionvector α ∈ Γ+(Q) as a problem of classifying orbits of a group action. More precisely tha

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space RepK(Q,α) of representations of Q of dimension vector α can be identified with theaffine space

RepK(Q,α) =∏

a∈Q1

HomK(Kα(ta),Kα(ha)).

The product of general linear groups GL(Q,α) =∏

x∈Q0GL(α(x),K) acts on RepK(Q,α)

in a natural way (by changes of bases in spaces Kα(x)). It is clear that two representationsV, V ′ ∈ RepK(Q,α) are isomorphic if and only if they belong to the same orbit of thisaction.

Sometimes we will use more equivariant notation. We will fix the vector spaces V (x)of dimensions α(x) for x ∈ Q0, and then

RepK(Q,α) =∏

a∈Q1

HomK(V (ta), V (ha)), GL(Q,α) =∏

x∈Q0

GL(V (x)).

Ideally the representation theory should give the classification of all isomorphismclasses of representations of a quiver Q. This is possible for some simple quivers.

Example 1.3b). The classification of isomorphism classes of linear maps amounts to

classifying the m × n matrices up to row and columns operations. It is known from

the basic linear algebra that the only invariant of matrices is rank. Let us look at this

classification from the point of view of representations. The rank classification means

that every representation of dimension vectoer (m,n) is a direct sum of three types of

indecomposable representations: ε1 : K → 0, ε2 : 0 → K and ε1,2 : K → K (with the

identity map). The m× n matrix has rank r when the summand ε1,2 occurs r times in its

decomposition.

Example 1.4b). The classification of isomorphism classes is equivalent to the classifica-

tion of endomorphisms of a vector space up to conjugation. This is a Jordan classification.

In terms of indecomposable objects this means that in dimension vector (n) there is a one

parameter family of indecomposable endomorphisms Jn(λ) given by an n× n matrix

Jn(λ) =

λ 1 0 . . . 0 00 λ 1 . . . 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . λ 10 0 0 . . . 0 λ

.

Example 1.5b). The classification of isomorphism classes is the classification of m-tuple

of endomorphisms of a vector space up to simultaneous conjugation. This is a well known

“wild” problem which seems to be very difficult.

Example 1.6b). The representation of m-subspace quiver ξ(m) with injective arrows

gives an m-tuple of subspaces V (ai)(V (i)) in the vector space V (m+ 1), for i = 1, . . . ,m.

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The classification of indecomposable representations with injective arrows amounts to clas-

sifying the m-tuples of subspaces up to simultaneous change of basis.

Example 1.7b). A representation of Aeqn is a sequence of linear maps

V (1)→ V (2)→ V (3)→ . . .→ V (n− 1)→ V (n).

A path p = a1 . . . an is an oriented cycle if ha1 = tan. Sometimes we will be makingan assumption that the quiver Q has no oriented cycles.

Proposition 1.11. Quiver Q has no oriented cycles if and only if the algebra KQ is finite

dimensional over K.

The basic result in theory of quiver representations is Gabriel Theorem classifying thequivers Q of finite representation type.

Definition 1.12. A quiver Q has finite representation type if the category RepK(Q)contains finitely many isomorphism classes of indecomposable objects. For a quiver Q we

define the graph Γ(Q) to be the graph we get when we forget the orientation of arrows in

Q.

Theorem 1.13 (Gabriel, 1973). A quiver Q has finite representation type if and only

if the graph Γ(Q) is a Dynkin quiver ,i.e. it is on the following list.

An : x1—x2— . . .—xn

Dn :x1 — x2 — . . . — xn−2 — xn−1

|xn

E6 :x1 — x2 — x3 — x4 — x5

|x6

E7 :x1 — x2 — x3 — x4 — x5 — x6

|x7

E8 :x1 — x2 — x3 — x4 — x5 — x6 — x7

|x8

.

In particular the representation finite type property does not depend on the orientation of

the arrows.

In fact Gabriel result gives more detailed information about the indecomposable rep-resentations for Dynkin quivers. Actually if Q is a Dynkin quiver Gabriel also proved that

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the number of indecomposable representations is equal to the number of positive roots,and their dimension vectors are the positive roots written in terms of simple roots. Thuseven the dimension vectors of indecomposables do not depend on the orientation.

Let us look at some examples.

Example 1.7c). Let Q = Aeqn . Denote by ei the dimension vector which is 1 at the vertex

i and 0 otherwise. For each 1 ≤ i ≤ j ≤ n define the representation Ei,j with dimension

vector εi,j = ei +ei+1 + . . .+ej by setting the maps between one dimensional vector spaces

to be the identity maps. Thus Ei,j is the representation

. . . 0 → K → K → . . . → K → 0 → . . .i i+ 1 j

.

There are(n+1

2

)of such representations. Looking at any book in Lie algebras you will see

that the simple positive roots for type An are the roots αi = εi − εi+1 and positive roots

are εi,j = εi − εj . Thus εi,j = αi + αi+1 + . . .+ αj . So these are all indecomposables.

Example 1.6c). Let us look at the 3-subspace quiver ξ(3). This is a quiver of type D4.

If you look in the book on Lie algebras you will see that the root system of type D4 has 12positive roots. So let’s try to find out the 12 indecomposable representations. Here they

are: all connected subgraphs of Γ(Q) give us an indecomposable representation with the

dimension vector being the characterisitc function of this subgraph (with the identity maps

between one dimensional spaces everywhere). This gives 4 representations for subgraph

with one vertex, 3 representations for subgraphs with two vertices, 3 representations for

subgraph with three vertices and one representation for subgraph with 4 vertices. This

gives 4+3+3+1 = 11 indecomposable representations. The remaining one (after looking

onto a book on root systems) has to have dimension vector

1 2 11 .

Thus we can take the representation

K

(10

)−→ K2

(01

)←− K

↑(

11

)K

to be the twelfth indecomposable.

Gabriel Theorem was generalized to arbitrary quivers by Kac ([K1], [K2], 1980), whoproved that indecomposable representations occur in dimension vectors that are the roots

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of so-called Kac-Moody Lie algebra associated to a given graph. In particular, thesedimension vectors do not depend on the orientation of the arrows in Q.

Of particular interest are so-called tame quivers. For these in each dimension vec-tor there is at most one parameter family of indecomposable representations. Nazarovaand Roiter (see also Donovan and Freislich (1976)) showed that the quiver Q is of tamerepresentation type if and only if Γ(Q) is the extended Dynkin quiver.

Remarks on the proof of Gabriel Theorem.

It is not difficult to show that if Q is of finite representation type, then Γ(Q) isDynkin. In fact, suppose this is true. Consider a dimension vector α and the action ofthe group GL(Q,α) on RepK(Q,α). If RepK(Q) has finitely many isomorphism classes ofindecomposable objects then this action has finitely any orbits. Thus one of these orbitshas to be open, so in particular

dim GLK(Q,α) ≥ dim RepK(Q,α).

In fact the inequality is sharp since the scalar subgroup consisting of the homothety by thesame scalar at each vertex takes every representation to an isomorphic one. We introducea quadratic form (so-called Tits form)

TQ(α) =∑

x∈Q0

α2(x)−∑

a∈Q1

α(ta)α(ha).

Notice that TQ does not depend on the orientation of Q, so in fact it depends only onΓ(Q). We might denote it by TΓ.

Thus if Q is of finite representation type then the quadratic form TΓ(Q) has to bepositive definite. Looking into Humphreys book on Lie algebras ([Hu]) we see that thisproperty characterizes Dynkin graphs (in the context of Lie algebras the quadratic formcomes from the Killing form).

Thus we established that if Q is of finite representation type, then it has to be Dynkin.The proof that every Dynkin quiver is of finite representation type is best understood

in terms of Auslander-Reiten techniques. Let us consider the case of equioriented quiverAeq

n (Example 1.7). We know that indecomposable representations are the representationsEi,j for 1 ≤ i ≤ j ≤ n. Let us investigate the maps between these representations. It isclear that there is a non-zero map f : Ei,j → Ek,l if and only if k ≤ i ≤ l ≤ j, and its imageis Ei,l. Thus evey map which is not an isomorphism can be written as a composition ofinjections Ei,j → Ei−1,j or surjections Ei,j → Ei,j−1. These are so-called irreducible mapsbetween indecomposable modules. They can be fit into the so-called Auslander-Reitenquiver, which, in case of Aeq

4 looks like this.

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E1,4

E2,4 E1,3

E3,4 E2,3 E1,2

E4,4 E3,3 E2,2 E1,1

This AR quiver has the following features:• Each square Ei,j → Ei−1,j⊕Ei,j−1 → Ei−1,j−1 and triangles Ei,i → Ei−1,i → Ei−1,i−1

give exact sequences of representations (almost split sequences),• There is a translation τ sending Ei,j to Ei+1,j+1 which preserves the structure of the

AR quiver except at the ends (it takes value zero at the modules Ei,n). In fact thetranslation τ can be defined as a functor on RepK(Q). The modules Ei,n are theindecomposable projective representations (see below). We also have the translationτ−1 taking Ei,j to Ei−1,j−1. This takes value zero at the modules E1,i (the injectiveindecomposables),• The left end of the almost split sequence having Ei,j at the right end is τ(Ei,j). The

right end of the almost split sequence having Ei,j at the left end is τ−1(Ei,j).The AR quiver allows to construct the indecomposables systematically. We start

with the indecomposable injectives, which correspond to the vertices of the quiver Q (withirreducible maps between them corresponding to arrows in Q) and we apply the translationfunctor τ repeatedly to get all the indecomposables.

Example 1.6.c). Here are the AR quivers of the 2 and 3-subspace quivers ξ(2) and ξ(3).Instead of indecomposables we write their dimension vectors

ξ(2) : 1→ 3← 2.

AR(ξ(2)) :

110 001

010 111

011 100

.

ξ(3) :1 → 4 ← 2

↑3

The quiver AR(ξ(3)) is

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1 1 00

0 1 11

1 0 00

0 1 0

0 → 0 1 10 → 1 2 1

1 → 1 1 01 → 1 1 1

1 → 0 0 10

0 1 0

11 1 1

00 0 0

1

Again the translation functors τ and tau−1 are just the right and left translation in theAR graph.

2. The Klyachko inequalities and quiver representations.

Let us start with four problems seemingly unrelated to each other.

1) Let A,B,C be abelian p-groups. We can associate to A,B,C three partitions λ, µ, νsuch that

A = ⊕iZ/pλiZ, B = ⊕iZ/pµiZ, C = ⊕iZ/pνiZ

with λ1 ≥ λ2 ≥ . . . ≥ λn, µ1 ≥ µ2 ≥ . . . ≥ µn, ν1 ≥ ν2 ≥ . . . ≥ νn. The question is:what conditions for λ, µ, ν need to be satisfied to have an exact sequence

0→ B → C → A→ 0.

In fact one might introduce the coefficient eνλ,µ which is the number of subgroups of

type µ inside of an abelian group C of type ν with factor of type λ. Then the questionbecomes when do we have eν

λ,µ > 0.

2. Let A,B,C be n × n Hermitian matrices. They have real eigenvalues that can bearranged in non-increasing order. Let eigenvalues of A be λ1 ≥ λ2 ≥ . . . ≥ λn, theeigenvalues of B be µ1 ≥ µ2 ≥ . . . ≥ µn and let the eigenvalues of C be ν1 ≥ ν2 ≥. . . ≥ νn. What are the conditions for λ, µ, ν when C = A+B ?

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3. Irreducible representations of SL(nC) are parametrized by the highest weights λ =(λ1, . . . , λn) with λ1 ≥ λ2 ≥ . . . ≥ λn = 0. For which λ, µ, ν the representation V (ν)occurs in the tensor product V (λ) ⊗ V (µ). In fact here again we can introduce thecoefficients cνλ,µ which are the multiplicities of V (ν) in V (λ)⊗ V (µ). Then again thequestion is when cνλ,µ > 0 ?

4. Schubert calculus. Let X = Grass(n,Cm). This is a complex manifold of dimensionn(m− n). Let us fix a basis e1, . . . , em of Cm and let us fix a flag

0 = F0 ⊂ F1 ⊂ . . . ⊂ Fm−1 ⊂ Fm = Cm

where Fi is the span of e1, . . . , ei. Now, for a sequence of numbers P = (p1, . . . , pn)such that 1 ≤ p1 < p2 < . . . < pn ≤ m we denote

ΩP (F ) = L ∈ X | dim(L ∩ Fpi) ≥ i for i = 1, . . . , n.

The subset ΩP (F ) is closed, irreducible, of dimension∑n

i=1(pi − i). For a partitionλ such that m − n ≥ λ1 ≥ . . . ≥ λn ≥ 0 we denote by |λ| =

∑i λi. We define

σλ ∈ H2|λ|(X) the cohomology class whose cap product with the fundamental classof X gives ωP = [ΩP (F )], where pi = m − n + i − pi for i = 1, . . . , n. Thus identifyhomology and cohomology via Poincare duality we have

σλ = ωP

where λi = m− n+ i− pi (1 ≤ i ≤ n).The classes σλ give a Z-basis of H∗(X) so we have (the nonnegative) coeeficients dν

λ,µ

such thatσλσµ =

∑ν

dνλ,µσν .

again we can ask when dνλ,µ > 0.

The link between the four problems is that they all have the same solution. In factthe link between 1), 3) and 4) was known some time ago, because the coefficients cνλ,µ,dν

λ,µ, eνλ,µ are all equal. They are given by the Litllewood-Richardson rule. In particular

they are zero unless |ν| = |λ|+ |µ|.The additional conditions were given conjecturally by Horn for 2). They are in the

form of inequalities ∑k∈K

νk ≤∑i∈I

λi +∑j∈J

µj (∗)I,J,K

where the subsets I, J,K of [1, n] have the same cardinality r (which can take any valuebetween 1 and n). See [F] for the history of these inequalities.

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Example 2.1. For n = 2 the inequalities are

ν1 ≤ λ1 + µ1, ν2 ≤ λ1 + µ2, ν2 ≤ λ2 + µ1

together with equality ν1 + ν2 = λ1 + λ2 + µ1 + µ2.

It remains to determine the set Tnr of triples (I, J,K) of subsets of cardinality r in

[1, n] which give needed inequalities.Horn’s conjecture was formulated as follows. One defines the set Un

r of triples (I, J,K)

Unr = (I, J,K), I, J,K ⊂ [1, n], |I| = |J | = |K| = r,

∑i∈I

i+∑j∈J

j =∑k∈K

k +r(r + 1)

2.

Then the set Tnr can be defined recursively as follows.

a) If r = 1 we have Tn1 = Un

1 ,b)

Tnr = (I, J,K) ∈ Un

r | ∀p < r,∀(F,G,H) ∈ T rp ,

∑f∈F

if +∑g∈G

jg ≤∑h∈H

kh +p(p+ 1)

2.

Horn conjectured in ([H], 1962) that the set of inequalities (∗)(I,J,K) with (I, J,K) ∈Tn

r gives (together with equality |ν| = |λ|+ |µ|) the solution of 2).Klyachko in ([K], 1998) proved the Horn conjecture by linking it to 1), 3) 4) (in fact

to 3)) and using the fact that the symplectic geometry quotients are the same as geometricinvariant theory quotients. See [F] for the details. However, the correspondance was notperfect. Klyachko proved in fact that

(I, J,K) ∈ Tnr ⇐⇒ ∃N ∈ Z+c

NνNλ,Nµ > 0.

Therefore in case 3) the problem that remained was whether the set of triples (λ, µ, ν)such that cνλ,µ > 0 is saturated, i.e. whether cNν

Nλ,Nµ > 0 for some N implies cνλ,µ > 0.Surprisingly the Littlewood-Richardson rule is not very helpful in proving such result. Thesaturation was proved by Knutson and Tao in ([KT], 1999) by combinatorial argument.For the remainder of this lecture we discuss how this result is a special case of a generalresult on semi-invariants of quiver representations.

Let Q = (Q0, Q1) be a quiver. We assume that Q has no oriented cycles, i.e. pathswith the same head and tail. We briefly investigate the structure of projective and injectiveresolutions in RepK(Q).

First of all let us notice that the simple representations are in a bijection with ver-tices Q0. The simple representation Sx is just the representation for which Sx(x) is one

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dimensional and Sx(y) = 0 for x 6= y. For each vertex x ∈ Q0 we define an indecomposableprojective representation Px as follows. Let [x, y] denotes a vector space over K with abasis labelled by all paths from x to y in Q. We denote by [p] the basis element in [x, y]corresponding to the path p.

Px(y) = [x, y], Px(a) = a : [x, ta]→ [x, ha]

Here a denotes a linear map taking the basis element element [p] to [a p].

Proposition 2.2. The representations Px are indecomposable, projective. More precisely,

there is a functorial isomorphism

HomQ(Px, V ) = V (x),

and therefore the functor HomQ(Px,) is exact. Every indecomposable projective represen-

tation is isomorphic to Px for x ∈ Q0.

Proof. The equivalence HomQ(Px, V ) = V (x) takes a morphism φ to φ([e(x)]). The pointis that for every v ∈ V (x) there exists unique homomorphism φv : Px → V such thatφv([e(x)] = v. Indeed, we define φ([p]) := V (p)v. To prove that every indecomposableprojective module is isomorphic to some Px we recall that the module is projective if andonly if it is a direct summand of a free module. But considering KQ as a left module overitself, we notice that KQ = ⊕x∈Q0Px where we identify Px with KQex. Then the resultfollows from Corollary 1.11.•

Similarly we define the the indecomposable injective modules Qx. We set

Qx(y) = [y, x]∗, Qx(a) = (a)∗ : [ta, x]∗ → [ha, x]∗

where ∗ denotes te dual space, and a : [ha, x] → [ta, x] is a linear map taking basiselement [p] into [p a].

Proposition 2.3. The representations Qx are indecomposable, injective. More precisely,

there is a functorial isomorphism

HomQ(V,Qx) = V (x)∗,

and therefore the functor HomQ(−, Qx) is exact. Every indecomposable projective repre-

sentation is isomorphic to Qx for x ∈ Q0.

Exercise 2.3. Let Sx be a simple module corresponding to a vertex x ∈ Q0. There is a

natural map px : Px → Sx sending the generator [ex] to the element 1 ∈ K = Sx(x). Prove

that the kernel of this map is ⊕a∈Q1;ta=xPha. The resulting complex

0→ ⊕a∈Q1;ta=xPha → Px → Sx → 0

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gives a projective resolution of length 1 of a simple object Sx.

This suggests that for quivers without oriented cycles the category RepK(Q) is hered-itary. It turns out this result is true for general quiver.

Theorem 2.4. The category RepK(Q) is hereditary, i.e. the subrepresentation of the

projective representation is projective.

Theorem 2.4. follows from the following proposition

Proposition 2.5. The spaces HomQ(V,W ) and Ext1Q(V,W ) are the kernel (resp. coker-

nel) of the following map

dVW : ⊕x∈Q0HomK(V (x),W (x))→ ⊕a∈Q1HomK(V (ta),W (ha)).

The map dVW is given by the formula

dVW (φ(x))x∈Q0 = (φ(ha)V (a)−W (a)φ(ta))a∈Q1 .

Proof of Theorem 2.4 It follows from the proposition 2.5 and the snake lemma that thefunctor Ext1Q(−,W ) is right exact, which means that the ExtiQ(V,W ) = 0 for i ≥ 2. This,by standard results from homological algebra implies that the category is hereditary.•Proof of Proposition 2.5. The kernel of dV

W consists of families φ(x)x∈Q0 such that forevery a ∈ Q1 we have φ(ha)V (a) −W (a)φ(ta) = 0. This, by definition is HomQ(V,W ).Let us look at the cokernel of the map dV

W . It consists of the family of linear mapsψ(a) : V (ta) → W (ha) for every a ∈ Q1. To each such family we associate the extensionE of W by V . We define E(x) := V (x) ⊕W (x) and the differential E(a) is given in theblock form by the matrix (

V (a) 0ψ(a) W (a)

).

It is clear that the subspaces W (x)x∈Q0 form a subrepresentation of E and that thefactor is isomorphic to V . Thus we have an exact sequence

0→W → E → V → 0.

Let us see when the family ψ(a) defines a zero element in CokerdVW . This happens

precisely when we have a family of linear maps φ(x)x∈Q0 for which ψ(a) = φ(ha)V (a)−W (a)φ(ta) for all a ∈ Q1. We define the morphism

η : E → V ⊕W

given over the vertex x ∈ Q0 by the matrix

η(x) =(

1V 0−φ(x) 1W

).

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It follows from the conditions on φ and ψ that this is a morphism of representations. Thismorphism is obviously an isomorphism. We also have a commutative diagram of exactsequences

0 → W → E → V → 0↓ 1W ↓ η ↓ 1V

0 → W → V ⊕W → V → 0.

This process can be reversed and every η satisfying the commutative diagram above hasto be of the triangular form written above, with the linear maps map φ(x) appearing inthe lower left corner related to linear maps ψ(a) by the relation dV

W ([φ(x)) = ψ(a).Thus we identified Coker dV

W with Ext1Q(V,W ) in the Yoneda description.•We define the Euler characteristic χ(V,W ) = dim HomQ(V,W )− dim ExtQ(V,W ).

Corollary 2.6. The Euler characterstic of χ(V,W ) depends only on the dimension vectors

of V,W and it defines a nonsymmetric bilinear form (Euler form)

〈α, β〉 =∑

x∈Q0

α(x)β(x)−∑

a∈Q1

α(ta)β(ha).

Remark 2.7. Notice that the Tits form is just the symmetrization of the Euler form.

In dealing with quivers that are not Dynkin or extended Dynkin one needs to usegeometric methods, as the modules occur in more complicated families. One such approachcomes from Invariant Theory.

We know that when GL(Q, β) acts on Rep(Q, β) the orbits of this action give the iso-morphism classes of representations. However the scalar group K∗ ⊂ GL(Q, β) (consistingof identity matrices multiplied by the same nonzero scalar at each vertex) acts triviallyon the set of isomorphism classes. This means that it is more convenient to consider thesubgroup SL(Q, β) :=

∏x∈Q0

SL(β(x),K) of GL(Q, β).The action of GL(Q, β) on RepK(Q, β) induces the action of GL(Q, β) on the coordi-

nate ring of polynomial functions on RepK(Q, β) (which is just a polynomial ring generatedby the entries of all matrices in a representation) by the rule

(g f)(v) := f(g−1v).

The factor of Rep(Q, β) by the action of SL(Q, β) is described by the ring of SL-invariants

SI(Q, β) = f ∈ K[RepK(Q, β)] | ∀g ∈ SL(Q, β) g f = f

which is isomorphic to the ring of GL- semi-invariants, i.e.

SI(Q, β) = ⊕χ∈charGL(Q,β)SI(Q, β)χ

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whereSI(Q, β)χ = f ∈ K[RepK(Q, β)] | ∀g ∈ GL(Q, β) g f = χ(g)f .

The characters of the group GL(Q, β) can be identified with the space ZQ0 becausethe character group of GL(n,K) is isomorphic to Z. We treat characters of GL(Q, β) asthe functions on the dimension vectors. We refer to these characters as weights.

Let us explain one more point of view explaining our interest in semi-invariants. Sincethe scalar group K∗ acts trivially on isomorphism classes of representations it is natural toask about the factor of the projectivisation of P(RepK(Q, β)) by the action of GL(Q, β).According to Geometric Invariant Theory the way to do it is to start with the GL(Q, β)linearized line bundle. These bundles correspond to weights σ. Thus each weight σ allowsto construct a Geometric Invariant Theory quotient which is just Proj(SI(Q, β, σ) where

SI(Q, β, σ) := ⊕n≥0SI(Q, β)nσ.

This means that the investigation of rings of semi-invariants gives information aboutthe variation of this quotient with the weight σ. In particular finding the weights wheresemi-invariants appear tells us for which σ the quotient is nonempty.

Before we start let us look at some examples.

Example 2.8. Q = Aeq1 . Then easy calculation shows that if β(1) 6= β(2) then SI(Q, β) =

K. When β(1) = β(2) then SI(Q, β) = K[det V (a)], where det V (a) is a semi-invariant

sending representation V to the determinant of the map V (a). The weight of det V (a)is (1,−1) because it corresponds to the representation

∧nV (1) ⊗

∧nV (2)∗ inside of

K[RepK(Q, β)]. The weight σ is the first row for the Euler matrix

E =(

1 −10 1

)i.e. it can be written in the form 〈α,−〉 for α = (1, 0).

Example 2.9. More generally, let Q = Aeqn

Q : 1a1→2a2→ . . .an−1→ n

The only possible semi-invariants are the semi-invariants V 7→ det(V (aj−1 . . . V (ai)) of

weights σi,j := εi = εj for 1 ≤ i < j ≤ n. These are the weights σi,j = 〈αi,j ,−〉 where

αi,j = εi + . . .+ εj−1.

Example 2.10. Let Q = ξ(3). There are several possibilities for the semi-invariants:

If β(1) = β(4) we have the semi-invariant V 7→ det V (a) of weight (1, 0, 0,−1). There

are three symmetric possibilities. If β(1) + β(2) = β(4) we have the semi-invariant V 7→

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det (V (a), V (b)) of weight (1, 1, 0,−1). There are three symmetric possibilities. If β(1) +β(2) + β(3) = β(4) we have the semi-invariant V 7→ det (V (a), V (b), V (c)) of weight

(1, 1, 1,−1). Notice that the Euler matrix is

E =

1 0 0 −10 1 0 −10 0 1 −10 0 0 1

.

All the weights of semi-invariants are the positive sums of rows of E. This means these

weights are of the form 〈α,−〉 for some dimension vectors. These dimension vectors turn

out to be the dimensions of indecomposable representations for Q.

Moreover, if we write the weight σ in the form 〈α,−〉 we see that the semi-invariantsin the examples above occur only in weights for which α is a dimension vector of anindecomposable representation.

This turns out to be the general pattern. In fact it is easy to produce the semi-invariants of such weights.

Let us recall that in Proposition 2.5. we defined the linear map

dVW : ⊕x∈Q0HomK(V (x),W (x))→ ⊕a∈Q1HomK(V (ta),W (ha))

defined by the formula

dVW (φ(x))x∈Q0 = (φ(ha)V (a)−W (a)φ(ta))a∈Q1

whose kernel and cokernel were HomQ(V,W ) and Ext1Q(V,W ) respectively. Let us denoteα = d(V ), β = d(W ). Let us assume that 〈α, β〉 = 0. Then the linear map dV

W is a squarematrix. This allows to consider its determinant.

Let ux fix a representation V . We define the function cV : RepK(Q, β) → K definedby the formula

cV (W ) := det(dVW ).

Similarly, if we fix the representationW we can define the function cW : RepK(Q,α)→K by the formula

cW (V ) := det(dVW ).

Proposition 2.11.a) The function cV is a semi-invariant of weight 〈α,−〉, i.e. cV ∈ SI(Q, β)〈α,−〉,

b) The function cW is a semi-invariant of weight −〈−, β〉, i.e. cV ∈ SI(Q,α)−〈−,β〉,

c) If

0→ V ′ → V → V ′′ → 0

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is an exact sequence of representations, with α′ := d(V ′), α′′ := d(V ′′), then if

〈α′, β〉 = 0, then cV = cV′cV

′′. If 〈α′, β〉 > 0, then cV = 0,

d) If

0→W ′ →W →W ′′ → 0

is an exact sequence of representations, with β′ := d(V ′), β′′ := d(V ′′), then if 〈α, β′〉 =0, then cW = cW

′cW

′′. If 〈α, β′〉 < 0, then cW = 0.

Corrolary 2.12.a) If V = V ′ ⊕ V ′′ then cV is either a product cV

′cV

′′or is identacally zero,

b) If W = W ′ ⊕W ′′ then cW is either a product cW ′cW ′′ or is identically zero.

Corrolary 2.13 (Reciprocity). Let α and β be the dimension vectors satisfying 〈α, β〉 =0. Then

dim SI(Q, β)〈α,−〉 = dim SI(Q,α)−〈−,β〉.

Proof. Let V1, . . . , Vs be the representations of dimension α such that cV1 , . . . , cVs form abasis of SI(Q, β)〈α,−〉. These are linearly independent polynomial functions on Rep(Q, β),so there exist s represntations W1, . . . ,Ws from Rep(Q, β) such that det(cVi(Wj))1≤i,j≤s isnonzero. But cVi(Wj) = cWj (Vi). This proves that cW1 , . . . , cWs are linearly independenton Rep(Q,α). This proves

dim SI(Q, β)〈α,−〉 ≤ dim SI(Q,α)−〈−,β〉.

The other inequality is shown in the same way.•The main result on semi-invariants is

Theorem 2.14. Let Q be a quiver with no oriented cycles.

a) For any dimension vector γ and a weight σ the weight space SI(Q, γ)σ can be nonzero

only for weights satisfying σ(γ) = 0,

b) If the weight σ is not of the form σ = 〈α,−〉, for some dimension vector α, then

SI(Q, β)σ = 0. If sigma = 〈α,−〉 with 〈α, β〉 = 0, then SI(Q, β)σ is spanned as a

vector space by the semi-invariants cV for V ∈ RepK(Q,α),c) If the weight σ is not of the form σ = −〈−, β〉, for some dimension vector β, then

SI(Q, β)σ = 0. If sigma = −〈−, β〉 with 〈α, β〉 = 0, then SI(Q,α)σ is spanned as a

vector space by the semi-invariants cW for W ∈ RepK(Q, β).

Before we prove the theorem we give some consequences. The first application is thedescription of the cone of weights in which the semi-invariants occur.

Let Q be a quver with no oriented cycles and β - a dimension vector. We define thecone of weights

Σ(Q, β) = σ ∈ Char(GL(Q, β) | SI(Q, β)σ 6= 0.

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We also define the generic hom, ext and the generic subrepresentation. Let Q be aquiver, α, β two dimension vectors. Notice that the sets

Zt0(α, β) = (V,W ) ∈ Rep(Q,α)×Rep(Q, β) |dimHomQ(V,W ) ≥ t

Zt1(α, β) = (V,W ) ∈ Rep(Q,α)×Rep(Q, β) |dimExtQ(V,W ) ≥ t

are closed, in fact they are defined by the determinantal condition for the matrix dVW .

Thus on the open set of Rep(Q,α) × Rep(Q, β) the dimensions of HomQ(V,W ) andExtQ(V,W ) are constant. We define homQ(α, β) and extQ(α, β) to be the generic val-ues of HomQ(V,W ) (resp. ExtQ(V,W )) on RepQ(Q,α)× RepQ(Q, β). We refer to themas generic hom and generic ext respectively.

We say that a dimension vector β′ is a sub-dimension vector of β (notation: β′ → β)if every representation from Rep(Q, β) has a subrepresentation of dimension vector β′.Similarly, we say that a dimension vector β′′ is a factor-dimension vector of β (notation:β′ →> β)if every representation of dimension vector β has a factor representation ofdimension vector β′′.

The connection between these notions is expressed by the following result.

Theorem 2.15 (Schofield). Let α, β be two dimension vectors for the quiver Q without

oriented cycles.

a) extQ(α, β) = 0 if and only if α → α+ β,

b) extQ(α, β) 6= 0 if and only if β′ → β and 〈α, β − β′〉 < 0 for some dimension vector

β′.

The main consequence of Theorem 2.14 is the explicit description of Σ(Q, β) by meansof one homogeneneous linear equality and homogeneous linear inequalities.

Theorem 2.16. The set Σ(Q, β) is a rational polyhedral cone described as follows

Σ(Q, β) = σ = 〈α,−〉 | 〈α, β〉 = 0,∀β′ → β, 〈α, β′〉 ≤ 0.

In particular Σ(Q, β) is saturated, i.e. nσ ∈ Σ(Q, β) for some n ∈ Z+ implies σ ∈ Σ(Q, β).

Remark 2.17.a) Theorems 2.14 implies that the ring SI(Q, β) is generated (as an K-algebra) by the

semi-invariants cV where V is an indecomposable representation of Q with 〈d(V ), β〉 =0.

b) In Theorem 2.16. it is enough to consider the subrepresentations β′ → β for which

general representation of dimension β′ is indecomposable. With this restriction still a

lot of inequalities σ(β′) ≤ 0 are redundant.

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Now we apply Theorem 2.16 to the problem 3). Take the quiver Q to be the followingquiver

x1a1→ x2

a2→ . . . xn−1an−1→ u

bn−1← yn−1 . . .b2← y2

b1← y1↓ cn−1

zn−1

↓ cn−2

. . .z2↓ c1z1

and the dimension vector β(xi) = β(yi) = β(zi) = i, β(u) = n. A standard calculationfrom representation theory shows that the ring SI(Q, β) can be described as

SI(Q, β) = ⊕λ,µ,ν(V (λ)∗ ⊗ V (µ)∗ ⊗ V (ν))SL(n,C).

with the space (V (λ)∗ ⊗ V (µ)∗ ⊗ V (ν))SL(n,C) occurring in the weight σ with σ(xi) =λi − λi−1, σ(yi) = µi − µi−1, σ(zi) = −νi + νi−1, σ(u) = 0. Here SL(n,C) is the speciallinear group at the central vertex u. Notice that the space of invariants (V (λ)∗⊗ V (µ)∗⊗V (ν))SL(n,C) is non-zero if and only if V (ν) occurs in the tensor product V (λ) ⊗ V (µ).Thus Theorem 2.16 implies that the set of triples (λ, µ, ν) such that V (ν) occurs in thetensor product V (λ)⊗ V (µ) is saturated.

3. The geometry of orbit closures for the Dynkin quivers.

We are interested in orbit closures of representations of Dynkin quivers. The archetypecase are the orbit closures for the orbits of the quiver Aeq

1 .

Example 3.1. Q : 1→ 2. For a fixed dimension vector β = (n,m) the space RepK(Q, β)is just a set of m× n matrices where the product of two general linear groups acts by row

and column operations. The orbits are the matrices of fixed rank. This means that the

orbit closures are just determinantal varieties

Xr = φ : Kn → Km | rank(φ) ≤ r.

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The determinantal varieties have many nice properties which are interesting for commu-

tative algebraists. In particular:

• They are normal. This means the coordinate rings K[Xr] are integrally closed in their

fields of fractions.

• They are Cohen-Macaulay. This means by cutting by hypersurfaces we can reduce

the study of Xr to the study of zero dimensional schemes (i.e. the coordinate ring

K[Xr] divided by well chosen regular sequence is finite dimensional over K).

• They have rational singularities. This means that on the desingularization of Xr

the higher cohomology of the structure sheaf vanishes. This property implies Cohen-

Macaulay. In particular the varieties Xr have two nice desingularizations

Zr = (φ,R) ∈ HomK(Kn,Km)×Grass(n− r,Kn) | R ⊂ Ker(φ),

Z ′r = (φ, S) ∈ HomK(Kn,Km)×Grass(r,Km) | S ⊃ Im(φ)

for which one can prove rational singularities rather easily.

• The determinantal varieties are Gorenstein (meaning Cohen-Macaulay plus selfduality

of free resolution ofK[Xr] as a module over the coordinate ring of RepK(Q, β)) if r = 0or if r ≥ 1 and m = n.

In recent years there have been important progress made in the direction of general-izing these properties to orbit closures of representations of Dynkin quivers.

There are even more basic algebraic questions: Let me list some of them:1) When is one orbit contained in the closure of another orbit ?2) What are the defining ideals of the orbit closures (in the case of determinantal variety

this is the ideal of (r + 1)× (r + 1) minors of a generic m× n matrix).Let us employ the following notation. For a representation V of dimension vector β

we denote by OV its orbit in RepK(Q, β). We also denote Ind(Q) the set of isomorphismclasses of indecomposable representations of Q.

Question 1) was settled by Bongartz.

Theorem 3.2. Let V,W be two representations of dimension vector β of a Dynkin quiver

Q. Then

OV ⊂ OW ⇐⇒ ∀U ∈ Ind(Q) dim(HomQ(U, V )) ≥ dim(HomQ(U,W )).

This means we can give set-theoretic conditios to be in the orbit closure OV . LetUα be an indecomposable representation corresponding to the positive root α. Let rα =dim(HomQ(Uα, V ). Then

OV = W |∀α dim(HomQ(Uα,W ) ≥ rα.

This means set-theoretically the orbit OV is given by the vanishing of rα + 1 sizeminors of the matrix duα

W .

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Example 3.3. Let Q = Aeqn . Then the rank conditions are just conditions for the rank

ri,j of the compositions V (i) → V (j) for each 1 ≤ i ≤ j ≤ n. Here ri,i := dim(V (i)).We can determine the representation V by multiplicities of the indecomposable summands,

V = ⊕mi,jEi,j or by the ranks ri,j of the maps V (i)→ V (j). These two types of conditions

easily translate to each other. We have

ri,j =∑

k≤i≤j≤l

mk,l,mi,j = ri,j − ri,j+1 − ri−1,j + ri−1,j+1.

The last equation tells us the nessesary conditions for the set of ranks ri,j actually occur-

ring. The condition in terms of mi,j ’s is obviously mi,j ≥ 0.

The orbit closures of Dynkin quivers contain many interesting varieties. For examplethe Schubert varieties in partial flag avrieties are among them.

Example 3.4. Let us consider the partial flag variety Flag(r1, r2, . . . , rs;Kn) of flags

0 = F0 ⊂ F1 ⊂ . . . ⊂ Fs ⊂ Kn

with dim(Fi) = ri. The Schubert varieties are the orbits of the group B of upper-triangular

matrices in GL(n,C) acting on Flag(r1, . . . , rs;Kn) (we identify Flag(r1, . . . , rs;Kn) with

GL(n,C)/P (r1, . . . , rs)). We take a quiver

Q : x1 → x2 → . . . xn−1 → u← ys ← . . .← y2 ← y1

and the dimension vectors β(xi) = i, β(u) = n, β(yj) = rj . Then the intersection of the

orbit closure with the open set of representations with all linear maps being injective gives

the fibered product Y ×BGl(n,C) for some Schubert variety Y . All Schubert varieties can

be obtained in this way. Thus the study of the singularities of Schubert varieties of type

An is part of the study of the singularities of orbit closures for quivers of type An.

Remark 3.5. For some time in the seventies there was hope that among different Dynkin

quivers one would find the singularities of types An, Dn, E6,7,8 as defined in singularities

theory. This did not materialize, Bongartz showed that the minimal degenerations for all

quivers of Dynkin type are the same (m× n matrices of rank ≤ 1 at the origin).

Here are the results known for orbit closures of Dynkin quivers right now.

Theorem 3.6. (Abbeassis, Del Fra, Kraft [ADK], 1982, characteristic zero case; Laksh-

mibai, Magyar [LM], 1999, general case) If Q = Aeqn then all orbit closures are Cohen-

Macaulay and the rank conditions generate the defining ideals. If Char(K) = 0 all these

orbit closures have rational singularities.

The method of Lakshmibai and Magyar is to exhibit all orbit closures as open subsetsof Schubert varieties of type An and use the conclusions of standard monomial theory.

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Theorem 3.7. (Bobinski, Zwara). If Q is a quiver of type An then all orbit closures are

Cohen-Macaulay and the rank conditions generate the defining ideals. If Char(K) = 0 all

these orbit closures have rational singularities.

This result is based on the general result of Zwara on the representations of Artinalgebras.

Theorem 3.8 (Zwara). Let φ : A → B be a homomorphism of Artin algebras. Let

φ : B −mod→ A−mod be the natural functor of the restriction of scalars. Assume that

φ is

a) exact,

b) Hom-controlled, i.e. there exists a bilinear form ξ : K0(A)×K0(A)→ Z such that

dim(HomA(φ(V ), φ(W ))− dim(HomB(V,W )) = ξ([V ], [W ]).

Then the singularities of OV and of Oφ(V ) are smoothly equivalent. In particular

Cohen-Macaulay, rational singularities properties are preserved.

In the case B = KQ, A = KQ′ Bobinski and Zwara construct needed functorφ for allembeddings of AR quiver of Q into the AR quiver of Q′. Thus to get the orbit closures forquiver of type An with strange orientation, we imbed its AR-quiver into the AR quiver ofbigger equioriented An quiver.

Example 3.9.

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