SEMI-INVARIANTS OF QUIVERS AND SATURATION OF …€¦ · of some quiver, with some relations. We...

SEMI-INVARIANTS OF QUIVERS AND SATURATION OF LITTLEWOOD-RICHARDSONCOEFFICIENTS

BY

AMELIE SCHREIBER

A Thesis Submitted to the Graduate Faculty of

WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES

in Partial Fulfillment of the Requirements

for the Degree of

MASTER OF ARTS

Mathematics

May 2015

Winston-Salem, North Carolina

Approved By:

Ellen Kirkman, Ph.D., Advisor

Jeremy Rouse, Ph.D., Chair

Frank Moore, Ph.D.

Acknowledgments

I would like to thank several people for their patience, endurance, guidance, support,and interest in my success. For one, and most obviously, my advisor, Dr. Ellen Kirk-man. She never let me settle for anything less than my best. She has pushed me tobecome a better, more confident, precise, and disciplined student and mathematician.Every young woman, especially in a STEM field, needs other strong women in her life.I’m thankful I’ve had that and more. As an advisor, one couldn’t ask for more, and hersupport has helped me take the next step in my career. I would also like to thank Dr.Frank Moore, who always had an open door. We had countless afternoon discussions.He always encouraged curiosity and a kind of playfulness that helps one get at the morecreative side of mathematics. I would like to thank Dr. Andrew Conner for first intro-ducing me to how interesting, difficult, and rewarding algebra can be. I credit him forgetting me hooked.

I would also like to thank my partner, Jae Southerland, who has been supportive, kind,proud of me, and understanding when I had a lot of overwhelming and stressful workto do. They have been a better partner than I could have ever hoped for and none ofthis would have been possible without them. It is in large part because of Jae that I evenhad this opportunity. Finally, I would like to thank Dr. Jerzy Weyman. Most of the workin this thesis is based off of work he and Dr. Harm Derksen have done over the pastdecade. He has been very helpful, and has provided email correspondence and helpfulcomments that made certain parts of this thesis much more clear, where otherwise theylikely would not have been. I am humbled and gracious that he has taken an interest inme and that I will have the privilege of working with him as his student.

ii

Table of Contents

Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 Quiver Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Representations and Morphisms of Quivers . . . . . . . . . . . . . . . . . . 2

2.2 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Indecomposable Representations . . . . . . . . . . . . . . . . . . . . . . . . 5

Chapter 3 The Ring of Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 4 Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Zariski and Euclidean Dense Sets . . . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 5 The Ring of Semi Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.1 Semi-Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2 The Sato-Kimura Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 Varieties in Rep(Q,α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.3.1 Conditions for Open Orbits of Representations . . . . . . . . . . . . 25

5.4 Semi-invariants of Selected Quivers of Type ADE . . . . . . . . . . . . . . . 26

5.4.1 A Quiver with Dynkin Diagram An . . . . . . . . . . . . . . . . . . . . 26

5.4.2 A Quiver with Dynkin Diagram D4 . . . . . . . . . . . . . . . . . . . . 27

5.4.3 A Quiver with Dynkin Diagram E6 . . . . . . . . . . . . . . . . . . . . 29

Chapter 6 Schofield Semi-Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.1 Defining the Schofield Semi-Invariants . . . . . . . . . . . . . . . . . . . . . 34

6.2 More on Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3 Schofield Semi-Invariants for a Quiver with Dynkin Diagram D4 . . . . . . 55

6.4 Schofield Semi-Invariants for a Quiver with Dynkin Diagram E6 . . . . . . 58

6.4.1 cV1W and cV2

W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.4.2 cV3W and cV4

W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.4.3 cV5W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.4.4 The Kronecker 2-Quiver . . . . . . . . . . . . . . . . . . . . . . . . . . 65

iii

Chapter 7 Construction of the Irreducible Polynomial Representations of GL(V ) . . 68

7.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.2 Polynomial Representations and Schur Modules . . . . . . . . . . . . . . . 73

7.3 Irreducible Polynomial Representations of GL(V ) . . . . . . . . . . . . . . . 77

Chapter 8 Computations with Schur Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Chapter 9 Examples Using the Cauchy Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

9.1 Skew Tableaux, Semi-invariants of GL(V ), and Products of Schur Modules 90

9.2 Some Computations and Examples . . . . . . . . . . . . . . . . . . . . . . . 92

9.2.1 Triple Flag Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Chapter 10 Application to Littlewood-Richardson Coefficients . . . . . . . . . . . . . . . . . . . . . . 114

10.1 Saturation and Rational Cones . . . . . . . . . . . . . . . . . . . . . . . . . . 114

10.2 Saturation of Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

10.3 Saturation of the Littlewood-Richardson Coefficients . . . . . . . . . . . . . 127

Appendix A The Path Algebra and CQ-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

A.1 The Path Algebra CQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

A.2 The Correspondence Between Quiver Representations and CQ-modules . 139

Appendix B Auslander-Reiten Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143

B.1 A Quiver with Graph A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

B.2 A Quiver with Graph A3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

B.3 A Quiver with Graph D4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

B.4 A Quiver with Graph E6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Appendix C The Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148

Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158

iv

Abstract

Semi-invariants of Quivers and Saturation of Littlewood-Richardson Coefficients.

Using Schofield semi-invariants, and showing a correspondence between weight spacesof semi-invariant rings for a special class of quivers, and the Littlewood-Richardson co-efficients, we show that the space of Littlewood-Richardson numbers is saturated, i.e. ifcNν

Nλ,Nµ6= 0 then cν

λ,µ 6= 0, by showing that weights σ ∈Σ(Q,β) are saturated.

v

Chapter 1: Introduction

Quivers and their representations are a central object of study in the representation the-

ory of associative algebras. The study of quivers is equivalent to the study of a large

class of algebras and their representations since every quiver has an associated algebra

known as the path algebra, and every finite dimensional algebra over an algebraically

closed field and many infinite dimensional algebras can be realized as the path algebra

of some quiver, with some relations. We begin with an introduction to representations

of quivers with a heavy focus on ADE-Dynkin quivers, which are the quivers of finite

representation type as proven by P. Gabriel [10]. We then discuss invariants and semi-

invariants of quivers under the action of a product of general linear groups matching

the dimensions of the vector spaces assigned to the vertices of the quiver. This leads us

in particular to the discussion of Schofield semi-invariants. These semi-invariants are

generators of rings of semi-invariants, denoted SI(Q,β), and these rings decompose in a

particular way into weight spaces, SI(Q,β)σ, of weight σ. We then show that the weights

in the set

Σ(Q,β) = {σ : SI(Q,β)σ 6= 0}

are saturated, i.e. if for n ∈ N, nσ ∈ Σ(Q,β) then σ ∈ Σ(Q,β). Finally, we construct a

correspondence between weights and triples of partitions {(λ,µ,ν) : cνλ,µ 6= 0}, where cν

λ,µ

is the Littlewood-Richardson coefficient. From this we prove that Littlewood-Richardson

coefficients are in fact also saturated. We finish with a few examples of the utility of

this correspondence of weights and partitions, and provide some computations showing

how the weights and partitions are related and how to get one from the other. We assume

familiarity with some basic concepts from algebra such as basic properties of modules,

algebras, and rings. We also assume familiarity with multilinear algebra.

1

Chapter 2: Quiver Representations

We begin with an introduction to quivers and their representations.

2.1 Representations and Morphisms of Quivers

Definition 2.1.1. A quiver Q, is a directed graph Q = (Q0,Q1), where Q0 is the set of

vertices and Q1 the set of arrows. The maps h : Q1 →Q0 and t : Q1 →Q0 take the arrows

to their heads and tails respectively.

Remark 2.1.2. Quivers may have arrows in any direction or combination of directions,

loops, and may be disconnected.

Example 2.1.3.

1

g

��

a//

f''

2bvv c // 3

d��

e��

4

i

YY 5

Here Q0 = {1,2,3,4,5}, Q1 = {a,b,c,d ,e, f , g , i }, t a = t f = t g = hg = hb = 1, ha = tc =tb = 2, hc = td = te = 3, hd = he = h f = 5, t i = hi = 4.

Definition 2.1.4. Suppose Q is a quiver. A representation V (Q) of the quiver Q is a set,

{V (x) : x ∈Q0}

of finite dimensional C vector spaces together with a set,

{V (a) : V (t a) →V (ha) : a ∈Q1}

of C-linear maps.

2

Example 2.1.5. Consider the quiver,

1 a // 2 .

Suppose A ∈ Hom(Cn ,Cm), i.e. A is an m×n matrix. Then we can define a representation

V (Q) by V (1) =Cn ,V (2) =Cm , and V (a) = A. So V (Q) is

Cm A // Cn .

Remark 2.1.6. We will denote V (Q) as simply V from now on if it is clear by the context

which quiver we are representing, and which representation we are referring to.

Definition 2.1.7. Suppose V and W are two representations of the same quiver Q. A

morphism of quiver representations, or a Q-morphisms, φ : V → W is a collection of

C-linear maps,

{φ(x) : V (x) →W (x)|x ∈Q0}

such that for every arrow a ∈Q1 the following diagram commutes,

V (t a)V (a) //

φ(t a)��

V (ha)

φ(ha)��

W (t a)W (a)

//W (ha)

.

This means φ(ha)V (a) = W (a)φ(t a). If φ(x) is invertible for every x ∈ Q0 then φ is an

isomorphism of the quiver representations V and W .

Definition 2.1.8. If V and W are representations of the quiver Q, we denote the space of

all Q-morphisms from V to W by HomQ (V ,W ).

Remark 2.1.9. At this point it would be prudent to mention that if V and W are repre-

sentations of the quiver Q, then HomQ (V ,W ) is a subspace of

⊕x∈Q0

HomC(V (x),W (x)) = HomC(V ,W ),

3

the direct sum of the spaces of C-linear maps from each V (x) to W (x). It is important

to make this distinction as a map in HomC(V ,W ) need not be a quiver morphism, only

a linear map φ = ⊕x∈Q0

φ(x), of vector spaces φ :⊕

x∈Q0V (x) → ⊕

x∈Q0W (x). The im-

portance of this will be more apparent later on when we talk about the Euler form and

define Schofield semi-invariants in §6.

Definition 2.1.10. The dimension vector α of the quiver Q with representation V is,

α= (α(x1),α(x2), ...,α(xn)) = (dimV (x1),dimV (x2), ...,dimV (xn))

where {x1, x2, ..., xn} =Q0.

Remark 2.1.11. In general, changing the labeling of the vertices will permute the com-

ponents of α. In some cases later on it will be helpful and sometimes necessary to label

Q0 in a particular way.

2.2 Irreducible Representations

Definition 2.2.1. Suppose V and W are both representations of the quiver Q. The rep-

resentation W is a subrepresentation of V if

1. For all x ∈Q0, W (x) is a subspace of V (x).

2. For all a ∈ Q1, the restriction of V (a) : V (t a) → V (ha) to the subspace W (t a) is

equal to W (a) : W (t a) →W (ha).

Remark 2.2.2. Every quiver has a trivial representation Z where Z (x) = 0 for all x ∈Q0,

and Z (a) = 0 for all a ∈Q1.

Definition 2.2.3. A nonzero representation V is called irreducible or simple if the only

subrepresentations are V and the trivial representation.

4

Example 2.2.4. For any x ∈ Q0, define Ex by Ex(x) = C and Ex(y) = 0 for all y 6= x in Q0.

For all a ∈Q1, let Ex(a) = 0. These Ex are all irreducible representations of the quiver Q.

We denote the dimension vector of the simple representations Ex by εx .

2.3 Indecomposable Representations

Definition 2.3.1. If V and W are representations of some quiver Q, then the direct sum

representation V ⊕W is given by

(V ⊕W )(x) =V (x)⊕W (x)

for every x ∈Q0 and for each a ∈Q1

(V ⊕W )(a) : V (t a)⊕W (t a) →V (ha)⊕W (ha),

which is given by the matrix (V (a) 0

0 W (a)

).

Definition 2.3.2. A representation V is decomposable if it is isomorphic to the direct

sum X ⊕Y of nonzero representations. A nonzero representation is indecomposable if

it is not isomorphic to a direct sum of nontrivial representations.

Remark 2.3.3. By the Krull-Schmidt theorem, since it is assumed that every vector space

in a representation V of a quiver is finite dimensional, every representation can be writ-

ten as a finite sum of indecomposable representations, which is unique up to isomor-

phism and permutation of the factors.

5

Chapter 3: The Ring of Invariants

Definition 3.0.4. The representation space over the fieldC of a quiver Q, with dimension

vector α= (α(x1),α(x2), ...,α(xn)) is defined as

RepC(Q,α) = ⊕a∈Q1

Hom(Cα(t a),Cα(ha)).

We will denote the representation space of Q as Rep(Q,α) in the future, as we will always

work over the complex numbers.

Example 3.0.5. Let Q be the following quiver,

•1 a1// •2 a2

// •3

If we choose α= (n,m, l ) then a general representation (see Definition 10.1.4) of dimen-

sion α is,

CnA1

// CmA2

// Cl .

So,

Rep(Q,α) = Hom(Cn ,Cm)⊕Hom(Cm ,Cl ) = {(A1, A2) : A1 is an m×n matrix , and A2 is an l×m matrix }.

Example 3.0.6. Let Q be the Kronecker 2-quiver,

• ((66 •

with representation

CnA1 **

A2

44 Cn .

Then α= (n,n) and

Rep(Q,α) = Hom(Cn ,Cn)⊕Hom(Cn ,Cn) = {(A1, A2) : A1, A2 are n ×n matrices }.

6

Example 3.0.7. Let Q be the following quiver

x1

a

��

with representation

Cn

A

��.

Then α= (n), and Rep(Q,α) = Hom(Cn ,Cn) = {A : A is an n ×n matrix}.

Definition 3.0.8. Let GL(n) denote the general linear group. For a quiver Q with dimen-

sion vector α we define the group

GL(α) = ∏x∈Q0

GL(α(x)).

Definition 3.0.9. Let G = (gx1 , gx2 , ..., gxn ) ∈ GL(α), (where n = |Q0| and each gx ∈ GL(α(x))).

Let A = (Aa1 , Aa2 , ..., Aak ) ∈ Rep(Q,α), (where k = |Q1| and each Aai ∈ Hom(Ct ai ,Chai )).

Define a group action GL(α) æ Rep(Q,α) by,

G · A = (gha1 Aa1 g−1t a1

, gha2 Aa2 g−1t a2

, ..., ghak Aak g−1t ak

).

Definition 3.0.10. The coordinate ringC[Rep(Q,α)] of the representation space Rep(Q,α)

is the ring of all polynomials in dim(Rep(Q,α)) commuting variables that represent the

coordinates of the matrices in Rep(Q,α), after some choice of basis.

Example 3.0.11. If Q is the quiver

• ((66 •

with representation

CnA ++

B33 C

m

then C[Rep(Q,α)] =C[x1,1, x1,2, ..., xm,n , y1,1, y1,2, ..., ym,n], where A = (xi , j ) and B = (yi , j ).

7

Note 3.0.12.

dim(Rep(Q,α)) = ∑a∈Q1

dimHom(Cα(t a),Cα(ha)) = ∑a∈Q1

(α(t a) ·α(ha)).

Remark 3.0.13. The action defined in 3.0.9 induces a second group action

GL(α) æC[Rep(Q,α)]

given by

G · f (A) = f (G−1 · A) = f (g−1ha1

Aa1 g t a1 , g−1ha2

Aa2 g t a2 , ..., g−1hak

Aak g t ak ).

or equivalently, G acts on the right by

G · f (A) = f (A ·G).

Denote the identity of GL(α) by 1α. Let G1,G2 ∈ GL(α), A ∈ Rep(Q,α) and f ∈C[Rep(Q,α)].

Then clearly

1α · f (A) = f (A)

and

(G1,G2) · f (A) = f (A · (G1,G2)) = f ((A ·G1) ·G2) =G2 · f (A ·G1) =G1 · (G2 · f (A))

or equivalently

(G1,G2) · f (A) = f ((G−12 G−1

1 ) · A) =G1 · (G2 · f (A)).

Definition 3.0.14. A polynomial invariant f ∈ C[Rep(Q,α)], is a polynomial such that

G · f = f for any G ∈ GL(α). The ring of invariants

I (Q,α) =C[Rep(Q,α)]GL(α)

is the subring ofC[Rep(Q,α)] of polynomials that are invariant under the action of GL(α).

8

Example 3.0.15. Let Q be the quiver

x1

a

��

with representation

Cn

A

��.

One invariant polynomial function f ∈C[Rep(Q,α)] is

f : Rep(Q,α) →C; given by A 7→ det(A).

The action of GL(n) on Rep(Q,α) is g · A = g Ag−1, for g ∈ GL(n) and A ∈ Rep(Q,α). The

action of GL(n) onC[Rep(Q,α)] is g · f (A) = f (g−1 Ag ) = det(g−1 Ag ) = det(A) since deter-

minants are invariant under a change of basis. Thus, f is indeed a polynomial invariant.

For example, if n = 2 then C[Rep(Q,α)] =C[x1,1, x1,2, x2,1, x2,2] and f (x1,1, x1,2, x2,1, x2,2) =x1,1x2,2 −x2,1, x1,2 is a GL(2) invariant.


•1 a1// •2 with representation Cn

A// Cn .

Then Rep(Q,α) = Hom(Cn ,Cn), GL(α) = GL(n)×GL(n) and for G = (g1, g2) ∈ GL(n)×GL(n) we have G · A = g2 Ag−1

1 . Again let f (A) = det(A), then G · f (A) = f (g−12 Ag1) =

det(g1)det(g2)−1 det(A). In this case f is no longer an invariant under all elements of

GL(α).


•1

a1 )) •2a2

ii with representation CnA1 ++

Cm

A2

jj .

9

Then α= (n,m) and Rep(Q,α) = Hom(Cn ,Cm)⊕Hom(Cm ,Cn), A1 is m×n, and A2 is n×m, so we have A1 A2 is m×m and A2 A1 is n×n. In this case we have G = (g1, g2) ∈ GL(n)×GL(m) and (A1, A2) ∈ Hom(Cn ,Cm)⊕Hom(Cm ,Cn), so G · (A1, A2) = (g2 A1g−1

1 , g1 A2g−12 ).

Then for f1(A1, A2) = det(A1 A2) we have G· f (A1, A2) = det(g−12 A1g1g−1

1 A2g2) = det(A1 A2).

This works similarly for f2(A1, A2) = det(A2 A1), which is also GL(α)-invariant, thus f1

and f2 are polynomial invariants.


•1a1 // •2

a2

��•3

a3

`` with representation Cn A1 // Cm

A2��Cr

A3

aa .

Then α= (n,m,r ) and Rep(Q,α) = Hom(Cn ,Cm)⊕Hom(Cm ,Cr )⊕Hom(Cr ,Cn). We have

an action of GL(α) = GL(n)×GL(m)×GL(r ) on Rep(Q,α). For G = (g1, g2, g3) ∈ GL(α) and

A = (A1, A2, A3) ∈ Rep(Q,α) we have G·A = (g2 A1g−11 , g3 A2g−1

2 , g1 A3g−13 ). Let f1(A1, A2, A3) =

det(A1 A3 A2). Then G · f (A1, A2, A3) = det(g−12 A1g1g−1

1 A3g3g−13 A2g2) = det(A1 A3 A2), and

thus f1 ∈C[Rep(Q,α)]GL(α). Similarly if we choose

f2(A1, A2, A3) = det(A2 A1 A3) and f3(A1, A2, A3) = det(A3 A2 A1)

we get three polynomial invariants f1, f2, f3 ∈ I(Q,α).

Lemma 3.0.19. For any quiver Q without oriented cycles we can label the vertices and

edges such that t a < ha for all a ∈Q1.

Proof. The proof is done by induction on n = |Q0|. For n = 1 it is trivial. Let |Q0| = k.

If Q has no oriented cycles then there is some v ∈ Q0, such that v has no arrows such

that t a = v and ha = w ∈ Q0. To see this is true, assume to the contrary that such a

vertex does not exist, then start at any vertex in Q0 and follow an "out" arrow. Since

the quiver is finite, repeat until we have returned to a repeated vertex. This gives an

10

oriented cycle, thus a contradiction. Now, let v ∈Q0 such that v has out degree 0. Label

v with the number k. Note, there may be more than one such v , but any such v will

work. Now, let Q ′ be the quiver obtained by deleting v and any arrows such that ha = v .

Now |Q ′0| = k −1. By induction we may continue to label vertices so that t a < ha for all

a ∈Q1.

Theorem 3.0.20. If Q is a quiver without oriented cycles, then I (Q,α) =C. In other words,

there are no nontrivial invariants.

Proof. By Lemma 3.0.19 we can assume that Q0 = {1,2,3, ...,n} and that t a < ha for all

a ∈Q1 without any loss of generality. Now, define φλ ∈ GL(α) by

φλ(k) =λk idα(k) ∈ GL(α(k))

for k = 1,2, ...,n. In the general case where Q is any quiver without oriented cycles, if

t a < ha for every a ∈Q1, then we have that φλ · Ak , for Ak ∈ Hom(Ct a ,Cha) is given by,

φλ · Ak =λha idα(ha) Akλt a idα(t a) =λha−t a Ak

where t a < ha =⇒ φλ ·Ak =λl Ak , with l ∈Z>0. Thus, we must have that each Ak = 0α(k),

the α(k)×α(k) zero matrix, for all indices k of the xk ∈ Q0. So, φλ · Ak = λl Ak = Ak , i.e.

Ak is invariant under the action ofφλ restricted to each Hom(Cα(t ak ),Cα(hak )) if and only

if Ak = 0 since φλ · Ak =λl Ak = Ak for all λ if and only if Ak = 0.

Similarly, for the actionφλ· f (A) we have that ifφλ· f (Ak ) = f (λ−ha idα(ha) Akλt a idα(t a)) =

f (λs Ak ) = f (Ak ) with s ≤−1, s ∈Z (since t a < ha,∀a ∈Q1), for every

Ak ∈ ⊕a∈Q1

Hom(Cα(t a),Cα(ha)),

then we must have that f is constant on each Ak if it is to be invariant under the ac-

tion of φλ since each Ak will be zero. So the only invariants will be constants (constant

polynomials) in the base field C.

11

So we see that unless a quiver Q has oriented cycles, we get no interesting polyno-

mial invariants I(Q,α) = C[Rep(Q,α)]GL(α). In §5 we discuss the ring of semi-invariants

SI(Q,α) = C[Rep(Q,α)]SL(α). First we introduce some basic concepts of algebraic geom-

etry, to be applied in the study of rings of semi-invariants in §5, in the next chapter. We

then proceed to discuss the ring of semi-invariants for representations of Dynkin quiv-

ers, using some methods from algebraic geometry, which allows us to describe when the

coordinate ringC[Rep(Q,α)] yields nontrivial semi-invariants. General requirements for

the ring SI(Q,α) to be nontrivial for arbitrary quivers without oriented cycles are dis-

cussed in Theorem 10.2.3.

12

Chapter 4: Algebraic Geometry

4.1 Definitions and Examples

The representation space Rep(Q,α) is isomorphic to some CN as a vector space. Solu-

tions to polynomials f ∈ C[Rep(Q,α)], define algebraic varieties. Thus techniques from

algebraic geometry can be useful tools in studying the representation spaces of quivers.

Here we introduce some basics of algebraic geometry and then show how to use some

algebraic geometry to study representation spaces of quivers.

Definition 4.1.1. A variety V (S) of some S ⊂C[x1, ..., xn] is the set,

V (S) = {(a1, ..., an) ∈Cn : f (a1, ..., an) = 0,∀ f ∈ S}.

Remark 4.1.2. The variety V (S) is equal to the variety V ((S)), where (S) is the ideal of

C[x1, ..., xn] generated by S. This is true since for any f , g ∈ S we have that ( f + g )(a) = 0

for all a ∈ V (S), and for any f ∈ S and any h ∈ C[x1, ..., xn] we have that (h · f )(a) = 0 for

any a ∈V (S).

Note 4.1.3. By the Hilbert Basis Theorem, ideals of C[x1, ..., xn] are finitely generated, so

varieties are the zero sets of a finite number of polynomials.

Example 4.1.4. The following are all examples of varieties.

1. Any point a = (a1, ..., an) ∈Cn , is the variety V (x1 −a1, x2 −a2, ..., xn −an).

2. The variety V (xn , ym) = V (x, y), since the solution set to the polynomial xn is the

same as the solution set to the polynomial x, and likewise the solution set to ym is

the solution set to the polynomial y .

3. V (2x2 +3y2 −11, x2 − y2 −3) = V (x2 − y, y2 −1) = {±2,±1}, so very different sets of

polynomials can produce the same variety.

13

Remark 4.1.5. V defines a map,

V :{Ideals of C[x1, ..., xn]

}→ {subvarieties of Cn}

given by,

I 7→V (I )

which is inclusion reversing, i.e. if I ⊂ J then V (J ) ⊂V (I ).

Definition 4.1.6. Define the ideal I (Z ) for some Z ⊆Cn to be,

I (Z ) = { f ∈C[x1, ..., xn] : f (z) = 0,∀z ∈ Z }.

Remark 4.1.7. I (Z ) is an ideal of C[x1, ..., xn], and I defines a map

I : {subvarieties V of Cn} → {Ideals of C[x1, ..., xn]}

given by,

Z 7→ I (Z )

which is also inclusion reversing.

Lemma 4.1.8. If X is a subvariety of Cn then V (I (X )) = X .

Proof. Clearly X ⊆ V (I (X )) since any polynomial in I (X ) is zero on X . Conversely, if

y ∈ V (I (X )), then for any g ∈ I (X ), g (y) = 0. In particular, X = V (S) therefore S ⊆ I (X )

and s(y) = 0 for all s ∈ S. Therefore y ∈ X =V (S).

Example 4.1.9. Let S = {x3}. Then the ideal (S) generated by S is just (x3) ⊂ C[x]. The

variety generated by (S) is just V ((x3)) = {0}, but the ideal I (V (x3))of C[x], is not (x3), but

rather (x).

Lemma 4.1.10. Let Z ⊆Cn be any subset. If X =V (I (Z )) is the variety defined by the ideal

I (Z ), then I (X ) = I (Z ) and X is the smallest variety in Cn containing Z .

14

Proof. Let X = V (I (Z )). First we want to show I (Z ) ⊂ I (X ). Take f ∈ I (Z ), then by def-

inition of X , f must be zero on X since it is the solution set to any f ∈ I (Z ). Since f

vanishes on X , f ∈ I (X ), so I (Z ) ⊂ I (X ). Conversely, Z is a subset of V (I (Z )) = X , so any

polynomial vanishing on X must also vanish on Z , thus I (X ) ⊂ I (Z ). Finally, if Y is a

variety such that Z ⊂ Y ⊂ X , then we must have that I (X ) ⊂ I (Y ) ⊂ I (Z ) = I (X ), which

implies I (Y ) = I (X ). Applying V and using the previous Lemma, we get Y = V (I (Y )) =V (I (X )) = X .

Definition 4.1.11. The Zariski topology on Cn is the topology in which closed sets are

varieties V ((S)), and open sets are the complements of varieties. For some subset Z ⊂Cn , the Zariski closure of Z , is the smallest variety containing Z , i.e. V (I (Z )), by Lemma

4.1.10. Subvarieties inherit their topology from the Zariski topology on Cn , and closed

subsets of a variety X are just the subvarieties of X . A subset Z ⊂ X is Zariski dense in

the variety X if its closure in the Zariski topology is X , i.e. if X is the smallest variety

containing Z .

Definition 4.1.12. A hypersurface is the set V ( f ), where f is some non constant poly-

nomial in C[x1, ..., xn].

Theorem 4.1.13. Any variety V ⊆Cn is the intersection of finitely many hypersurfaces.

Proof. By the Hilbert Basis Theorem, we know that every ideal I ⊂C[x1, ..., xn] is finitely

generated. Let Z = V (I ), be the variety defined by the ideal I . I is then generated by

some set f1, f2, ..., fm of polynomials in C[x1, ..., xn]. Then Z = V ( f1, f2, ..., fm) = V ( f1)∩V ( f2)∩·· ·∩V ( fm), and Z is the intersection of finitely many hypersurfaces.

4.2 Zariski and Euclidean Dense Sets

Theorem 4.2.1. 1. Any Zariski closed set is closed in the Euclidean topology.

15

2. Any Zariski open set is open in the Euclidean topology.

3. A nonempty Euclidean open set is Zariski dense.

4. A nonempty Zariski open set is dense in the Euclidean topology.

Proof.

We first show that the hypersurface V ( f ) is closed in the Euclidean topology. If f ∈C[x1, ..., xn] is any non constant polynomial, then f is continuous in the Euclidean topol-

ogy since polynomials are continuous in the Euclidean topology. Then the set V ( f ) is

the preimage f −1(0) of zero. Continuity implies the preimage of a closed set is closed,

thus V ( f ) is closed. The fact that varieties V ((S)) are closed follows from the previous

theorem and the fact that the intersection of closed sets is closed.

A Zariski open set is the complement of a Zariski closed set, so from the previous state-

ment, a Zariski open set must be open in the Euclidean topology.

Let U 6= ; be an open set it the Euclidean topology. Then U contains some ball B(z,ε) ⊆U . We will show that the smallest subvariety containing B(z,ε), i.e. V (I (B(z,ε))) is all of

Cn . Let f ∈ C[x1, ..., xn] be some polynomial such that f ∈ I (B(z,ε)). We would like to

show that f ≡ 0. The Taylor series of f is defined as,

T (x1, ..., xn) =∞∑

k1=0

∞∑k2=0

· · ·∞∑

kn=0

n∏i=1

(xi − zi )ki

ki !

(∂

∑ni=1 ki f

∂xk11 · · ·∂xkn

n

)(z1, ..., zn).

Since f ∈ I (B(z,ε)) is identically zero on B(z,ε), all of its partial derivatives are also zero

on B(z,ε), and therefore the Taylor series must be zero since it is just the sum of partial

derivatives of f . The Taylor series of f , and f itself being identically zero on some dense

open set B(z,ε) implies f must be the zero polynomial. So, we have that V (I (B(z,ε))) =V (0) =Cn , thus V (I (U )) =Cn .

16

Let U 6= ; be some open subset of Cn in the Zariski topology. Then

U =Cn −V (S) =Cn − (⋂V ( fi )

)=⋃(Cn −V ( fi )

).

If fi is a nonzero constant polynomial then V ( fi ) = ; =⇒ Cn −V ( fi ) = Cn . If fi ≡ 0

then V ( fi ) = Cn =⇒ Cn −V ( fi ) =;. Let f be a non constant polynomial. We claim the

interior of V ( f ), in the Euclidean topology, is empty. To see this, let x ∈ Int(V ( f )). Then

x ∈ B(x,ε) ⊂V ( f ). Then f ≡ 0 by the above argument, a contradiction, so Int(V ( f )) =;.

Let Cn −V ( f ) ⊂C , a closed set in the Euclidean topology. Then V ( f ) ⊃ Cn −C , an open

set. Thus Cn −C =; =⇒ C =Cn . Therefore, the closure of U in the Euclidean topology

is Cn , i.e. U is dense in Cn with the Euclidean topology.

Definition 4.2.2. An algebraic group is a variety G that is also a group such that the

maps defining the group structure µ : G ×G → G , with µ(x, y) = x y , and i : x 7→ x−1 are

morphisms of varieties. If the underlying variety is a variety of the type we have already

defined (known as affine varieties), then we call G a linear algebraic group. Some exam-

ples are GL(n) and subgroups of GL(n), as well as the groups GL(α) and SL(α). We can

realize GL(n) as a closed subset

{(g ,λ) ∈Cn2 ×C : g ∈ Mn(C),λ ∈C;det(g ) ·λ= 1}.

In this way the general linear group is a linear algebraic group. We can extend this to

GL(α) =∏x∈Q0 GL(α(x)).

Remark 4.2.3. In the next chapter we define the ring of semi-invariants SI(Q,α) ⊂C[Rep(Q,α)]

under the action of GL(α). The results in the next chapter apply to dense orbits (open

orbits in the Zariski topology) of the action of the linear algebraic group GL(α) on the

variety Rep(Q,α). We use a theorem of M. Sato and T. Kimura proven in [14] for pre-

homogeneous vector spaces, which are defined by Sato and Kimura as triples (G ,ρ,V )

where G is a connected linear algebraic group, ρ is a rational representation of G (to be

17

defined and discussed in §7), on a finite dimensional complex vector space V , and such

that V has a Zariski dense G-orbit. This is adapted and used by A. Skowronski and J.

Weyman in [24] to the case of the orbit of a representation in Rep(Q,α) under the ac-

tion of GL(α) for Dynkin and Euclidean quivers. This gives us a way to describe the

algebras of semi-invariants in the coordinate ring C[Rep(Q,α)] and their generators, for

ADE-Dynkin quivers Q. We show that the orbits of representations for the finite repre-

sentation type (ADE-Dynkin) quivers are dense.

18

Chapter 5: The Ring of Semi Invariants

5.1 Semi-Invariants

Definition 5.1.1. A character of the group GL(α) is a group homomorphism

χ : GL(α) →C∗

Remark 5.1.2. If χ : GL(α) →C∗ is a character, then χ will always be of the form

Gα = (gx1 , ..., gxn ) ∈ GL(α) 7→ ∏x∈Q0

det(gx)σ(x) =χ(G) ∈C∗

where σ : Q0 →Z is called the weight. Weights σ are dual to dimension vectors α by the

following definition.

Definition 5.1.3. Define

σ(α) = ∑x∈Q0

σ(x)α(x).

In this way, we can think of σ as a function on dimension vectors α.

Remark 5.1.4. Here we are viewing weights as functions on dimension vectors. In par-

ticular, denote the space of all integer valued functions on Q0 by Γ= Hom(Q0,Z). Then

dimension vectors α ∈ Γ, x 7→ α(x) = dimV (x) are (nonnegative) integer valued func-

tions on Q0. We think of weights σ ∈ Γ∗ = Hom(Γ,Z) as being in the dual space. We

also think of σ ∈ Γ as integer valued functions on vertices in Q0 as well. Further, we

sometimes think of σ= (σ(x1), ...,σ(xn)), where Q0 = {x1, ..., xn}, as vectors similar to di-

mension vectors. When we are thinking of σ as an element of the dual Γ∗ we always

write σ(α), to denote σ evaluated at the dimension vector α. When we are thinking of

σ as an element of Γ we always write σ(xi ) to denote the weight at the vertex xi ∈Q0, or

the i th component of the weight vector σ.

19

Definition 5.1.5. A polynomial semi-invariant f ∈ C[Rep(Q,α)], is a polynomial such

that

Gα · f =χ(Gα) f

for all Gα ∈ GL(α), and some fixed character χ.

Definition 5.1.6. We denote the ring of semi-invariants under the action of GL(α) by

SI(Q,α). Semi-invariants of GL(α) are invariants under the action of SL(α), since the

characters are products of determinants. We denote the ring of semi-invariants by

C[Rep(Q,α)]SL(α) = SI(Q,α)

Remark 5.1.7. There is a direct sum decomposition, i.e. a grading of SI(Q,α) by charac-

ters,

SI(Q,α) =⊕χ

SI(Q,α)χ

or equivalently by weight vectorsσ corresponding to each characterχ=∏x∈Q0 det(gx)σ(x),

SI(Q,α) =⊕σ

SI(Q,α)σ.

5.2 The Sato-Kimura Theorem

Lemma 5.2.1. Let G be a linear algebraic group acting regularly on an affine variety X .

Let f1, ..., fr ∈ C[X ] be nonzero semi-invariants with distinct characters χ1, ...,χr . Then

f1, ..., fr are linearly independent.

Proof. The proof is by induction on r . It’s clear for the case where there is only one semi-

invariant f , that the set { f } is a linearly independent set. Now suppose that we have a

linearly independent set f1, ..., fk of nonzero semi-invariants with characters χ1, ...,χk ,

where χi 6= χ j for all i 6= j . Now let f1, ..., fk , fk+1 be the set of semi-invariants with

20

the additional semi-invariant fk+1 added, and with corresponding distinct characters

χ1, ...,χk ,χk+1. Suppose

a1 f1 +·· ·+ak+1 fk+1 = 0

where ai ∈C. Now let g ∈G , then

g · (a1 f1 +·· ·+ak+1 fk+1) = a1χ1(g ) f1 +·· ·+ak+1χk+1(g ) fk+1 = 0

and

χk+1(g )(a1 f1 +·· ·+ak+1 fk+1) =χk+1(g )a1 f1 +·· ·+χk+1(g )ak+1 fk+1 = 0.

Subtracting we have,

a1(χ1(g )−χk+1(g )) f1 +·· ·+ak (χk (g )−χk+1(g )) fk = 0

applying the assumption that f1, ..., fk were linearly independent and that χi (g ) 6= χ j (g )

for all i 6= j and i , j ∈ {1,2, ...,k,k +1}, we have that a1(χ1(g )−χk+1(g )) = ·· · = ak (χk (g )−χk+1(g )) = 0, and therefore a1 = ·· · = ak = 0. This means ak+1 fk+1 = 0 and thus ak+1 = 0,

proving the claim.

Lemma 5.2.2. Suppose that a connected linear algebraic group G acts on a variety X . If

f is a semi-invariant, and h divides f , then h is also a semi-invariant.

Proof. Let G = GL(α) act on the variety V ( f ). Then g · f = χ(g ) f for any g ∈G . Without

loss of generality assume h is an irreducible factor of the polynomial semi-invariant f ,

say f = hq . Then V (h) ⊂ V ( f ) and for irreducible h we have V (h) irreducible. Since

G is connected, G stabilizes each irreducible component of a variety X by proposition

8.2 of [13], so G must stabilize V (h), so it must be that g ·h = λg h, for λg ∈ C, therefore

h is also a semi-invariant. The map g 7→ λg ∈ C is a character of G since it defines a

homomorphism

χ : G →C∗

21

given by,

g 7→χ(g ) =λg .

Definition 5.2.3. Define the orbit of the representation V in Rep(Q,α) under the action

of GL(α) to be

Orb(V ) = {φ ·V : φ ∈ GL(α)}.

Proposition 5.2.4. ([24] Theorem 2, Sato-Kimura Theorem) Let GL(α) have a dense orbit

in Rep(Q,α). Let S be the set of all σ such that there exists an fσ ∈ SI(Q,α) that is nonzero

and irreducible. Then,

1. For every weight σ we have that dimSI(Q,α)σ ≤ 1.

2. All weights in S are linearly independent overQ.

3. SI(Q,α) is the polynomial ring generated by all fσ : σ ∈ S.

Proof. 1. Suppose that f ,h ∈ SI(Q,α)σ. Since f /h is constant on the open dense or-

bit, and since f and h are polynomials and thus continuous, the quotient f /h is

continuous wherever h 6= 0. We then must have f /h is constant on Rep(Q,α), and

f and h must be linearly dependent, i.e. f =λh for some λ ∈C.

2. Suppose that ∑σ∈S

aσσ= 0

with aσ ∈Z for all σ. Then we have

∑σ∈S

aσσ= ∑aσ>0

aσσ+ ∑aσ<0

aσσ =⇒ ∑aσ>0

aσσ= ∑aσ<0

|aσ|σ

and therefore ∏aσ>0

f aσσ =λ ∏

aσ<0f |aσ|σ

22

for some nonzero λ ∈C. From unique factorization in C[Rep(Q,α)], it follows that

aσ = 0 for all σ.

3. Every semi-invariant is a product of irreducible semi-invariants by Lemma 5.2.2.

This shows that the fσ,σ ∈ S generate SI(Q,α). Also, all monomials in the fσ’s have

distinct weights, so all monomials in the fσ’s are linearly independent by Lemma

5.2.1. This shows that the fσ’s are algebraically independent.

5.3 Varieties in Rep(Q,α)

Here we give an example of a representation space with an open orbit under the GL(α)-

action. We would now like to define polynomials fi ∈ C[Rep(Q,α)] such that the solu-

tions of the collection of { fi } will give us a variety in Rep(Q,α). The complement of this

variety will then be an open set, by definition of the Zariski topology. We want to choose

{ fi } so that this complement is an orbit in Rep(Q,α), giving a dense orbit, by the previous

theorem.

Recall 5.3.1. We defined the orbit of the representation V in Rep(Q,α) under the action

of GL(α) to be

Orb(V ) = {φ ·V : φ ∈ GL(α)}.

Example 5.3.2. We first look at the case where Q is the following quiver,

•1 a1// •2

with representation

CnA// Cm .

23

Claim 5.3.3. Let r = min{m,n}. We let { fi } ⊂C[Rep(Q,α)] be defined as the determinant

of all r × r minors Mi of the matrix A. For example if

A =(

x11 x12 x13

x21 x22 x23

)∈ Hom(C3,C2).

Then the set of zeroes of the collection

f1(A) = x11x22 −x21x12

f2(A) = x11x23 −x21x13

f3(A) = x12x23 −x22x13

of polynomials defines a variety consisting of all matrices A ∈ Hom(Cn ,Cm) that do not

have full rank. So in general, if we choose the collection { fi } to be the determinants of all

maximal minors, then we have a variety V ⊂ Rep(Q,α) of all matrices that are not of full

rank, and the complement of V is all matrices with full rank.

Proof. Let A ∈ Hom(Cn ,Cm) be a matrix without full rank, i.e. Rank(A) = s < r . Since

transposition of a matrix does not change the determinant, det(A) = det(AT ), we can

assume that m < n without any loss of generality. If A does not have full rank and has

more columns than rows we have that A′, the reduced row echelon form of A, must have

at least one row of zeroes. This means any m ×m minor of A′ will have at least one

row of zeros, and thus det(Mi ) = 0 for all m ×m minors Mi . From this we see that the

m×m minors of A must all have zero determinants for any A such that Rank(A) < m. So

defining the collection { fi (A)} = {det(Mi ) : Mi is an m ×m minor of A} will give a set of

polynomials vanishing on any matrix A that does not have full rank. Since the collection

of matrices that do not have full rank is the zero set to the polynomials { fi }, we have that

it is a variety. The orbit of a matrix A with full rank is all matrices in Hom(Cn ,Cm) of full

rank, and we have a dense orbit as desired.

24

5.3.1 Conditions for Open Orbits of Representations

Describing explicitly the dense orbits of a representation variety Rep(Q,α) may not al-

ways be so simple, so we would now like to describe conditions for quivers giving dense

orbits of representations in Rep(Q,α), without needing to find explicit open orbits.

Recall 5.3.4. The orbit of the representation V in Rep(Q,α) under the action of GL(α) to

be

Orb(V ) = {φ ·V : φ ∈ GL(α)}.

Note 5.3.5. The orbits of the action GL(α) æ Rep(Q,α) define isomorphism classes of

representations V ∈ Rep(Q,α).

Lemma 5.3.6. A subset Y ⊆ X = Rep(Q,α) is Zariski dense in X if and only if Y has the

property that if f ∈C[Rep(Q,α)] is any polynomial that is zero for all y ∈ Y , then f is the

zero polynomial.

Proof. (⇐) : Let Z = Y the closure of Y , be the smallest variety containing Y , i.e. Z =V (I (Y )) = V ( f1, ..., fn) for some fi ∈ C[X ]. Since Y ⊆ Z , each fi is zero on Y , so by as-

sumption fi ≡ 0. Hence Z =V (0) = X and Y is Zariski dense.

(⇒) : Since Y is Zariski dense, V (I (Y )) = X . Let f ∈ C[X ] be a polynomial that is zero

on Y ; show f ≡ 0. Then f ∈ I (Y ), and since X is the zero set of I (Y ), f ≡ 0.

Theorem 5.3.7. If X = Rep(Q,α) has only finitely many indecomposable representations,

then X has a Zariski dense orbit under the action of G = GL(α).

Proof. By hypothesis, X is the union of a finite number of orbits Orb(Vi ), i = 1, ...,r .

Suppose no orbit is Zariski dense. Then by Lemma 5.3.6, for each i there is a polynomial

fi ∈ C[X ] with fi zero on Yi = Orb(Vi ) but fi 6= 0. Let f = f1 · · · fr . Then f is a nonzero

polynomial that is zero on all of X , a contradiction.

25

Remark 5.3.8. It is well known that Dynkin quivers of type An ,Dn ,E6,E7,E8, are exactly

the quivers of finite representation type, i.e. with finitely many indecomposable repre-

sentations. So, if a quiver Q is of finite representation type, i.e. if the underlying graph

of Q is an ADE-Dynkin graph then for each dimension vector α, there is a dense orbit

Orb(V ) ⊂ Rep(Q,α).

5.4 Semi-invariants of Selected Quivers of Type ADE

We will now compute some of the rings of semi-invariants for some quivers of type ADE.

We compute the generators of the rings SI(Q,α) and then show, with some minor as-

sumptions to be proven in §6, that these are indeed a complete set of generators of the

rings SI(Q,α).

5.4.1 A Quiver with Dynkin Diagram An

Now we will find the generators for the ring SI(Q,α) of semi-invariants for the quiver,

•1 a1// •2 a2

// •3 · · · a3// •n−1 an−1

// •n

with representation,

Cn A1 // Cn A2 // Cn · · · A3 // Cn An−1 // Cn .

It is easy to see that

fσi (X ) = det(Ai )

will give a semi-invariant with χi (B) = det(Bi )−1 det(Bi+1) corresponding to

σi = (σi (1),σi (2), ...,σi (n −1),σi (n)) = (0,0, ...,0,−1,1,0, ...,0,0),

where the i th component of σi , σi (i ) = −1 and σi (i +1) = 1. This gives a set of n −1 =|Q0|−1 different σi such that σi (α) = 0 for each i = 1,2, ...,n −1. The set { fσi }n−1

i=1 is a lin-

early independent set since each characterχi corresponding to fσi is distinct, by Lemma

26

5.2.1. Now, we want to show that this is also a spanning set for SI(Q,α) using Theo-

rem 5.2.4. GL(α) has a dense orbit in Rep(Q,α) since it is of finite representation type

by Lemma 5.3.7, so by Theorem 5.2.4, every weight σ corresponding to an irreducible

fσ ∈ SI(Q,α) must be linearly independent overQ of every other such σ. Our set { fσi }n−1i=1

with corresponding weights {σi }n−1i=1 is a maximal linearly independent set over Q. The

reason is as follows; we will prove later on when we discuss Schofield semi-invariants

that the weights must all satisfy the conditionσ(α) =∑x∈Q0 σ(x)α(x) = 0. Assuming this,

and by the fact that we have n −1 linearly independent weights (over Q) {σi }n−1i=1 ⊂ Zn ,

we can add at most one more weight and still have a linearly independent set of weights.

The condition that all n − 1 weights found so far be linearly independent, along with

the assumption that σ(α) = 0, i.e. all σ are orthogonal to the dimension vector α ∈ Zn≥0

makes it impossible to add another weight. Thus, we have found a maximal Q-linearly

independent set of weights, and by Theorem 5.2.4, the fσ span SI(Q,α), thus we have

found a complete set of generators for the ring SI(Q,α), i.e. SI(Q,α) =C[ fσ1 , ..., fσn−1 ].

5.4.2 A Quiver with Dynkin Diagram D4

Next we construct the generators for the ring of semi-invariants for the quiver,

•1 a1// •4 •3a3oo

•2

a2

OO


CnA1

// C2n CnA3

oo

Cn

A2

OO .

From the representation we have dimension vector β= (n,n,n,2n), and we want to find

fσ ∈ SI(Q,β)σ so that the set { fσ ∈ SI(Q,β)σ} is a generating set for SI(Q,β). An element

27

in Rep(Q,β) is of the form A = (A1, A2, A3) and an element G ∈ GL(β) is of the form G =(g1, g2, g3, g4). The action of GL(β) on the augmented matrix [A1|A2] is as follows,

(g1, g2, g3, g4) · [A1|A2] = [g4 A1g−11 |g4 A2g−1

2 ]

= g−14 [A1|A2]

(g−1

1 00 g−1

2

)= g4[A1|A2](g−1

1 ⊕ g−12 )

= g4[A1|A2](g−11 , g−1

2 ).

Let f (A1, A2, A3) = det[A1|A2] and let G = (g1, g2, g3, g4) ∈ GL(α). Then we have,

G · f (A1, A2, A3) = f (G−1 · (A1, A2, A3)) = det(g−14 A1g1, g−1

4 A2g2, g−14 A3g3)

= det([g−14 A1g1|g−1

4 A2g2])

= det(g−14 [A1|A2](g1 ⊕ g2))

= det(g1)det(g2)det(g4)−1 f (A1, A2, A3) =χ(G) · f (A).

So our choice of f (A1, A2, A3) = det([A1|A2]) is a semi-invariant under the GL(β) action.

The choice of f (A1, A2, A3) = det([A1|A3]) and f (A1, A2, A3) = det([A2|A3]) will also work

by the same argument as above for f (A) = det([A1|A2]), and they will correspond to σ=(1,0,1,−1) and σ= (0,1,1,−1) respectively. So we have the set

{ fσ(A) = det([Ai |A j ]) : i < j , for i , j ∈ {1,2,3}}

with

σ1 = (1,1,0,−1),

σ2 = (1,0,1,−1),

σ3 = (0,1,1,−1).

Since determinants of matrices of independent variables are irreducible polynomials,

the polynomials det([Ai |A j ]) given above are irreducible. The fσi for i = 1,2,3 are all

28

linearly independent, as Lemma 5.2.1 predicts, since they all have distinct characters.

By the argument in the previous example, this quiver is of finite representation type,

and thus has a dense orbit. By Theorem 5.2.4 we have that the weights must all be Q-

linearly independent. Again, assuming the condition σ(α) = 0 must be fulfilled, we have

a set of three weights each lying in Z4, and each must be orthogonal to the dimension

vector α ∈ Z4≥0. Thus, we can add no more weights, and we have a maximal Q-linearly

independent set of weighs and therefore SI(Q,α) =C[ fσ1 , fσ2 , fσ3 ].

5.4.3 A Quiver with Dynkin Diagram E6

Next we find the generators for the ring of semi-invariants for the quiver,

•1 a1// •3 a2

// •6 •4a4oo •2a2

oo

•5

a5

OO


CnA1

// C2nA3

// C3n C2nA4

oo CnA2

oo

C2n

A5

OO

.

So we have α= (n,n,2n,2n,2n,3n). Let A = (A1, A2, A3, A4, A5) ∈ Rep(Q,α), where A1, A2

are 2n ×n matrices, A3, A4, A5 are 3n ×2n. Let G = (g1, g2, g3, g4, g5, g6) ∈ GL(α), where

g1, g2 ∈ GL(n), g3, g4, g5 ∈ GL(2n) and g6 ∈ GL(3n). Then for f ∈C[Rep(Q,α)] we have

G · f (A) = f (g−13 A1g1, g−1

4 A2g2, g−16 A3g3, g−1

6 A4g4, g−16 A5g5).

Semi-invariant 1

We first choose f1(A) = f1(A1, A2, A3, A4, A5) ∈C[Rep(Q,α)] to be

f1(A) = det

(A3 A1 id3n 0

0 id3n A4

).

29

From the action of GL(α) on Rep(Q,α) we get

G · f1(A) = det

(g−1

6 A3 A1g1 id3n 0

0 id3n g−16 A4g4

)

= det

(g−1

6 0

0 g−16

)(A3 A1 id3n 0

0 id3n A4

) g1 0 0

0 g6 0

0 0 g4

= det

((g−1

6 ⊕ g−16 )

(A3 A1 id3n 0

0 id3n A4

)(g1 ⊕ g6 ⊕ g4)

)

= det

((g−1

6 , g−16 )

(A3 A1 id3n 0

0 id3n A4

)(g1, g6, g4)

)

= det(g1)det(g4)det(g6)−1 det

(A3 A1 id3n 0

0 id3n A4

).

This gives us χ1(G) = det(g1)det(g4)det(g6)−1 and σ1 = (1,0,0,1,0,−1).

Semi-invariant 2

Next, we choose

f2(A) = det

(A4 A2 id3n 0

0 id3n A3

).

for our second semi-invariant. From the action of GL(α) on Rep(Q,α) we get

G · f2(A) = det

(g−1

6 A4 A2g2 id3n 0

0 id3n g−16 A3g3

)

= det

((g−1

6 , g−16 )

(A4 A2 id3n 0

0 id3n A3

)(g2, g6, g3)

)


(A4 A2 id3n 0

0 id3n A4

).


30

Semi-invariant 3

We choose

f3(A) = det

(A3 A1 id3n 0

0 id3n A5

).

for our third semi-invariant. From the action of GL(α) on Rep(Q,α) we get

G · f1(A) = det

(g−1

6 A3 A1g1 id3n 0

0 id3n g−16 A5g5

)

= det

((g−1

6 , g−16 )

(A3 A1 id3n 0

0 id3n A5

)(g1, g6, g5)

)


(A3 A1 id3n 0

0 id3n A5

).


Semi-invariant 4

For our fourth semi-invariant we choose,

f4(A) = det

(A4 A2 id3n 0

0 id3n A5

).

From the action of GL(α) on Rep(Q,α) we get

G · f4(A) = det

(g−1

6 A4 A1g2 id3n 0

0 id3n g−16 A5g5

)

= det

((g−1

6 , g−16 )

(A4 A2 id3n 0

0 id3n A5

)(g2, g6, g5)

)


(A4 A2 id3n 0

0 id3n A5

).


31

Semi-invariant 5

Finally, we choose

f5(A) = det

A3 id3n 0 0

0 id3n A4 0

0 id3n 0 A5

.

for our fifth and final semi-invariant. Under the action of GL(α) on Rep(Q,α) we get

G · f5(A) = det

g−16 A3g3 id3n 0 0

0 id3n g−16 A4g4 0

0 id3n 0 g−16 A5g5

= det

(g−16 , g−1

6 , g−16 )

g−16 A3g3 id3n 0 0

0 id3n g−16 A4g4 0

0 id3n 0 g−16 A5g5

(g3, g6, g4, g5)

= det(g3)det(g4)det(g5)det(g6)−2 det

A3 id3n 0 0

0 id3n A4 0

0 id3n 0 A5

.

This gives us χ5(G) = det(g3)det(g4)det(g5)det(g6)−2 and σ5 = (0,0,1,1,1,−2).

We now have { f1, f2, f3, f4, f5 : fi ∈ SI(Q,α)} where the first four fi can be simplified to

f1(X ) = det[A3 A1|A4]

f2(X ) = det[A4 A2|A3]

f3(X ) = det[A3 A1|A5]

f4(X ) = det[A4 A2|A5].

32

These five semi-invariants correspond to the characters {χ1,χ2, ...,χ5}, and weights,

σ1 = (1,0,0,1,0,−1)

σ2 = (0,1,1,0,0,−1)

σ3 = (1,0,0,0,1,−1)

σ4 = (0,1,0,0,1,−1)

σ5 = (0,0,1,1,1,−2).

The first four polynomials are determinants of matrices of independent variables, and

thus is irreducible in C[Rep(Q,α)]. The fifth polynomial is equivalent to the determi-

nant det([A3|A4]⊕ A5) which is again a matrix over independent variables, and thus is

an irreducible polynomial in C[Rep(Q,α)]. Again, the fσi are all linearly independent,

as Lemma 5.2.1 predicts, since they all have distinct characters. This quiver is of finite

representation type, and thus has a dense orbit. By Theorem 5.2.4 we have that the

weights must all be Q-linearly independent. Again, assuming the condition σ(α) = 0,

which we will prove later must be fulfilled, we have a set of five weights each lying in

Z6, and each must be orthogonal to the dimension vector α ∈Z6≥0. Thus, we can add no

more weights, and we have a maximal Q-linearly independent set of weights and there-

fore SI(Q,α) =C[ fσ1 , fσ2 , fσ3 , fσ4 , fσ5 ].

33

Chapter 6: Schofield Semi-Invariants

6.1 Defining the Schofield Semi-Invariants

Here we introduce the Schofield semi-invariants, semi-invariants associated to repre-

sentations of quivers that were introduced by Schofield in [21]. These semi-invariants

are defined for arbitrary finite quivers without oriented cycles, and thus do not depend

on whether or not there is an open orbit. It has been shown in [5] and independently in

[23] that the Schofield semi-invariants in fact generate SI(Q,α) when Q has no oriented

cycles. Thus we have a more general way of finding semi-invariants and generators for

the ring of semi-invariants.

Definition 6.1.1. Let V and W be two representations of a quiver Q. The map

⊕x∈Q0

Hom(V (x),W (x))dV

W //⊕a∈Q1

Hom(V (t a),W (ha))

is given by,

dVW : (φ(1),φ(2), ...,φ(k)) 7→ ⊕

a∈Q1

(W (a)φ(t a)−φ(ha)V (a)),

where {1,2, ...,k} =Q0 and (φ(1),φ(2), ...,φ(k)) =⊕x∈Q0

φ(x).

Definition 6.1.2. Define the Euler form or the Ringel form ⟨,⟩ on dimension vectors α

and β by

⟨α,β⟩ = ∑x∈Q0

α(x)β(x)− ∑a∈Q1

α(t a)β(ha).

The value∑

x∈Q0 α(x)β(x)−∑a∈Q1 α(t a)β(ha) ∈Z is called the Euler characteristic of the

representations V and W of Q. The Euler characteristic of two representations V and W

of Q thus depends only on the dimension vectors α and β.

34

Remark 6.1.3. If α is the dimension vector of the representation V and β is the dimen-

sion vector of the representation W , then dVW can be written as a square matrix if,

⟨α,β⟩ = ∑x∈Q0


α(t a)β(ha)

= dim

( ⊕x∈Q0

Hom(V (x),W (x))

)−dim

( ⊕a∈Q1

Hom(V (t a),W (ha))

)

= 0.

Definition 6.1.4. When Q has representations V and W of dimensions α and β respec-

tively, and ⟨α,β⟩ = 0, define cVW = c(V ,W ) := det(dV

W ) ∈ C[Rep(Q,α)×Rep(Q,β)]. These

are known as the Schofield semi-invariants. Fixing a representation V and allowing W

to vary, we get cV : W →C as a polynomial function, in the coordinate ring C[Rep(Q,β)],

of representations W . Similarly, fixing a representation W and allowing V to vary, we

get cW : V → C as a polynomial function, in C[Rep(Q,α)], of representations V . We

then show in Theorem 6.1.8 that cV ∈ SI(Q,β) are semi-invariants in the coordinate ring

C[Rep(Q,β)], and cW ∈ SI(Q,α) are semi-invariants in the coordinate ring C[Rep(Q,α)].

It will eventually be shown in §6.2 that weights of semi-invariants σ of cV and σ′ of cW

are related to dimension vectors α and β of the representations V and W respectively.

The rings SI(Q,α) and SI(Q,β) can be decomposed by weights as a direct sum

SI(Q,α) =⊕σ

SI(Q,α)σ and SI(Q,β) =⊕σ′

SI(Q,β)σ′ .

Recall 6.1.5. We defined for dimension vectors β and weights σ,

σ(β) = ∑x∈Q0

σ(x)β(x)

Each weightσ of a cV will correspond to the dimension vectorα of a fixed representation

V by

σ(β) = ⟨α,β⟩

35

i.e. σ 7→ ⟨α,•⟩, and similarly each weight σ′ of cW will correspond to a dimension vector

β of a fixed representation W by

σ′(α) =−⟨α,β⟩

i.e. σ′ 7→ −⟨•,β⟩. So we have cV ∈ SI(Q,β)⟨α,•⟩, and cW ∈ SI(Q,α)−⟨•,β⟩, and the decompo-

sitions of SI(Q,α) and SI(Q,β) become

SI(Q,α) =⊕α

SI(Q,β)⟨α,•⟩ and SI(Q,β) =⊕β

SI(Q,α)−⟨•,β⟩.

It is proven in [5] in Lemma 1 that if we take the representations V to be all indecompos-

able representations of Q, and choose all indecomposable V with dimension vectors α

such that ⟨α,β⟩ = 0, then this gives a complete set of generators {cV } of the ring SI(Q,β).

Remark 6.1.6. In §10 we give conditions on representations V and W of a quiver Q so

that the cVW are nontrivial. In particular, these conditions rely on the dimension vectors

α and β of V and W respectively. This gives us conditions on weights σ, that tell us what

weights give us a nontrivial Schofield semi-invariant.

Remark 6.1.7. The definition of the Schofield semi-invariants comes from what is known

as the Ringel resolution, and an equivalent definition can be made using minimal projec-

tive resolutions of GL(β)-modules (representations). We leave the reader to investigate

these further, as venturing into techniques of homological algebra would take us too far

afield. Information on this can be found in [4].

Theorem 6.1.8. The cVW , i.e. c(V ,W ) ∈ C[Rep(Q,α)×Rep(Q,β)] are invariant under the

action of SL(α)×SL(β) and semi-invariant under the action of GL(α)×GL(β) on

C[Rep(Q,α)×Rep(Q,β)]

induced by the action on Rep(Q,α)×Rep(Q,β).

36

Proof. To see this we describe the action of GL(β) and GL(α) on each Hom(V (x),W (x))

and on Hom(V (t a),W (ha)) in⊕

x∈Q0Hom(V (x),W (x)) and

⊕a∈Q1

Hom(V (t a),W (ha))

respectively, which gives an action of GL(α)×GL(β) on Rep(Q,α)×Rep(Q,β) (and thus

on C[Rep(Q,α)×Rep(Q,β)]). Since

dVW ∈ Hom

( ⊕x∈Q0

Hom(V (x),W (x)),⊕

a∈Q1

Hom(V (t a),W (ha))

),

if we know the action on the domain and co-domain of dVW , then we can describe the

action on dVW . From this we derive an action on cV

W = det(dVW ). From this action we com-

pute the character, and are then able to show that cVW is indeed semi-invariant under the

action of GL(α)×GL(β).

First we look at the action of GL(β) on dVW . Let {v1, ..., vn} be a basis for the vector space

V (x), and let {w1, ..., wm} be a basis for W (x), for some x ∈Q0. A basis for Hom(V (x),W (x))

is then { fi j } for i = 1, ...,n and j = 1, ...,m where

fi j (vi ) = w j and fi j (vk ) = 0,∀i 6= k.

Then

f =n∑

i=1

m∑j=1

ai j fi j

is represented by

(ai j ) =

a11 a12 · · · a1n...

......

am1 am2 · · · amn

.

Using the ordered basis ( f11, f21, ..., fm1, f12, f22, ..., fm2, ..., f1n , ..., fmn), the coordinate vec-

37

tor of f is

a11...

am1

a12...

am2...

a1n...

amn

.

Now, let Gβ = (gx1 , gx2 , ..., gxk ) ∈ GL(β), where |Q0| = k, and each gx ∈ GL(β(x)). Now set

Gx =

gx

gx. . .

gx

where there are n =α(x) blocks on the diagonal that areβ(x)×β(x) matrices gx ∈ GL(β(x)).

We have det(Gx) = (det(gx))α(x). Gβ = (gx)x∈Q0 acts on Hom(V (x),W (x)) as G−1x [ f ], i.e.

by left multiplication of the coordinate vector of f by the inverse G−1x , or equivalently by

post composition (left multiplication) of the matrix (ai j ) of f by the matrix g−1x . We have

det(G−1x ) = det(gx)−α(x).

Now, G acts on⊕

x∈Q0Hom(V (x),W (x)) as

⊕x∈Q0

G−1x =

G−1

x1

G−1x2

. . .G−1

xk

.

Gβ acts on Hom(V (t a),W (ha)) as left multiplication of elements of Hom(V (t a),W (ha))

38

by Gha where

Gha =

gha

gha. . .

gha

.

Here there are α(t a) blocks gha ∈ GL(β(ha)) on the diagonal. So, we have

det(Gha) = det(gha)α(t a),

and G acts on⊕

a∈Q1Hom(V (t a),W (ha)) as

⊕a∈Q1

Gha ,

⊕a∈Q1

Gha =

Gha1

Gha2

. . .Ghan

.

In order for dVW to be a homomorphism of representations we must have the action of

GL(α)×GL(β) commute with the linear map dVW , i.e.

Gβ ·dV (W ) = dV (Gβ ·W )

for all Gβ ∈ GL(β), and

Gα ·dW (V ) = dW (Gα ·V )

for all Gα ∈ GL(α). We discuss G-maps, i.e. linear maps of representations of a group G

that respect the action of the group, in more detail in §7, and in Definition 7.2.5. To show

that the action does in fact commute, let us look at the most basic case, that of a square

diagram

V (x1)

φ(t a)��

V (a) // V (x2)

φ(ha)��

W (x1)W (a)

//W (x2)

39

Let α be the dimension vector of V and β the dimension vector of W . Assume that

⟨α,β⟩ = α(1)β(1)+α(2)β(2)−α(1)β(2) = 0, and thus that dVW can be represented as a

square α(1)β(2) ×α(1)β(2) matrix. Fix the representation V , and let Gβ = (gx1 , gx2 ) ∈GL(β). Gβ acts on Hom(V (x1),W (x1))⊕Hom(V (x2),W (x2)) as

Gβ · (φ(x1),φ(x2)) = (g−1x1φ(x1), g−1

x2φ(x2))

As matrices the action is given by

Gβ · (φ(x1),φ(x2)) = (G−1x1

[φ(x1)],G−1x2

[φ(x2)])

where Gx1 =⊕α(1)

i=1 gx1 and Gx2 =⊕α(2)

i=1 gx2 , and [φ(x1)] ∈ Cα(1)β(1) and [φ(x2)] ∈ Cα(2)β(2)

are the coordinate vector representations of φ(x1) and φ(x2) respectively. Gβ ∈ GL(β)

acts on Hom(V (x1),W (x2)) as

Gβ · f = gx2 f .

As matrices the action is

Gβ · f =G ′x2

[ f ]

where G ′x2=⊕α(1)

i=1 gx2 , and [ f ] ∈Cα(1)β(2) is the coordinate vector of f . Hence, Gβ ∈ GL(β)

acts on dV (W ) as

Gβ ·dV (W ) =G ′x2

(dV (W ))(Gx1 ⊕Gx2 )−1.

Since Gβ ∈ GL(β) acts by the inverse on functions we get

Gβ · cV (W ) =Gβ ·det(dV (W )) = det(G−1β ·dV (W )) = det

((G ′

x2)−1(dV (W ))(Gx1 ⊕Gx2 )

).

Now, we show dV (Gβ ·W ) =G ′x2

(dV (W ))(Gx1 ⊕Gx2 )−1 =Gβ ·dV (W ), i.e. that dV (W ) com-

mutes with the GL(β)-action. First, let Gβ act on W so that we now have the following

square diagram

V (x1)V (a) //

φ(t a)��

V (x2)

φ(ha)��

W (x2)gx2W (a)g−1

x1

//W (x2)

.

40

Then dV (Gβ ·W ) is given by

(φ(x1),φ(x2)) 7→ gx2W (a)g−1x1φ(x1)−φ(x2)V (a).

Next, begin instead with the action of Gβ on (φ(x1),φ(x2)) to get

Gβ · (φ(x1),φ(x2)) = (g−1x1φ(x1), g−1

x2φ(x2)).

Now apply dV (W ) to get

dV (W )(g−1x1φ(x1), g−1

x2φ(x2)) =W (a)(g−1

x1φ(x1))− (g−1

x2φ(x2))V (a).

Now let Gβ act on Hom(V (x1),W (x2)) to get

Gβ ·dV (W )(φ(x1),φ(x2)) =Gβ ·(W (a)(g−1

x1φ(x1)− (g−1

x2φ(x2))V (a)

)= gx2

(W (a)(g−1

x1φ(x1))− (g−1

x2φ(x2))V (a)

)= gx2W (a)g−1

x1φ(x1)−φ(x2)V (a)

= dV (Gβ ·W ).

So, Gβ ·dV (W ) = dV (Gβ ·W ), and dV (W ) commutes with the GL(β)-action.

Next we show that dW (V ) commutes with the GL(α) action. In this case an element

Gα ∈ GL(α) acts on each Hom(V (x),W (x)) by right multiplication by the matrices

gx ∈ GL(α(x)).

So for the following square diagram

V (x1)V (a) //

φ(t a)��

V (x2)

φ(ha)��

W (x2)W (a)

//W (x2)

.

41

we have the action of Gα on Hom(V (x1),W (x1))⊕Hom(V (x2),W (x2)) is given by

Gα · (φ(x1),φ(x2)) = (gx1 , gx2 ) · (φ(x1),φ(x2)) = (φ(x1)gx1 ,φ(x2)gx2 )

where gx1 ∈ GL(α(x1)), gx2 ∈ GL(α(x2)), and Gα = (gx1 , gx2 ) ∈ GL(α). Now, Gα acts on

Hom(V (t a),W (ha)) by right multiplication by g−1t a . So if we use the square diagram

above, we get

Gα · f = f g−1x1

where f ∈ Hom(V (x1),W (x2)). To show that dW (V ) commutes with this action, first let

Gα act on Hom(V (x1),W (x1))⊕Hom(V (x2),W (x2)), giving

Gα · (φ(x1),φ(x2)) = (φ(x1)gx1 ,φ(x2)gx2 ).

Now, apply the map dW (V ),

dW (V )(φ(x1)gx1 ,φ(x2)gx2 ) =W (a)φ(x1)gx1 −φ(x2)gx2V (a).

Now let Gα act on Hom(V (x1),W (x2)),

Gα · (W (a)φ(x1)gx1 −φ(x2)gx2V (a)) = (W (a)φ(x1)gx1 −φ(x2)gx2V (a))g−1x1

=W (a)φ(x1)−φ(x2)gx2V (a)g−1x1

= dW (Gα ·V )(φ(x1),φ(x2)).

In the general case this can be realized as matrix multiplication if we coordinatize the

φ(x) ∈ Hom(V (x),W (x)) as row vectors, using the ordered basis

{e11,e12, ...,e1n ,e21,e22, ...,e2n , ...,em1,em2, ...,emn},

relative to the (m×n) = (β(x)×α(x)) matrix representation of eachφ(x) ∈ Hom(V (x),W (x)).

In this case the row vector [φ(x)] ∈ (Cα(x)β(x))∗, and gx acts by right multiplication by the

matrix

Gx =β(x)⊕i=1

gx =

gx

gx. . .

gx

.

42

Here there are β(x) blocks gx ∈ GL(α(x)) on the diagonal. The action of Gα on each

Hom(V (t a),W (ha)) is given by coordinatizing f ∈ Hom(V (t a),W (ha)) in the same way

as the φ(x), as a row vector [ f ] ∈ (Cα(t a)β(ha))∗, then right multiplying by the matrix

G−1t a =

β(ha)⊕i=1

g−1t a =

g−1

t ag−1

t a. . .

g−1t a

,

where there are β(ha) blocks g−1t a ∈ GL(α(t a)) on the diagonal. The Gα acts on

(φ(x1), ...,φ(xk )) ∈ ⊕x∈Q0

Hom(V (x),W (x))

by right multiplication of the row vector coordinatization of (φ(x1), ...,φ(xk )) by the ma-

trix

⊕x∈Q0

Gx =

Gx1

Gx2

. . .Gxk

.

The action of Gα ∈ GL(α) on

f ∈ ⊕a∈Q1

Hom(V (t a),W (ha))

is given by right multiplication of the coordinate row vector of f by the matrix

⊕a∈Q1

G−1t a =

G−1

t a1

G−1t a2

. . .G−1

t an

.

Given the way we have defined the action of GL(α) × GL(β), the action of GL(β) cer-

tainly commutes with the linear map dV (W ), and the action of GL(α) commutes with

43

the linear map dW (V ), and we do in fact have a GL(α)×GL(β)-homomorphism of rep-

resentations since the action commutes with dVW . Hence in the general case, Gβ ∈ GL(β)

acts by the inverse on functions cV ∈C[Rep(Q,β)],

Gβ · cV (W ) =Gβ ·det(dV (W )) = det(G−1β ·dV (W ))

= det

(( ⊕a∈Q1

G−1ha

)dV

W

( ⊕x∈Q0

Gx

))

= ∏a∈Q1

det(G−1ha)

∏x∈Q0

det(Gx) · cV (W )

= ∏a∈Q1

det(gha)−α(t a)∏

x∈Q0

det(gx)α(x) · cV (W )

= ∏x∈Q0

det(gx)σ(x)cV (W ).

Thus we have an expression for the characters

χ(Gβ) = ∏a∈Q1

det(gha)−α(t a)∏

x∈Q0

det(gx)α(x) = ∏x∈Q0

det(gx)σ(x)

under the action of GL(β), and we have shown that the cV (W ) are indeed SL(β) invariant

since these characters are determinants. Fixing a representation W , the characters of the

action of Gα ∈ GL(α) on cW (V ) is found in the same way. We compute that

Gα · cW (V ) = det(G−1α ·dW (V ))

= det

(( ⊕x∈Q0

G−1x

)dW (V )

( ⊕a∈Q1

Gt a

))

= ∏x∈Q0

det(gx)−β(x) det(dW (V ))∏

a∈Q1

det(g t a)β(ha).

Giving us a character

χ(Gα) = ∏x∈Q0

det(gx)−β(x)∏

a∈Q1

det(g t a)β(ha).

This gives us the full action of GL(α)×GL(β) on the cVW , and shows invariance under the

action of SL(α)×SL(β).

44


•1a // •2

Let β= (2,4) andα= (2,1) be the dimension vectors of representations W and V respec-

tively. So we have the representations,

V : C2 A // C and W : C2 B // C4 .

We allow A and B to be matrices of indeterminants and require that they each have full

rank. Then we have the noncommutative diagram

V : C2 A //

φ(1)��

C

φ(2)��

C2B// C4

where A is a 1×2 matrix in Rep(Q,α) and B is a 4×2 matrix in Rep(Q,β), after a choice

of basis. Note that ⟨α,β⟩ = 0. Now, GL(β) = {(gx1 , gx2 ) : gx1 ∈ GL(2), gx2 ∈ GL(4)}. Let

A = (a1 a2

) ∈ Rep(Q,α), and B =

b11 b12

b21 b22

b31 b32

b41 b42

∈ Rep(Q,β).

Then C[Rep(Q,α)×Rep(Q,β)] =C[a1, a2,bi j ]. Let

[φ(x1)] = X =(

x11 x12

x21 x22

)

represent φ(1) as a matrix, and let

[φ(x2)] = Y =

y1

y2

y3

y4

45

represent φ(2) as a matrix. If we wish to construct a homomorphism of representations

φ : V →W , we need to solve a system of equations given by the matrix equation

W (a)φ(t a)−φ(ha)V (a) = B [φ(1)]− [φ(2)]A

=

b11x11 +b12x21 b11x12 +b12x22

b21x11 +b22x21 b21x12 +b22x22

b31x11 +b32x21 b31x12 +b32x22

b41x11 +b42x21 b41x12 +b42x22

−

a1 y1 a2 y1

a1 y2 a2 y2

a1 y3 a2 y3

a1 y4 a2 y4

=

0 00 00 00 0

Since we have as many equations as unknowns (since ⟨α(x),β(x)⟩ = 0 and dVW is square),

we can attempt to solve this system of equations in order to findφ(x) ∈ Hom(V (x),W (x))

that actually give a homomorphism of representations φ : V → W . We have no con-

straints on the entries of B except that we require the columns to be linearly indepen-

dent so that B has full rank. Similarly, we require one of the columns of A to be nonzero

in order to have A of full rank. Now, for any B and any

Y =

y1

y2

y3

y4

such that Y is a linear combination of the columns of B , we have that the above equation

has a nontrivial solution. This means that the map

dVW :

⊕x∈Q0

Hom(V (x),W (x)) → ⊕a∈Q1

Hom(V (t a),W (ha))

46

has nontrivial kernel (and thus nontrivial cokernel), since the equations

b11x11 +b12x21 −a1 y1 = 0 b11x12 +b12x22 −a2 y1 = 0

b21x11 +b22x21 −a1 y2 = 0 b21x12 +b22x22 −a2 y2 = 0

b31x11 +b32x21 −a1 y3 = 0 b31x12 +b32x22 −a2 y3 = 0

b41x11 +b42x21 −a1 y4 = 0 b41x12 +b42x22 −a2 y4 = 0

which give the equations

x11

b11

b21

b31

b41

+x21

b12

b22

b32

b42

−a1

y1

y2

y3

y4

and x12

b11

b21

b31

b41

+x22

b12

b22

b32

b42

−a2

y1

y2

y3

y4

have nontrivial solutions for any A and B of full rank. In other words, we have con-

structed an element of HomQ (V ,W ) for any choice of general representation V and W ,

by choosing appropriate (φ(x1),φ(x2)) ∈ Hom(V (x1),W (x1))⊕Hom(V (x2),W (x2)). Since

ker(dVW ) = HomQ (V ,W ) this means the kernel of dV

W must be non trivial, meaning its

determinant must be zero, so

cVW = det(dV

W ) = 0

and the Schofield semi-invariants are necessarily trivial. In general if ⟨α,β⟩ = 0,

cVW = 0 ⇐⇒ HomQ (V ,W ) 6= 0 ⇐⇒ ExtQ (V ,W ) 6= 0

where ExtQ (V ,W ) is the cokernel of dVW . In §10 we give a theorem of Schofield (Theo-

rem 10.2.3) and a theorem of Derksen and Weyman (Theorem 10.2.7) that tell us exactly

when dVW has nontrivial kernel, and when it does not. We also give a description of when

nontrivial semi-invariants occur, i.e. when cVW = det(dV

W ) 6= 0, in §9.2. These theorems

only use properties of dimension vectors α and β and weights σ.


•1a1 // •2 with representation W : Cn A // Cn

47

where A is an n ×n matrix. For the general quiver representation V (1) → V (2), we find

the indecomposable representations V of Q in Appendix B. So we have the following

indecomposables,

C0 // 0 C

id // C 0 0 // C .

The only dimension vector such that ⟨α,β⟩ = 0 is α= (1,0), so take

V : C(0) // 0 .

We have β= (n,n) is the dimension vector of W and we have the following (noncommu-

tative) diagram,

C(0) //

φ(1)��

0

φ(2)=(0)��

CnA// Cn

dVW : Hom(C,Cn)⊕Hom(0,Cn) → Hom(C,Cn)

given by

(φ(1),0) 7→ (Aφ(1)−φ(2) ·0) = Aφ(1)

which is given by the matrix A, and cVW = det(A).


•1a // •2 •3

boo

with representations

V : C id // C Cidoo and W : C4 A // C9 C5Boo .

So, α= (1,1,1) and β= (4,9,5). We will show that

dVW =

(A − id 00 − id B

).

48

We have G = (G1,G2,G3), where G1 is 4×4, G2 is 9×9, and G3 is 5×5. Now,

dVW (A,B) : (φ(1),φ(2),φ(3)) 7→ (Aφ(1)−φ(2) id,Bφ(3)−φ(2) id).

So,

G · (φ(1),φ(2),φ(3)) = (G−11 ·φ(2),G−1

2 ·φ(2),G−13 ·φ(3))

and

G · (Aφ(1)−φ(2) id,Bφ(3)−φ(2) id) = (G2 AG−11 φ(1)−φ(2) id,G2BG−1

3 φ(3)−φ(2) id),

or in terms of the matrix dVW we have

G−1 ·det(dVW ) = det

(G−1

2 AG1 − id 00 − id G−1

2 BG3

)

= det

(G−1

2 00 G−1

2

)det

(A − id 00 − id B

)det

G1 0 00 G2 00 0 G3

.

So we have a character det(G1)det(G2)−1 det(G3) of G ∈ GL(β), corresponding to a weight

σ= (1,−1,1), which corresponds to the dimension vectorα= (1,1,1) of the indecompos-

able representation V .

Recall 6.1.12. We defined the Schofield semi-invariants as

cV := c(V , ·) ∈C[Rep(Q,β)]SL(β) = SI(Q,β),

and

cW := c(·,W ) ∈C[Rep(Q,α)]SL(α) = SI(Q,α).

It can be shown (see [5] Lemma 1), that it suffices to take V an indecomposable CQ-

module to compute SI(Q,β). In particular, [5] uses homological properties of quiver

representations to prove this. Next we compute two examples to find cV for SI(Q,β), for

a particular Q and a representation W of dimension β.

49


•1a1 // •2 •3

a2oo with representation Cn A1 // Cn+m CmA2oo

where A1 is (n +m)×n and A2 is (n +m)×m. Using the list of indecomposable mod-

ules obtained from the Auslander-Reiten quivers in Appendix B, the indecomposable

representations of this quiver are

V1 : C id // C Cidoo V2 : 0

(0) // C Cidoo V3 : C id // C 0

(0)oo

V4 : 0(0) // C 0

(0)oo V5 : 0(0) // 0 C

(0)oo V6 : C(0) // 0 0

(0)oo .

An indecomposable with ⟨α,β⟩ = 0 is V1, so we have the noncommutative diagram,

Cid //

φ(1)��

C

φ(2)��

Cidoo

φ(3)��

CnA1

// Cn+m CmA2

oo

and α= (1,1,1) and β= (n,n +m,m). So we have

dV1W : Hom(C,Cn)⊕Hom(C,Cn+m)⊕Hom(C,Cm) → Hom(C,Cn+m)⊕Hom(C,Cn+m)

given by

(φ(1),φ(2),φ(3)) 7→ (A1φ(1)−φ(2) id, A2φ(3)−φ(2) id)

which is given by the matrix

dV1W =

n n +m m

n +m A1 − idn+m 0

n +m 0 − idn+m A2

.

Thus cV1 (W ) = det(dV1W ) = det[A1|A2].

50

6.2 More on Weights

For computing the weights of cV and cW we recall the description of the actions of GL(β)

and GL(α) on Hom(V (x),W (x)), and the action on Hom(V (t a),W (ha)) from §6.1.4. From

this we get a formula for the weights of cV . For the weight at each vertex x ∈Q0 we have

σ(x) =α(x)− ∑a∈Q1,ha=x

α(t a) = ⟨α,εx⟩, (6.2.1)

where εx is the dimension vector of the simple module Ex as described in Example 2.2.4

and in Appendix A. We can also derive a formula for the weights of the cW

σ′(x) =−β(x)+ ∑a∈Q0,t a=x

β(ha).

We will not use this second formula for the weights σ′ of the cW , as most of the results

we prove are in terms of the semi-invariants cV .


•1a // •2

and give Q the following representation

V : C(0) // 0 .

Then the components of the weight vector corresponding to cV (and thus to the dimen-

sion vector α) are

σ(1) =α(1)− ∑a∈Q1: ha=1

α(t a) = 1

and

σ(2) =α(2)− ∑a∈Q1: ha=2

α(t a) = 1−1 = 0

so σ= (1,0).

51

Example 6.2.2. Let Q be the following quiver.

•5

a5

��•1 a1// •3 a3

// •6 •4a4oo •2

a2oo


V : C

id��

Cid// C

id// C 0

(0)oo 0(0)oo

.

Then the components of the weight vector of cV are

σ(1) =α(1)− ∑a∈Q1: ha=1

α(t a) = 1

σ(2) =α(2)− ∑a∈Q1: ha=2

α(t a) = 0

σ(3) =α(3)− ∑a∈Q1: ha=3

α(t a) = 0

σ(4) =α(4)− ∑a∈Q1: ha=4

α(t a) = 0

σ(5) =α(5)− ∑a∈Q1: ha=5

α(t a) = 1

σ(6) =α(6)− ∑a∈Q1: ha=6

α(t a) =−1

so σ= (1,1,0,0,1,−1). We sometimes write weights and dimension vectors in the shape

of the quiver. In this case

σ= 11 0 −1 0 0

.

One can check for this weight and for dimension vectors

β= 2nn 2n 3n 2n n

52

that

σ(β) = ∑x∈Q0

σ(x)β(x) = 0.

Remark 6.2.3. For weights σ, as stated previously, we can think of σ as being dual to

the dimension vectors β : Q0 → Zn . To be clear, β is a function on vertices β : Q0 → Z,

given by β(x) = dim(V (x)). We defined Γ= Hom(Q0,Z) to be the space of integer valued

functions on Q0, thus β ∈ Γ. We defineσ(β) =∑x∈Q0 σ(x)β(x), and we think ofσ as being

an element of the dual Γ∗ = Hom(Γ,Z).

Theorem 6.2.4. For every σ there is a dimension vector α ∈ Γ, so that we can express σ as

⟨α,•⟩ ∈ Γ∗, given by β 7→ ⟨α,β⟩, i.e. so that σ(β) = ⟨α,β⟩, where ⟨,⟩ is the Euler form on

dimension vectors.

Proof. We use the formula given in equation 6.2.1.

σ(β) = ∑x∈Q0

σ(x)β(x)

= ∑x∈Q0

α(x)− ∑a∈Q1ha=x

α(t a)

β(x)

= ∑x∈Q0

α(x)β(x)− ∑a∈Q1ha=x

α(t a)β(x)

= ∑

x∈Q0

α(x)β(x)− ∑x∈Q0

∑a∈Q1ha=x

α(t a)β(x)

= ∑x∈Q0


α(t a)β(ha)

= ⟨α,β⟩.

Definition 6.2.5. Denote the number of arrows from vertex x to vertex y in Q0 by bx,y ,

and denote the number of paths from x to y by px,y .

53

The following Lemma gives the weight vector of cV in terms of the dimension vector α

of V , and the dimension vector α of V in terms of the weight vector of cV .

Lemma 6.2.6. Suppose that σ= ⟨α,•⟩. Then we have the following equalities

1.

σ(x) =α(x)− ∑y∈Q0−{x}

by,xα(y).

2.

α(x) = ∑y∈Q0

py,xσ(y).

Proof. 1. The number of arrows from each y in Q0− {x} to x is denoted by,x , so by,x =|{a ∈Q1 : ha = x, t a = y}| ·α(t a), so by equation 6.2.1

σ(x) =α(x)− ∑a∈Q1: ha=x

α(t a)

=α(x)− ∑y∈Q0−{x}

by,xα(y).

2. The proof is by induction on the maximal path length of an in-path to the vertex.

Let x be a vertex with maximal path length of an in-path equal to zero; i.e. x is a

source vertex. Denote α(x) = nx . Clearly

nx = ∑y∈Q0

py,xσ(y) = 1 ·nx

since the only path into x is ex , the trivial path as defined in Appendix A. Now

assume the result is true for vertices of maximal in-path length strictly less than n

and let x be a vertex of maximal in-path length equal to n. For any arrow a with

ha = x by our induction hypothesis the formula holds for t a, and so,

α(t a) = ∑y∈Q0

py,t aσ(y).

54

By equation 6.2.1

σ(x) =α(x)− ∑a∈Q1: ha=x

α(t a)

=α(x)− ∑a∈Q1: ha=x

∑y∈Q0

py,t aσ(y)

which implies

α(x) = ∑y∈Q0: y 6=x

py,xσ(y)+σ(x) = ∑y∈Q0

py,xσ(y).

6.3 Schofield Semi-Invariants for a Quiver with Dynkin Diagram D4

Let Q be the following quiver,

•2

a2

��•1 a1// •4 •3a3oo


W : Cn

A′2 ��

CnA′

1

// C2n CnA′

3

oo

.

Let α be the dimension vector of some indecomposable representation of Q, and β be

the dimension vector β= (n,n,n,2n) of the representation W above. The indecompos-

ables of this quiver are given by the Auslander-Reiten quiver in example B.3.1. The gen-

eral indecomposable representations of the quiver Q with ⟨α,β⟩ = 0 that we need are,

V1 : C

id��

Cid// C 0

(0)oo

V2 : 0

(0)��

Cid// C C

idoo

V3 : C

id��

0(0)// C C

idoo

.

55

So we have that the map dVW : {φ(x)| x ∈Q0} → {W (a)φ(t a)−φ(ha)V (a)| a ∈Q1}, for each

Vi , is the map

dVW : Hom(C,Cn)⊕Hom(C,Cn)⊕Hom(C,C2n) → Hom(C,C2n)⊕Hom(C,C2n)

and for each representation Vi , dVW is given by a square matrix. The (noncommutative)

diagram for dV1W is as follows:

C

id

}}

φ(2)

��

Cid //

φ(1)

��

C

φ(4)

��

0(0)oo

φ(3)

��

Cn

A′2

~~Cn

A′1

// C2n CnA′

3

oo

.

This gives three sub-diagrams,

Cid //

φ(1)��

C

φ(4)��

CnA′

1

// C2n

Cid //

φ(2)��

C

φ(4)��

CnA′

2

// C2n

0(0) //

φ(3)��

C

φ(4)��

CnA′

3

// C2n

.

and we have dV1W (φ(1),φ(2),φ(3),φ(4))

= (A′1φ(1)−φ(4) id, A′

2φ(2)−φ(4) id, A′3φ(3)−φ(4) id) = (A′

1φ(1)−φ(4) id, A′2φ(2)−φ(4) id,0),

56

which gives the matrix

dV1W =

A′1 0 0 − id2n

0 A′2 0 − id2n

0 0 0 0

.

This matrix is singular as is, but if we restrict to the maps φ(1),φ(2),φ(4), which are not

forced to be identically zero, we get the matrix,

dV1W =

n n 2n

2n A′1 0 − id2n

2n 0 A′2 − id2n

,

and dVW as the map,

dVW : Hom(C,Cn)⊕Hom(C,Cn)⊕Hom(C,C2n) → Hom(C,C2n)⊕Hom(C,C2n).

By column operations we can get,

dV1W =

(A′

1 − id2n 00 − id2n A′

2

).

which is exactly one of the matrices we chose in Example 5.4.2 with this quiver. We also

know that the Schofield semi-invariant, cV1W = det(dV1

W ) = det([A′1|A′

2]), is in fact a semi-

invariant for the representation W from previous results. Similarly, we can obtain

dV2W =

(A′

1 − id2n 00 − id2n A′

3

), dV3

W =(

A′2 − id2n 0

0 − id2n A′3

)

from the other indecomposable representations V2 and V3, giving Schofield semi-invariants

cV2W = det([A′

1|A′3]), and cV3

W = det([A′2|A′

3]),

which are exactly the other two semi-invariants found previously.

Using equation 6.2.1 in §6.2 we get the same weights for these semi-invariants as was

found in §5.4.2.

57

6.4 Schofield Semi-Invariants for a Quiver with Dynkin Diagram E6

Now, let Q be the following quiver

•5

a5

��•1 a1// •3 a3

// •6 •4a4oo •2a2

oo


W : C2n

A′5 ��

CnA′

1

// C2nA′

3

// C3n C2nA′

4

oo CnA′

2

oo

Letαbe the dimension vector of an indecomposable representation of Q, andβ= (n,n,2n,2n,3n)

be the dimension vector of the representation W above. The Auslander-Reiten quiver

computed for this quiver gives all indecomposables, and the indecomposable represen-

tations of Q such that ⟨α,β⟩ = 0 that we need are,

V1 : C

id��

0(0)// 0

(0)// C C

idoo C

idoo

, V2 : C

id��

Cid// C

id// C 0

(0)oo 0

(0)oo

,

V3 : 0

(0)��

Cid// C

id// C C

idoo 0

(0)oo

, V4 : 0

(0)��

0(0)// C

id// C C

idoo C

(0)oo

,

V5 : C

id��

0(0)// C

id// C C

idoo 0

(0)oo

.

58

6.4.1 cV1W and cV2

W

For the map dV1W we have the following noncommutative diagram,

C

id

}}

φ(5)

��

0

φ(1)

��

(0) // 0(0) //

φ(3)

��

C

φ(6)

��

Cidoo

φ(4)

��

C

φ(2)

��

idoo

C2n

A′5

}}Cn

A′1

// C2nA′

3

// C3n C2nA′

4

oo CnA′

2

oo

Giving five sub-diagrams,

C

φ(1)��

C

φ(5)��

idoo

C3n C2nA′

5

oo

0(0) //

φ(1)��

0

φ(3)��

CnA′

1

// C2n

0(0) //

φ(3)��

C

φ(6)��

C2nA′

3

// C3n

C

φ(6)��

C

φ(4)��

idoo

C3n C2nA′

4

oo

C

φ(4)��

C

φ(2)��

idoo

C2n CnA′

2

oo

.

We have that φ(1) = 0 and φ(3) = 0, so we have

dV1W : Hom(C,Cn)⊕Hom(C,C2n)⊕Hom(C,C2n)⊕Hom(C,C3n)

→ Hom(C,C2n)⊕Hom(C,C3n)⊕Hom(C,C3n)

59

and

(A′1 ·0−φ(3) ·0, A′

2φ(2)−φ(4) id, A′3 ·0−φ(6) ·0, A′

4φ(4)−φ(6) id, A′5φ(5)−φ(6))

giving the matrix,

dV3W =

0 0 0 0 0 00 A′

2 0 − id2n 0 00 0 0 0 0 00 0 0 A′

4 0 − id3n

0 0 0 0 A′5 − id3n

.

Sinceφ(1) ∈ Hom(0,Cn) andφ(6) ∈ Hom(0,C2n) we want to restrict to the mapsφ(2),φ(4),φ(5),φ(6),

which are not forced to be the zero map, and we get

(A′2φ(2)−φ(4) id, A′

4φ(4)−φ(6) id, A′5φ(5)−φ(6))

giving the matrix

dV1W =

n 2n 2n 3n

2n A′2 − id2n 0 0

3n 0 A′4 0 − id3n

3n 0 0 A′5 − id3n

.

Now, as was previously shown, by a block-column operation, C2 ↔C3, on the matrix, we

can get,

dV1W =

n 2n 3n 2n

2n A′2 − id2n 0 0

3n 0 A′4 − id3n 0

3n 0 0 − id3n A′5

giving the Schofield semi-invariant cV1

W = det(dV1W ) = det([A′

4 A′2|A′

5]), which is exactly one

of the semi-invariants found previously in §5.4.3 for the representation W of Q. In a

60

similar fashion we can see that,



and

(A′1 ·φ(1)−φ(3) · id, A′

2 ·0−φ(4) ·0, A′3φ(3)−φ(6) id, A′

4 ·0−φ(6) ·0, A′5φ(5)−φ(6))

= (A′1 ·φ(1)−φ(3) · id,0, A′

3φ(3)−φ(6) id,0, A′5φ(5)−φ(6)).

So,

dV2W =

A′1 − id2n 0 0

0 A′3 − id3n 0

0 0 − id3n A′5

,

giving the Schofield semi-invariant cV2W = det([A′

3 A′1|A′

5]), again, one of our previously

found semi-invariants.

Using the formulas given in §6.2 we can compute σ1 and σ2 as follows,

σ(1) =α(1)− ∑a∈Q1,ha=x

α(t a) = 0−0 = 0

σ(2) =α(2)− ∑a∈Q1,ha=x

α(t a) = 1−0 = 1

σ(3) =α(3)− ∑a∈Q1,ha=x

α(t a) = 0−0 = 0

σ(4) =α(4)− ∑a∈Q1,ha=x

α(t a) = 1−1 = 0

σ(5) =α(5)− ∑a∈Q1,ha=x

α(t a) = 1−0 = 1

σ(6) =α(6)− ∑a∈Q1,ha=x

α(t a) = 1−2 =−1

thus σ1 = (0,1,0,0,1,−1), exactly as was computed in §5.4.3. Similarly we compute σ2 =(1,0,0,0,1,−1) as in §5.4.3.

61

6.4.2 cV3W and cV4

W

For the representation V3 we have the noncommutative diagram for the map dV3W as,

0

(0)

}}

φ(5)

��

0

φ(1)

��

(0) // Cid //

φ(3)

��

C

φ(6)

��

Cidoo

φ(4)

��

C

φ(2)

��

idoo

C2n

A′5

}}Cn

A′1

// C2nA′

3

// C3n C2nA′

4

oo CnA′

2

oo

giving five sub-diagrams,

C

φ(6)��

0

φ(5)��

(0)oo

C3n C2nA′

5

oo

0(0) //

φ(1)��

C

φ(3)��

CnA′

1

// C2n

Cid //

φ(3)��

C

φ(6)��

C2nA′

3

// C3n

C

φ(6)��

C

φ(4)��

idoo

C3n C2nA′

4

oo

C

φ(4)��

C

φ(2)��

idoo

C2n CnA′

2

oo

from this we see that φ(5) = 0 and φ(1) = 0, and we have



62

and

(A′1 ·0−φ(3) · (0), A′

2φ(2)−φ(4) id, A′3φ(3)−φ(6) id, A′

4φ(4)−φ(6) id, A′5 ·0−φ(6) ·0)

= (0, A′2φ(2)−φ(4) id,0, A′

4φ(4)−φ(6) id, A′5φ(5)−φ(6) · (0))

giving,

dV3W =

0 0 0 0 0 00 A′

2 − id2n 0 0 00 0 A′

3 0 0 − id3n

0 0 0 A′4 0 − id3n

0 0 0 0 0 0

.

Restricting to the maps φ(2),φ(4),φ(3),φ(6) we get

dV3W =

A′2 − id2n 0 0

0 A′3 0 − id3n

0 0 A′4 − id3n

.

As was previously shown, by a reordering of the vertices in Q we can get,

dV3W =

A′2 − id2n 0 0

0 A′4 − id3n 0

0 0 − id3n A′3

giving the Schofield semi-invariant cV3W = det(dV3

W ) = det([A′4 A′

2|A′3]), which is another

one of the semi-invariants we have already found for the representation W of Q. In a

similar fashion we can see that,



and

(A′1 ·φ(1)−φ(3) id, A′

2 ·0−φ(4) ·0, A′3φ(3)−φ(6) id, A′

4φ(4)−φ(6) id, A′5 ·0−φ(6) ·0)

= (A′1 ·φ(1)−φ(3) id,0, A′

3φ(3)−φ(6) id, A′4φ(4)−φ(6) id, A′

5φ(5)−φ(6) · (0)).

63

So,

dV4W =

A′1 − id2n 0 0

0 A′3 − id3n 0

0 0 − id3n A′4

giving the Schofield semi-invariant cV4

W = det([A′3 A′

1|A′4]), again, one of our semi-invariants

from previous work in §5.4.3. Computing the σ3 and σ4 using equation 6.2.1 in §6.2 we

get σ1 = (1,0,0,1,0,−1) and σ4 = (0,1,1,0,0,−1) as was found in §5.4.3.

6.4.3 cV5W

We have the noncommutative diagram,

C

id

}}

φ(5)

��

0

φ(1)=0

��

(0) // Cid //

φ(3)

��

C

φ(6)

��

Cidoo

φ(4)

��

0

φ(2)=0

��

(0)oo

C2n

A′5

}}Cn

A′1

// C2nA′

3

// C3n C2nA′

4

oo CnA′

2

oo

giving five sub-diagrams,

C

φ(6)��

C

φ(5)��

idoo

C3n C2nA′

5

oo

64

0(0) //

φ(1)=0��

C

φ(3)��

CnA′

1

// C2n

Cid //

φ(3)��

C

φ(6)��

C2nA′

3

// C3n

C

φ(6)��

C

φ(4)��

idoo

C3n C2nA′

4

oo

C

φ(4)��

0

φ(2)=0��

(0)oo

C2n CnA′

2

oo

If we compute the matrix of dV5W we have

dV5W =

A3 0 0 − id0 A4 0 − id0 0 A5 − id

.

So we have cV5W = det(dV5

W ). Using the equations in §6.2 we have σ5 = (0,0,1,1,1,−2) as

was found in 5.4.3.

6.4.4 The Kronecker 2-Quiver


• ((66 •

with representation

W : CnA **

B44 Cn

where A and B are n ×n matrices. Let Vλ (indecomposable) be the representation

V : Cid))

λ

55 C

for λ 6= 0 ∈C. Then we have the following noncommutative diagram,

C

φ(1)��

id))

λ

55 C

φ(2)��

CnA **

B44 Cn

65

and so we have

dVλW =

(A − idB −λ id

)

with id an n×n identity matrix. Each cVλ has weight (1,−1), giving a parametrized family

of cVλ ’s, where cVλ(W ) = det(dVλW ). Since λ is arbitrary, we may regard it as an indetermi-

nate, and consider cVλ(A,B) ∈C[Rep(Q, (n,n))][λ]. For n = 2 we have

det(B −λA) =λ2c2(A,B)−λc1(A,B)+ c0(A,B)

a polynomial in C[Rep(Q, (n,n))] = C[a11, a12, a21, a22,b11,b12,b21,b22][λ] of degree two

in λ, where each ci is a polynomial in C[ai j ,bi j ], where A = (ai j ) and B = (bi j ). In gen-

eral, expanding det(B −λA) using the definition of the determinant we have

det(B −λA) =λncn(A,B)+·· ·λc1(A,B)+ c0(A,B)

where the ci j (A,B) ∈C[ai j ,bi j ]. According to [19] p.7

1. ci (A,B) are also invariants of weight (1,−1).

2. ci (A,B) are linearly independent.

3. ci (A,B) span SI(Q, (n,n))(1,−1).

Hence, unlike the case where the quiver is a Dynkin diagram of type ADE, SI(Q,α)σ

can have vector space dimension ≥ 1, since we have n + 1 linearly independent semi-

invariants here.

Proof. 1. Let G = (g1, g2) ∈ GL((n,n)). Then G · (A,B) = (g2 Ag−11 , g2B g−1

1 ), so

G · cV (W ) = cV (g−12 Ag1, g−1

2 B g1)

=λncn(g−12 Ag1, g−1

2 B g1)+·· ·+λc1(g−12 Ag1, g−1

2 B g1)+ c0(g−12 Ag1, g−1

2 B g1).

Since cV (W ) is a semi-invariant of weight (1,−1),

G · cV (W ) = det(g1)det(g2)−1cV (W ) = det(g1)det(g2)−1(n∑

i=0λi ci (A,B)).

66

Regarding λ as an indeterminate and equating the coefficients of λi we get

Q · ci = ci (g−12 Ag1, g−1

2 B g1) = det(g1)det(g2)−1ci (A,B)

so that ci (A,B) is a semi-invariant of weight (1,−1).

2. The ci (A,B) are linearly independent because

cV (A,B) = det

(A − idB −λ id

).

Using the definition of the determinant ci (A,B) is a sum of monomials that have i

variables from {akl } and n − i variables from {bkl }; hence the total degree in {akl }

of ci (A,B) is i and the ci (A,B) are linearly independent.

3. The fact that the ci (A,B) span SI(Q, (n,n))(1,−1) is due to the fact that the Schofield

semi-invariants cV , where V is an indecomposable representation with dimension

vector α such that ⟨α,β⟩ = 0, span SI(Q,β)σ. Proving this would require an discus-

sion of homological techniques and the use of Lemma 1 in [5] which would take us

too far afield, thus we leave the reader to investigate the proof given in [5] Section

4. The proof that the Schofield semi-invariants span the ring of semi-invariants

for the Kronecker 2-quiver actually forms the basis of the proof in [5] Section 4 of

the main theorem that states for any quiver Q without oriented cycles, that the

Schofield semi-invariants actually span any SI(Q,β)σ.

67

Chapter 7: Construction of the Irreducible Polynomial

Representations of GL(V )

7.1 Partitions

Here we introduce the language of partitions and Young diagrams.

Definition 7.1.1. A partition of n ∈ N is a tuple λ = (λ1,λ2, ...,λr ) of positive integers

with λ1 ≥ λ2 ≥ ·· · ≥ λr and |λ| :=∑ri=1λi = n. We denote this by λ` n, and we say λ is a

partition of n. If λ is an r -tuple, i.e. λ has r components λi , then we say λ has r parts, or

that λ is of height r . We denote this by ht(λ) = r .

To every partition λ, we can associate a Young diagram, for example, the Young diagram

of the partition λ= (6,4,3,1) is

.

Taking the transpose of a Young diagram give the conjugate partition λ′. For example if

λ= (6,4,3,1) then λ′ = (4,3,3,2,1,1) and the corresponding Young diagram is,

.

Let V ×λ be the set of all maps from the boxes of a Young diagram to the vector space V .

The Cartesian product V ×λ is isomorphic to V n = V |λ| since it is just V ×n with compo-

nents indexed by the boxes of the Young diagram. A map from a Young diagram to V

68

can be thought of as a labeling of the Young diagram by elements of V . For example, if

a,b,c,d ,e, f , g ,h, i ∈V then

a b c d ef g hi

is one such labeling. This defines an element v ∈V ×λ.

Example 7.1.2. Let λ= (3,2,1). Then V ×(3,2,1) ∼=V 6 and the elements of V ×λ are given by

all labelings,

a b cd ef

with a,b,c,d ,e, f ∈V .

We would now like to construct a GL(V )-module Sλ(V ), for some GL(V )-module V (a

C-vector space) and some partition λ. Again, denote the Cartesian product of n copies

of V by V ×n . Indexing the components by λ we have V ×n ∼=V ×λ, where V ×λ is just V ×n

indexed by the boxes of a Young diagram of shape λ. So, an element in V ×λ will just

be a labeling of the Young diagram by elements of V . We now define relations on the

elements of V ×λ via exchanges.

Definition 7.1.3. Let v ∈ V ×λ. An exchange takes two subsets of elements of the same

size from two columns of the Young diagram of shapeλ, then exchanges the two subsets,

while maintaining their order (vertically). For example

a b cd e fgh

7−→c b ad e hgf

is an exchange on the first and third columns of the Young diagram of shapeλ= (3,3,1,1),

exchanging the elements {a,h} ↔ {c, f }.

69

Definition 7.1.4. Now, let φ : V ×λ → W be a map from V ×λ to some C-vector space W

with the following three properties:

1. φ is multilinear, i.e. linear in each component.

2. φ is alternating in the entries of any column of the Young diagram of shape λ,

i.e. a permutation σ of the entries introduces a coefficient of sgn(σ). So, if two

entries in the same column are equal, φ vanishes on this v ∈V ×λ. This along with

the previous condition implies that if v ′ is obtained by an exchange on v , then

φ(v) =−φ(v ′).

3. For all v ∈ V ×λ we have φ(v) = ∑φ(w) where the sum is over all w obtained from

v via an exchange of two given columns, with a given subset in the right chosen

column.

Definition 7.1.5. Define the Schur module Sλ(V ) to be the universal target C-vector

space for all φ with the properties of Definition 7.1.4.

Let λ= (λ1, ...,λr ) be a partition and let µ= (µ1, ...,µs) be the conjugate partition, i.e. µi

is the i th column of λ. Define a map,

∧: V ×λ→

µ1∧V ⊗

µ2∧V ⊗·· ·⊗

µs∧V

given by taking the exterior product (wedge) over all columns, and then taking the tensor

products of those antisymmetric products.

Example 7.1.6. The filling of the Young diagram λ= (6,4,2,1)

a b c d e fg h i jk lm

70

corresponds to

(a ∧ g ∧k ∧m)⊗ (b ∧h ∧ l )⊗ (c ∧ i )⊗ (d ∧ j )⊗e ⊗ f

which lies in the space,

4∧V ⊗

3∧V ⊗

2∧V ⊗

2∧V ⊗V ⊗V.

Remark 7.1.7. The universal target C-vector space mentioned in Definition 7.1.5 can

then be described as

Sλ(V ) =(µ1∧

V ⊗µ2∧

V ⊗·· ·⊗µs∧

V

)/(Qλ(V )

)where µ = (µ1, ...,µs) = λ′ is the conjugate partition to λ and Qλ(V ) is the subspace of⊗

i∧µi (V ) spanned by all ∧v −∑

w ∧w , where the sum is over all w obtained from v by

an exchange of two given columns and a given subset in the right chosen column. With

the first two properties of 7.1.4 we need only include the exchanges where the top box in

the right chosen column is selected (alone, or in any combination with the other boxes

in that column).

Example 7.1.8. Let λ= (2,2,1). Then, λ′ = (3,2), and

Sλ(V ) = S(2,2,1)(V ) =( 3∧

V ⊗2∧

V

)/(Q(2,2,1)(V )

)where Q(2,2,1)(V ) is spanned by all

∧ x uy vz

−∧ u xy vz

−∧ x yu vz

−∧ x zy vu

= (x ∧ y ∧ z)⊗ (u ∧ v)− (u ∧ y ∧ z)⊗ (x ∧ v)− (x ∧u ∧ z)⊗ (y ∧ v)− (x ∧ y ∧u)⊗ (z ∧ v)

and

∧ x uy vz

−∧ u xv yz

−∧ u xy zv

−∧ x yu zv

71

= (x ∧ y ∧ z)⊗ (u ∧ v)− (u ∧ v ∧ z)⊗ (x ∧ y)− (u ∧ y ∧ v)⊗ (x ∧ z)− (x ∧u ∧ v)⊗ (y ∧ z)

with all x, y, z,u, v ∈V .

Example 7.1.9. Taking λ= (n) we get

S(n)(V ) ∼= Symn(V )

the nth symmetric power of V , i.e. the quotient of the nth tensor product V ⊗n of V , by

the relations generated by all

v1 ⊗ v2 ⊗·· ·⊗ vn − vσ(1) ⊗ vσ(2) ⊗·· ·⊗ vσ(n)

for some permutation σ ∈ Sn , the symmetric group on n-letters.

Example 7.1.10. If we let λ= (1,1, ...,1) = (1n), then we have that

S(1n )(V ) =n∧

V.

Definition 7.1.11. Let λ be a partition. A tableau is a filling of the Young diagram corre-

sponding to λ with positive integers that are weakly increasing along rows, and strictly

increasing on columns. We call tableaux with these two properties row semi-standard

and column standard.

Example 7.1.12. For example,

1 1 1 2 22 2 3 33 4 45 6 .

Now, for any vector space V , choose a basis B = {e1,e2, ...,en}. To each tableau T of a

partition λ with entries in {1,2, ...,n} we can associate an element eT ∈ Sλ(V ) as in the

following example.

72

Example 7.1.13.

T =1 1 1 2 22 2 3 33 4 45 6

is associated to

eT = (e1 ∧e2 ∧e3 ∧e5)⊗ (e1 ∧e2 ∧e4 ∧e6)⊗ (e1 ∧e3 ∧e4)⊗ (e2 ∧e3)⊗e2 +Qλ(V ) ∈ Sλ(V ).

Theorem 7.1.14. (see [9] Theorem 1, 8.1) The elements

{eT : T is a tableau on λ with entries in {1,2, ...,n}}

form a basis for Sλ(V ).

Remark 7.1.15. From Theorem 7.1.14 it follows that

Sλ(V ) = 0 ⇐⇒ λ has more than n = dim(V ) parts (rows)

as this corresponds to an SλV with an exterior product of more than n = dim(V ) copies

of V ; and

dim(Sλ(V )) = 1 ⇐⇒ λ= (kn) for some k.

which corresponds to SλV = (∧n V )⊗k .

Remark 7.1.16. We have thus far written Sλ(V ) but we will often write SλV to mean

Sλ(V ) in the future.

7.2 Polynomial Representations and Schur Modules

Definition 7.2.1. For any vector space V and any group G , V is a G-module or a G-

representation if there is a group homomorphism

ρ : G → GL(V )

73

or equivalently, if there is left action, g · v , on elements of v by elements of G such that

the following four properties hold,

1. g · v ∈V .

2. g · (c1v1 + c2v2) = c1(g · v1)+ c2(g · v2) for any c1,c2 ∈C.

3. (g h) · v = g · (h · v).

4. idG ·v = v .

for all g ,h ∈G , v, v1, v2 ∈V .

Definition 7.2.2. Let G be a group acting on a set S. Define CS to be the vector space

given by all formal linear combinations of elements of S. Then CS is a G-module, where

the action is given by linear extension of the action on S,

g ·(∑

ici si

)=∑

ici (g · si ).

Definition 7.2.3. The group ring or group algebra, denoted CG is the G-module given

by letting G act on itself, i.e. G = S in the previous definition.

Remark 7.2.4. If V and W are two representations of a group G , then V ⊗W is also a

G-module with action

g · (v ⊗w) = g v ⊗ g w.

Further, if V is a G-representation, then V ∗ is as well. It is given by

ρ∗(g ) = ρ(g−1)t : V ∗ →V ∗

i.e. it is the transpose of the representation of ρ(g−1) = ρ(g )−1. From this we gather that

V ∗⊗W ∼= Hom(V ,W )

74

is also a G-module if V and W are. Moreover, any tensor power of a G-module V is a

G-module, as are symmetric and anti-symmetric (exterior) powers of V , i.e. Symn(V ) =V ⊗n/R and

∧n V =V ⊗n/R, with the relatons R generated by v1⊗·· ·⊗vn−vσ(1)⊗·· ·⊗vσ(n),

and v1 ⊗·· ·⊗ vn − sgn(σ)vσ(1) ⊗·· ·⊗ vσ(n) respectively.

Definition 7.2.5. Suppose V and W are G-representations (modules). A G-map, i.e. a

map of G-modules respecting the action of G is any linear map φ : V →W such that the

following diagram commutes

Vφ //

g��

W

g��

Vφ//W

for any g ∈G . For two G-modules V and W denote the set of all G-maps by

Hom(V ,W )G = {φ ∈ Hom(V ,W ) :φ is a G-module map}.

Definition 7.2.6. Schur’s Lemma tells us that for any two irreducible modules, any map

between them is either the zero map, or an isomorphism. If V is irreducible, by Schur’s

Lemma we have that dimHom(V ,W )G is the multiplicity of V in W , i.e. it is the number

of distinct isomorphisms g ∈ Hom(V ,W ), sending V to an isomorphic submodule of W .

Similarly if W is irreducible. If both V and W are irreducible then dimHom(V ,W )G = 1

if V ∼=W , and it is zero otherwise.

Definition 7.2.7. Now, suppose we have a Young diagram associated to the partition

λ = (λ1,λ2, ...,λk ) of n, define the subset of operators Rλ ⊂ Sn , of the symmetric group

on n letters, which permutes the elements of the rows, but not the columns of λ, as

Rλ = Sλ1 ×Sλ2 ×·· ·×Sλk ,

where Sλi is the symmetric group inside Sn that permutes only elements of row i in the

diagram and fixes all other elements. Similarly define the subset of operators Cλ ≤ Sn ,

75

permuting columns only, as

Cλ = Sλ′1×·· ·×Sλ′

r,

where each Sλ′j

permutes only column j of λ.

Example 7.2.8. Suppose we have λ= (6,4,3,1) with the associated Young diagram,

where the boxes are labeled as

a b c d e fg h i jk l mn .

The subgroup Rλ is

S{a,b,c,d ,e, f } ×S{g ,h,i , j } ×S{k,l ,m} ×S{n}

and the subgroup Cλ is

S{a,g ,k,n} ×S{b,h,l } ×S{c,i ,m} ×S{d , j } ×S{e} ×S{ f }.

Definition 7.2.9. Define the row stabilizer of λ as

rλ =∑σ∈Rλ

σ.

and the column stabilizer of λ as

cλ =∑τ∈Cλ

sgn(τ)τ.

These are each operators on the space V ⊗n .

Definition 7.2.10. Define the Young symmetrizer to be

eλ = cλrλ.

76

There is an action of Sn on V ⊗n given by

σ · (v1 ⊗ v2 ⊗·· ·⊗ vn) = vσ(1) ⊗ vσ(2) ⊗·· ·⊗ vσ(n).

So we have the following equivalent definition of Sλ(V ).

Definition 7.2.11. We can think of the construction of the Sλ(V ) in a slightly different

way. Quotienting out by the exchange relations Qλ(V ) can be thought of in terms of the

Young symmetrizer, namely we can define

Sλ(V ) = eλV ⊗n

i.e. the image under the action of the Young symmetrizer eλ on V ⊗n . SλV is a quotient

of

cλV ⊗n =µ1∧

V ⊗µ2∧

V ⊗·· ·⊗µr∧

V

by the relations Qλ(V ) as shown previously, where the µi are the columns of the Young

diagram. It is also a subspace of

rλV ⊗n = Symλ1 (V )⊗Symλ2 (V )⊗·· ·⊗Symλk (V )

7.3 Irreducible Polynomial Representations of GL(V )

There is a natural action of GL(V ) on V ⊗n given by

A · (v1 ⊗ v2 ⊗·· ·⊗ vn) = (Av1 ⊗ Av2 ⊗·· ·⊗ Avn).

where A ∈ GL(V ). Thus, there is an obvious action of GL(V ) on Sλ(V ) induced by this

action. So, in particular, the Schur modules Sλ(V ) are GL(V )-modules. We will now

discuss some of the properties of the Schur modules as representations of GL(V ).

We can think of maps between Schur modules Sλ(V ) and Sλ(W ) as a sequences of maps

Sλ(V ) = eλV ⊗n // V ⊗n A⊗n//W ⊗n eλ // eλW ⊗n = Sλ(W )

77

where A ∈ Hom(V ,W ), since the Sλ(V ) and Sλ(W ) are subspaces of V ⊗n and W ⊗n re-

spectively. We will show a correspondence between the (polynomial) representation

C[Hom(V ,W )] and the representation Sλ(V )⊗ Sλ(W ∗). This will allow us to pass from

the language of quivers to the language of polynomial representations of GL(V )×GL(W )

and thus to phrase results about semi-invariants of quivers in terms of the Littlewood-

Richardson coefficients. A more detailed discussion and definition of polynomial repre-

sentations in general can be found in [9] pg. 112.

Definition 7.3.1. The representations Sλ(V ) of GL(V ) where λ is a partition of any posi-

tive integer n, with at most m := dim(V ) parts (rows) are called the irreducible polyno-

mial representations of GL(V ). The proof that they constitute all irreducible polynomial

GL(V )-representations can be found in [9] Theorem 2 §8.2. Further, direct sums, tensor

products, quotients by polynomial subrepresentations, and invariant subspaces of poly-

nomial representations are again polynomial representations. So, in particular, we have

that C[Hom(V ,W )] is a polynomial representation of GL(V )×GL(W ), and(∧dimV V

)⊗k,

Sym(V ), V ⊗k are all polynomial representations of GL(V ).

Remark 7.3.2. By the way we have defined Sλ(V ), if λ has more than n = dim(V ) parts,

sayλ has n+i parts for some i > 0, then we will have at least one exterior power∧n+i V =

0, and thus we will have at least the first term of Sλ(V ) as zero so

Sλ(V ) = 0⊗k2∧

V ⊗·· ·⊗kr∧

V /Qλ(V ) = 0

where λ′ = (k1,k2, ...,kr ) is the conjugate partition λ′, to λ; making Sλ(V ) = 0.

Definition 7.3.3. Let g ∈ GL(V ), and let dimV = n. Then we have a map

n∧g :

n∧V →

n∧V

78

given by

v1 ∧·· ·∧ vn 7→ g v1 ∧·· ·∧ g vn = det(g )(v1 ∧·· ·∧ vn)

For any integer k we have a one dimensional representation of GL(V ) called the deter-

minant representation ( n∧V

)⊗k

given by,

g · v 7→ det(A)k v, g ∈ GL(V ), v ∈( n∧

V

)⊗k

This representation will sometimes be denoted by

det(V )k

The partition λ associated to det(V )k is an n ×k rectangle. The action of GL(V ) on the

representation V ∗ then gives rise to the representation∧n V ∗ given by

v∗1 ∧·· ·∧ v∗

n 7→ (g−1)t v∗1 ∧·· ·∧ (g−1)t v∗

n = det(g )−1(v∗1 ∧·· ·∧ v∗

n)

for v∗i ∈V ∗. This gives an action on the representation

( n∧V ∗

)⊗k

given by

v 7→ det(g )−k v

for v ∈ (∧n V ∗)⊗k .

Remark 7.3.4. If W is any GL(V )-representation we can "twist" the representation to

W ⊗det(V )k . Further, for k, l ∈Z,

W ⊗det(V )k ⊗det(V )l ∼=W ⊗det(V )k+l .

79

If W is an irreducible representation of GL(V ) then for some sufficiently large positive

integer k we have that W ⊗det(V )k is an irreducible polynomial representation, and thus

equal to some Sλ(V ) for some λ.

If W = Sλ(V ) is already an irreducible polynomial representation then W ⊗det(V )k is

also an irreducible polynomial representation, so W ⊗det(V )k ∼= Sµ(V ) for some µ. If

λ= (λ1, ...,λr ), then

Sλ(V )⊗det(V )k = S(λ1+k,λ2+k,...,λm+k)(V )

where λi = 0 for i > r .

Definition 7.3.5. The representations of GL(V )

Sλ(V )⊗det(V )k

where k ∈ Z and λ is a partition with at most m − 1 = dim(V )− 1 parts, are called the

irreducible rational representations. For negative values of k, and dim(V ) = n, we have

det(V )k =( n∧

V ∗)⊗k

.

The fact that these are all irreducible rational representations can be found in [9] §8.2.

Remark 7.3.6. For distinct positive integers k, the Sλ(V )⊗det(V )k are all isomorphic as

representations of SL(V ). This follows immediately from the fact that the determinant is

1 for elements of SL(V ).

Theorem 7.3.7. The irreducible representations of SL(V ) are exactly all Sλ(V ) with λ a

partition with at most n −1 parts (rows), where n := dim(V ) (see [8] §15.2).

Remark 7.3.8. If W = Sλ(V ) is an irreducible representation of SL(V ), then W ∗ is also

an irreducible representation of SL(V ), so W ∗ = Sµ(V ) for some partition µ. In fact, it is

80

well known (for example [9], and [8] §15.3 pg. 223) that the following must be true of λ

and µ

λ1 +µn =λ2 +µn−1 = ·· · =λn +µ1

i.e. λi −µi is constant over all i . This condition implies λ and µ must have the same

number of parts, and that the Young diagram for µ when rotated 180◦, fits together with

the Young diagram of λ to form a rectangle of size m ×k for some integer k.

In other words, as a consequence of Schur’s Lemma and definition 7.2.5,

(Sλ(V )⊗Sµ(V ))SL(V ) ∼= Hom((Sλ(V ))∗,Sµ(V ))SL(V ) 6= 0

if and only if Sµ(V ) ∼= (Sλ(V ))∗ as SL(V )-representations. So,

(Sλ(V )⊗Sµ(V ))SL(V ) 6= 0

if and only if λ and µ fit together to make an m ×k rectangle for some positive integer k

and m = dim(V ), in which case we have that

dim(SλV ⊗SµV )SL(V ) = dim(SλV ⊗SλV ∗)SL(V ) = dim(SλV ∗⊗SλV )SL(V )

= dimHom(SλV ,SλV )SL(V )

= 1

So, SλV ⊗SµV gives us exactly one SL(V )-invariant representation and it is one dimen-

sional. In particular it is obtained from the representation

S(λ1+µn ,...,λn+µ1)V =(

dimV∧V

)⊗(λ1+µn )

and is a power of the determinant representation. This gives a power of a polynomial in-

variant det(g )λ1+µn under the action of SL(V ), and we say this invariant has weight λ1 +µn . In the future, this will be shown to correspond to some weight σ(x), a determinantal

81

character of a factor GL(β(x)) of GL(β), as we have calculated previously in §6, where

we discussed Schofield semi-invariants cVW , characters χ(g ) = ∏

x∈Q0 det(gx)σ(x); g =(gx)x∈Q0 ∈ GL(β), and weights σ(x). We often abuse language and say that SλV ⊗ SµV

contains an SL(V )-invariant, when in fact what we mean is that it contains an irreducible

SL(V )-invariant representation (corresponding to the partition ν= (λ1+µn)dimV ), which

yields an SL(V )-polynomial invariant. The details of the computation of this particular

representation inside the tensor product SλV ⊗SµV yielding an SL(V )-invariant will be

addressed in §9, where we discuss the computations of the Littlewood-Richardson coef-

ficients.

82

Chapter 8: Computations with Schur Functors

The map Sλ : V → Sλ(V ), taking a vector space V to the Schur module (GL(V )-representation)

SλV , is what is known as a polynomial functor from the category of vector spaces to it-

self. The functor on maps A : V → W , induces maps between Schur modules Sλ(A) :

SλV → SλW . For a brief description of basic notions in category theory see Appendix

A. For a description of polynomial functors and how the Sλ(•) can be viewed as polyno-

mial functors see [17] pg. 273. Now that we have defined the Schur module Sλ(V ) and

have given a description of the irreducible representations of GL(V ) and SL(V ), we will

describe and use these Schur functors. We compute some specific examples for certain

quivers and representations, and we give an example computation for the triple flag

quiver, a quiver which will become important for discussing the saturation conjecture

for the Littlewood-Richardson coefficients, a result proven first by Tao and Knutson us-

ing so-called puzzles and honeycombs, mathematical gadgets invented by the authors,

in [15]. This result is also proven by Derksen and Weyman in [5], which we now describe.

Definition 8.0.9. The Littlewood-Richardson coefficients cνλ,µ are the nonnegative in-

tegers in the following equation

Sλ(V )⊗Sµ(V ) = ⊕|ν|=|λ|+|µ|

cνλ,µSν(V )

giving the multiplicity of the Schur module Sν(V ) in the tensor product of the Schur

modules Sλ(V )⊗Sµ(V ). Here cνλ,µSν(V ) denotes the direct sum of cν

λ,µ copies of Sν(V ).

8.1 Preliminaries

Here we establish some basic facts in order to compute examples for various quivers.

83

Recall 8.1.1. The vector spaces Hom(V ,W ) ∼= V ∗⊗W , are isomorphic. Choose a basis

BV = {e1,e2, ...,em}. Then a basis for V ∗ is BV ∗ = {e∗1 ,e∗

2 , ...,e∗m}, where e∗

i ∈ Hom(V ,C)

and e∗i (e j ) = δi

j . Choose some basis for W , BW = { f1, f2, ..., fn}. Now define a map

Hom(V ,W ) →V ∗⊗W

by x11 x12 · · · x1m

x21 x22 · · · x2m...

.... . .

...xn1 xn2 · · · xnm

7→n∑

i=1(

m∑j=1

xi j (e∗i ⊗ f j )).

A basis on Hom(V ,W ) is given by the matrices {Ei j }, where the (i , j ) entry is one and all

other entries are zero, so this map defines a bijection on bases of Hom(V ,W ) and V ∗⊗W ,

thus an isomorphism of vector spaces.

Lemma 8.1.2. There is an isomorphism of vectors spaces (V ∗⊗W )∗ ∼=V ⊗W ∗

Proof. This is clear since for any finite dimensional vector space X we have that (X ∗)∗ ∼=X .

Lemma 8.1.3. For vector spaces V ,W , and Z , we have that

((V ∗⊗Z )⊕ (W ∗⊗Z ))∗ ∼= (V ⊗Z∗)⊕ (W ⊗Z∗).

Lemma 8.1.4. For vector spaces X and Y we have

Sym(X ⊕Y ) ∼= Sym(X )⊗Sym(Y ).

Proof. This follows from the fact that C[V ⊕W ] ∼= C[V ]⊗C[W ]. Choosing a basis of V

and W we have

C[V ⊕W ] ∼=C[v1, ..., vn , w1, ..., wm] ∼=C[v1, ..., vn]⊗C[w1, ..., wm] ∼=C[V ]⊗C[W ].

84

Remark 8.1.5. Since Sλ(V )⊗Sµ(V ) is a polynomial representation of GL(V ), it must de-

compose into irreducible polynomial representations

Sλ(V )⊗Sµ(V ) = ⊕|ν|=|λ|+|µ|

cνλ,µSν(V )

for some nonnegative cνλ,µ ∈N∪ {0}.

Remark 8.1.6. Sν(V ⊕W ) gives a polynomial representation of GL(V )×GL(W ), and the

irreducible polynomial representations of GL(V )×GL(W ) are simply irreducible repre-

sentation of GL(V ) tensored with irreducible representations of GL(W ). So, we have a

decomposition

Sν(V ⊕W ) = ⊕|λ|+|µ|=|ν|

cνλ,µ(Sλ(V )⊗Sµ(W )).

Some special cases are λ= (n) giving

S(n)(V ⊕W ) ∼= Symn(V ⊕W ) ∼=⊕

a+b=n

S(a)(V )⊗S(b)(W )

and λ= (1n) = (1,1, ....,1) giving

S(1n )(V ) ∼=n∧

(V ⊕W ) ∼=⊕

a+b=n

a∧(V )⊗

b∧(W ).

The following formulas are known as the Cauchy formulas and can be found in [17]

§9.6.3.

S(n)(V ⊗W ) ∼= Symn(V ⊗W ) = ⊕|λ|=n

Sλ(V )⊗Sλ(W )

where Symn(X ) is the homogeneous degree n symmetric functions on a vector space X ,

and

S(1n )(V ⊗W ) ∼=n∧

(V ⊗W ) = ⊕|λ|=n

Sλ(V )⊗Sλ′(W )

where λ′ is the conjugate partition.

85

Remark 8.1.7. From the Cauchy formulas we can derive the following equalities,

C[Hom(V ,W )] =C[V ∗⊗W ] = Sym((V ∗⊗W )∗)

= Sym(V ⊗W ∗)

= ⊕n≥0

Symn(V ⊗W ∗)

= ⊕n≥0

⊕|λ|=n

Sλ(V )⊗Sλ(W ∗)

=⊕λ

Sλ(V )⊗Sλ(W )∗.

Remark 8.1.8. Since C[V ⊕W ] ∼=C[V ]⊗C[W ], we have a decomposition

C[Rep(Q,β)] =C[ ⊕

a∈Q1

Hom(V (t a),V (ha))

]

=C[ ⊕

a∈Q1

V (t a)∗⊗V (ha)

]

= ⊗a∈Q1

C[V (t a)∗⊗V (ha)]

= ⊗a∈Q1

Sym(V (t a)⊗V (ha)∗)

so we have that C[Rep(Q,β)] =⊗a∈Q1

Sym(V (t a)⊗V (ha)∗). A justification of the use of

equalities is given by [17] pg. 270-277. In this case we are identifying the above spaces as

spaces of functions, as well as isomorphic GL(V )×GL(W ) representations. We discuss

this in detail in Appendix C. Now, observing the application of the Cauchy formula above

86

we see that

C[Rep(Q,β)] =C[ ⊕

a∈Q1

V (t a)∗⊗V (ha)

]

= Sym

( ⊕a∈Q1

V (t a)∗⊗V (ha)

)

= ⊗a∈Q1

Sym(V (t a)⊗V (ha)∗)

= ⊕n≥0

⊗a∈Q1

Symn(V (t a)⊗V (ha)∗) using the grading of Sym(X )

= ⊕n≥0

⊗a∈Q1

[ ⊕λ(a)`n

Sλ(a)V (t a)⊗Sλ(a)V (ha)∗]

using the Cauchy formula Symn(V ⊗W ∗) = ⊕λ`n

Sλ(V )⊗Sλ(W )∗

= ⊕λ(a): a∈Q1

⊗a∈Q1

Sλ(a)V (t a)⊗Sλ(a)V (ha)∗.

So,

C[Rep(Q,β)] = ⊕λ:Q1→P

⊗a∈Q1

Sλ(a)V (t a)⊗Sλ(a)V (ha)∗

as GL(V )-representations, where we treat λ : Q1 →P given by

a 7→λ(a),

as a function from arrows in Q1 to partitions in P , the set of all partitions, and the sum is

over all such functions. The only one dimensional representations of GL(V ) are powers

of the determinant, i.e. (∧dimV V )⊗k , where g acts by det(g )k for all g ∈ GL(V ), k ∈Z.

Thus, for there to be an SL(V )-invariant in a tensor product

SλV ⊗SµV = ⊕|ν|=|λ|+|µ|

cνλ,µSνV

we need there to be a summand of the form S(kn )V = (∧n V )⊗k , where n = dimV , in the

summation on the right. This happens when λi −µi is constant for all i , i.e. λ and µ fit

87

together to make a rectangle when we rotate µ by 180◦. Each semi-invariant S(kn )V =

S(σ(x)β(x))V (x), for some x ∈Q0, and where σ(x) is the weight evaluated at x defined in §6

when we discussed the Schofield semi-invariants. From this we deduce that for there to

be a SL(β) =∏x∈Q0 SL(V (x))-invariant of weight σ= (σ(x1), ...,σ(xr )) in

C[Rep(Q,β)] = ⊕λ:Q1→P

⊗a∈Q1


we need to find an occurrence of

⊗x∈Q0

S(σ(x)β(x))V (x),

where the σ(x) are the determinantal weights discussed in §6. The dimension of the

weight space SI(Q,β)σ is then the multiplicity (see definition 7.2.6) of

⊗x∈Q0

S(σ(x)β(x))V (x)

in ⊕λ:Q1→P

⊗a∈Q1

Sλ(a)V (t a)⊗Sλ(a)V (ha)∗,

i.e.

dimSI(Q,β)σ = dimHom

( ⊗x∈Q0

S(σ(x)β(x))V (x),⊕

λ:Q1→P

⊗a∈Q1

Sλ(a)V (t a)⊗Sλ(a)V (ha)∗)SL(U )

.

We calculate these using methods that we develop in the next chapter to calculate Littlewood-

Richardson coefficients. We introduce skew Young diagrams and skew Young tableaux,

and an algorithm to compute the nonnegative integers cνλ,µ. We then provide some ex-

amples of computations with triple flag quivers.

We also use the Cauchy formulas to relate Schur modules and partitions to the coor-

dinate rings C[Rep(Q,β)], and describe what conditions are necessary for there to be a

88

determinantal semi-invariant. Namely we give requirement on dimension vectorsβ and

weights σ, and partitions λ,µ, and ν related to a triple flag quiver Q, in order for the ring

of semi-invariants SI(Q,β)σ to be nonzero. The tools and methods of §10.3 is necessary

and fundamental to proving the saturation conjecture using a bijection defined by H.

Derksen and J. Weyman in [6] between sets related to weight spaces

Σ(Q,β) = {σ : SI(Q,β)σ 6= 0}

and sets related to triples of partitions (λ,µ,ν) and the Klyachko cone, all of which we

define and discuss in detail in §10.

89

Chapter 9: Examples Using the Cauchy Formula

9.1 Skew Tableaux, Semi-invariants of GL(V ), and Products of Schur

Modules

Definition 9.1.1. Suppose λ,µ, and ν are Young diagrams. Suppose further that when

λ is aligned with the top left corner of ν that λ is fully contained in ν. Then we define

the skew diagram ν/λ to be the diagram left after deleting λ from ν. We say that ν/λ

has content µ if there are µ1 1’s, µ2 2’s, etc. in the boxes of ν/λ. The content is denoted

c(ν/λ) = (1µ1 ,2µ2 , ...,nµn ).

Example 9.1.2. Let ν= (4,3,2) and λ= (3,1). The following diagram shows λ labeled by

dots, sitting inside of ν,

• • ••

.

Then ν/λ= (4,3,2)/(3,1) looks like

.

If ν/λ is filled in the following way

11 2

3 3

then ν/λ has content (12,21,32). There are many ways to have such content, so this is

not the only filling of ν/λ with this content.

Definition 9.1.3. Suppose we have a filling of a skew Young diagram by positive integers.

Define the word of the Young diagram ν/λ, denoted w(ν/λ), to be the sequence of pos-

90

itive integers obtained when the entries are read from right to left, and top to bottom.

For example,

11 2

3 3

has word w(ν/λ) = 12133.

Definition 9.1.4. Define a lattice permutation to be a sequence of positive integers,

such that when one reads from left to right and stops at each point in the sequence, one

always has at least as many 1’s as 2’s, at least as many 2’s as 3’s, and so on.

Example 9.1.5. The filling of ν/λ

11 2

3 3

gives a word 12133. This is not a lattice permutation, as there are more 3’s than 2’s when

one reaches the last position in the word. The filling

11 2

2 3

gives the word 12132, which is a lattice permutation.

Definition 9.1.6. Let ν/λ be a skew Young diagram. Suppose that the diagram is row

semi-standard, and column standard, that is, the entries in the rows are weakly increas-

ing, and the entries in the columns are strictly increasing. Suppose further that the filling

gives a word that is a lattice permutation. Then we call this filling of ν/λ a Littlewood-

Richardson skew tableau.

Theorem 9.1.7. (see for example [8] pg. 82-83, pg. 455-456, [17] pg. 498, and [9] pg. 62-

71) The number of Littlewood-Richardson skew tableaux of shape ν/λ and of content µ is

91

exactly the Littlewood-Richardson coefficient cνλ,µ. This gives a direct and straightforward

way of calculating cνλ,µ in the equation

Sλ(V )⊗Sµ(V ) = ⊕|ν|=|λ|+|µ|

cνλ,µSν(V ).

Recall 9.1.8. (See for example [8] pg. 223, [17], [9] pg. 114) (Sλ(V )⊗ Sµ(V ))SL(V ) 6= 0 if

and only if λi −µi is constant on all i , with dim(V ) rows and λ1+µn columns. Further, if

this is the case, (S(λ1,...,λn )Cn ⊗S(µ1,...,µn )Cn)SL(V ) is one dimensional, and contains exactly

one semi-invariant of weight λ1 +µn . This is a general fact often found in discussions

on semi-invariants of GL(V ) and highest weight theory (for example in [8] Chapter 8).

It follows from the First Fundamental Theorem of Invariant Theory which says all SL(n)

invariants are generated by determinants of maximal minors of A ∈ Hom(Cm ,Cn), (n ≥m), where

dim(SλV ⊗SµV )SL(V ) = dimHom(SλV ∗,SµV )SL(V )

which is equal to 1 if and only if SλV ∗ ∼= SµV as SL(V ) modules, and zero otherwise.

9.2 Some Computations and Examples

As we have seen, there is a decomposition of the GL(β) representation C[Rep(Q,β)] into

Schur modules SλV (x). For a Schur module SλV (x) to have a GL(V ) semi-invariant, or

equivalently an SL(V ) invariant of weight σ, we need λ = (σ(x)β(x)). In this case we get

the determinant representation raised to some power S(σ(x)β(x))V (x) = (∧β(x) V (x))⊗σ(x).

The action of GL(V ) (and thus SL(V )), is given by multiplication by (det g )σ(x), for g ∈GL(V ).

Example 9.2.1. Let Q be the following quiver

•1a // •2

92

with dimension vector β= (m,n). Now, let V =Cm and W =Cn . Then we have

SI(Q, (m,n)) =C[Hom(V ,W )]SL(V )×SL(W ) =C[V ∗⊗W ]SL(V )×SL(W )

= Sym((V ∗⊗W )∗)SL(V )×SL(W )

= Sym(V ⊗W ∗)SL(V )×SL(W )

= ⊕n≥0

Symn(V ⊗W ∗)SL(V )×SL(W )

= ⊕n≥0

(⊕|λ|=n

Sλ(V )⊗Sλ(W ∗))SL(V )×SL(W )

=⊕λ

(Sλ(V )⊗Sλ(W ∗))SL(V )×SL(W )

=⊕λ

Sλ(V )SL(V ) ⊗Sλ(W ∗)SL(W ).

The last equality is given by the fact that GL(V ),GL(W ), and GL(V )×GL(W ) are linearly

reductive groups, and thus irreducible GL(V )×GL(W ) modules are simply tensor prod-

ucts of irreducible GL(V ) modules with irreducible GL(W ) modules [27]. For there to be

an SL(V ) invariant, i.e. for SλV SL(V ) 6= 0, λ must be a rectangular partition of the form

(σ(1)β(1)), where dimV =β(1) = m, and σ(1) is obtained from the determinantal charac-

ter χ(g ) = ∏x∈Q0 det(gx)σ(x). In this case we must have that σ(1)β(1) = −σ(2)β(2) given

by the equality σ(β) = 0 from our discussion of Schofield semi-invariants, otherwise we

get a representation of the form

(β(1)∧

V

)⊗σ(1)

⊗(β(1)∧

W ∗)⊗σ(1)

.

Now, If β(1) < β(2) we do not get a power of a determinant representation of W , and if

β(1) >β(2) then we get∧β(1) W ∗ = 0. So there is an SL(V )×SL(W ) invariant only if m = n,

i.e. β(1) =β(2) and V =W . In §10 we give Theorem 10.2.3 due to Schofield, and Theorem

10.2.7 due to H. Derksen and J. Weyman in [5], that give an alternate explanation of why

we must have a dimension vector β= (n,n) for there to be nontrivial semi-invariants.

93

So, assuming we have β(1) = β(2) we have (SλV ⊗ SλV ∗)SL(V ) 6= 0, i.e. there is a sin-

gle one dimensional representation of SL(V ) yielding an SL(V )-invariant det(g )σ(1) for

g ∈ SL(V ), corresponding to some power of the determinant representation of V . In this

case, the action of GL(V ) (and SL(V )) on SλV is just multiplication by some power of the

determinant det(g ), and thus the action of SL(V ) is trivial (as it is multiplication by 1).

Explicitly, we have an action of SL(V ) on

SλV ⊗SλV ∗ ∼=(

dimV∧V

)⊗σ(x)

⊗(

dimV∧V ∗

)⊗σ(x)

giving a semi-invariant det(g ), for g ∈ GL(V ), of weight (σ(1),−σ(1)) = (σ(1),σ(2)) (Re-

call det(g )−1 acts on∧β(x) V (x)∗). At this point, it might be prudent to remind our-

selves of how to interpret partitions (−σ(x)β(x)) for weights σ(x) < 0. A negative weight

σ(x) = −σ(x) < 0 is obtained from the dual of a Schur module, i.e. something of the

form S(σ(x)β(x))V (x)∗. In this case we can interpret the weight as a negative integerσ(x) =−σ(x) < 0, and −σ(x) > 0 gives us a valid partition µ= (−σ(x)β(x)) = (σ(x)β(x)) such that

Sλ(V )∗ = S(σ(x)β(x))(V )∗ ∼= S(−σ(x)β(x))(V ) = Sµ(V ), giving a negative weight σ(x) < 0; where

we have tensored with powers of determinants to get an SL(V )-isomorphic represen-

tation S(σ(x)β(x))(V ) = S(−σ(x)β(x))(V ) (see [8] pg. 231-232, [9] pg. 114). This may seem

confusing at first, as we are thinking of σ(x) and σ(x) as, in a sense, the same. In the fu-

ture, we often omit the details of tensoring with powers of determinant representations

in order to pass from σ(x) to σ(x), and we simply use σ(x) throughout computations.

Thus the ring of semi-invariants is generated by the determinant and

SI(Q,β)σ = SI(Q, (n,n))(k,−k) =C[

det(V )k]

.


•1a1 // •3 •2

a2oo

94

and suppose we have some representation with dimension vector β = (β(1),β(2),β(3)).

Using the Cauchy formula as in the previous example we can see that

C[Rep(Q,β)] = ⊕λ(a1),λ(a2)

Sλ(a1)V (1)⊗Sλ(a1)V (3)∗⊗Sλ(a2)V (2)⊗Sλ(a2)V (3)∗.

From this we need several conditions to hold simultaneously in order for there to be a

semi-invariant:

1. Sλ(a1)V (1)SL(V (1)) 6= 0 ⇐⇒ λ(a1) = (σ(1)β(1)) where dimV (1) =β(1),σ(1) ∈Z≥0.

2. Sλ(a2)V (2)SL(V (2)) 6= 0 ⇐⇒ λ(a2) = (σ(2)β(2)) where dimV (2) =β(2),σ(2) ∈Z≥0.

If one of V (1) or V (2) are zero, we have reduced to the case of the previous example, and

we follow all of the same arguments. Thus we get semi-invariants for dimension vectors

β = (β(1),β(2),β(3)) = (n,0,n) and β = (β(1),β(2),β(3)) = (0,n,n). They are exactly the

same semi-invariants found in the previous example, with the same weights. Now, sup-

pose V (1) and V (2) are both nonzero. Then Sλ(a1)V (3)∗⊗Sλ(a2)V (3)∗ 6= 0 if and only if

|λ(a1)|+ |λ(a2)| =σ(3)β(3), i.e. the number of boxes |λ(a1)|+ |λ(a2)| must be a multiple

of an integer, namely β(3), in order for the product

Sλ(a1)V (3)∗⊗Sλ(a2)V (3)∗ = ⊕|ν|=|λ(a1)|+|λ(a2)|

cνλ(a1),λ(a2)SνV (3)∗

to yield a rectangular partition ν= (σ(3)β(3)) in the summation on the right. So,

Sλ(a1)V (1)SL(V (1)) ⊗Sλ(a2)V (2)SL(V (2)) ⊗ (Sλ(a1)V (3)∗⊗Sλ(a2)V (3)∗)SL(V (3)) 6= 0

if and only if there is some

S(σ(1)β(1))V (1)⊗S(σ(2)β(2))V (2)⊗S(σ(3)β(3))V (3)∗.

This only happens ifσ(β) =∑3i=1σ(i )β(i ) = 0, by our discussion of Schofield semi-invariants

in 6. In this section we stated that to each weight σ there is a dimension vector α such

95

that σ = ⟨α,•⟩, and that it suffices to take α for indecomposable representations of the

quiver Q. So we have σ(1)β(1)+σ(2)β(2) = σ(3)β(3). Further, for a non-trivial semi-

invariant to exist we get the dimension vectors β= (p, q, p+q). and, we get the following

weight vector

σ= (σ(1),σ(2),σ(3)) = (1,1,−1)

corresponding to the dimension vector of an indecomposable representation,

α= (α(1),α(2),α(3)) = (1,1,2)

via the formula α(x) = ∑y∈Q0 py,xσ(y). giving the semi-invariant det(V (1),V (2)), i.e.

det[A1|A2] where A1, A2 are the linear maps assigned to a1 and a2 respectively. If σ′ =(k,k,−2k) = kσ, then we get powers of this representation, so SI(Q,β)kσ =C[det(V (1),V (2))]k .

These match the Schofield semi-invariants. If one of the vertices has the zero vector

space assigned to it, then we have reduced to the previous example 9.2.1, and the invari-

ants match those already calculated.

9.2.1 Triple Flag Quivers

Here we discuss what we call the triple flag quiver. These quivers will be important

in proving the saturation conjecture of Littlewood-Richardson coefficients for GL(V ).

Triple flag quivers have underlying graphs of the form,

•

...

•

• · · · · · · • • • · · · · · · •

96

were each arm can in general be of any length. We will however restrict ourselves to a

specific orientation of the arrows.

Example 9.2.3. The Triple Flag Quiver T1,1,1

We first look at the simplest case of a nontrivial triple flag quiver, a quiver which we have

already studied. Let Q be the quiver T1,1,1,

•2

��•1// •4 •3oo


Cq

A2��

CpA1

// Cn CrA3

oo

.

97

Let U =Cp , V =Cq , W =Cr , and Z =Cn . Using Lemma 8.1.3 and Lemma 8.1.4, we have

SI(Q,α) =C[Hom(U , Z )⊕Hom(V , Z )⊕Hom(W, Z )]SL(U )×SL(V )×SL(W )×SL(Z )

=C[(U∗⊗Z )⊕ (V ∗⊗Z )⊕ (W ∗⊗Z )]SL(U )×SL(V )×SL(W )×SL(Z )

= Sym(((U∗⊗Z )⊕ (V ∗⊗Z )⊕ (W ∗⊗Z ))∗)SL(U )×SL(V )×SL(W )×SL(Z )

= Sym((U ⊗Z∗)⊕ (V ⊗Z∗)⊕ (W ⊗Z∗))SL(U )×SL(V )×SL(W )×SL(Z )

= (Sym(U ⊗Z∗)⊗Sym(V ⊗Z∗)⊗Sym(W ⊗Z∗))SL(U )×SL(V )×SL(W )×SL(Z )

= ⊕n≥0

(Symn(U ⊗Z∗)⊗Symn(V ⊗Z∗)⊗Symn(W ⊗Z∗))SL(U )×SL(V )×SL(W )×SL(Z )

= ⊕n≥0

[( ⊕|λ|=n

Sλ(U )⊗Sλ(Z∗)

)⊗

( ⊕|µ|=n

Sµ(V )⊗Sµ(Z∗)

)

⊗( ⊕|ν|=n

Sν(W )⊗Sν(Z∗)

)]SL(U )×SL(V )×SL(W )×SL(Z )

= ⊕λ,µ,ν

(Sλ(U )⊗Sλ(Z∗)⊗Sµ(V )⊗Sµ(Z∗)⊗Sν(W )⊗Sν(Z∗))SL(U )×SL(V )×SL(W )×SL(Z )

= ⊕λ,µ,ν

Sλ(U )SL(U ) ⊗Sµ(V )SL(V ) ⊗Sν(W )SL(W ) ⊗ (Sλ(Z∗)⊗Sµ(Z∗)⊗Sν(Z∗))SL(Z ).

Recall 9.2.4. In the above computation, we can think of the triples of partitions (λ,µ,ν)

in the direct sum as being associated to the arrows of the quiver a1, a2, and a3 respec-

tively. In each triple of partitions (λi ,µ j ,νk ), we then think of the partitions λi ,µ j ,νk

as functions on the arrows. In general we can think of partitions λ as functions on the

arrows of a quiver Q to the set of all partitions P ,

λ( ) : Q1 →P

given by

ai 7→λ(ai )

where ai is the arrow we have associated with the partition λ, and we can write the

98

equation

C[Rep(Q,β)] = ⊕λ(a): a∈Q1

⊗a∈Q1

Sλ(a)V (t a)⊗Sλ(a)V (ha)∗.

This will help us with notation and indexing for quivers with many arrows, and it will

help us keep track of what vector spaces in the quiver representation each partition is

associated to. For the triple flag quivers this will help picture the way in which the tensor

product of Schur modules interact with one another, and how to find partitions giving

non-trivial semi-invariants. In the right hand side of the above equation, we are looking

for tensor products of the form

⊗x∈Q0

S(σ(x)β(x))V (x)

giving a GL(β) =∏x∈Q0 GL(β(x)) semi-invariant (and thus an SL(β) invariant).

As before, if one of the vector spaces V (1),V (2), or V (3) are zero, we reduce to the pre-

vious example for the quiver •→ •← •, and obtain all of the same semi-invariants and

corresponding weights. If all three are nonzero, each partition λ, µ, and ν must be such

that,

1. λ= (σ(1)β(1)).

2. µ= (σ(2)β(2)).

3. ν= (σ(3)β(3)).

in order for SλV (1)SL(V (1)),SµV (2)SL(V (2)),SνV (3)SL(V (3)) 6= 0. We then get a copy of

S(σ(1)β(1))V (1)⊗S(σ(2)β(2))V (2)⊗S(σ(3)β(3))V (3)

⊗ (S(σ(1)β(1))V (4)∗⊗S(σ(2)β(2))V (4)∗⊗S(σ(3)β(3))V (4)∗).

In this case, we need to have the product yield a rectangle with |λ| + |µ| + |ν| boxes,

implying |λ| + |µ| + |ν| must be a multiple of some nonnegative integer, in particular

99

|λ| + |µ| + |ν| = σ(4)β(4), so that we get a representation S(σ(4)β(4))V (4)∗ inside the ten-

sor product

(S(σ(1)β(1))V (4)∗⊗S(σ(2)β(2))V (4)∗⊗S(σ(3)β(3))V (4)∗)

The condition σ(β) = 0 means β(1)+β(2)+β(3) = β(4), and the new weight that is not

found by running the same argument as the previous two examples where one or more

of the V (i ) are zero is

σ= 11 −1 1

or in terms of an indecomposable dimension vector, σ= ⟨α,•⟩, we have

α= 11 2 1

with the dimension vector

β= qp (p +q + r ) r

.

This gives us ⟨α,β⟩ = 0 = σ(β). We get a semi-invariant for several possible families of

dimension vectors in addition to this, which are reductions to previous examples for

quivers of the form •→•←• and •→•. They are as follows:

β= qp (p +q) 0

β= q0 (q + r ) r

β= 0p (p + r ) r

β= 00 n n

β= n0 n 0

β= 0n n 0

.

The weight σ must be such that σ(β) = ∑x∈Q0 σ(x)β(x) = 0, for each possible β in order

to have a semi-invariant, as proven by our study of Schofield semi-invariants. So we get

the following corresponding families of weights,

σ= kk −2k 0

σ= k0 −2k k

σ= 0k −2k k

100

σ= 00 −k k

σ= k0 −k 0

σ= 0k −k 0

.

These all yield weights σ such that σ(β) = 0 = ⟨α,β⟩ for some dimension vectors α of

indecomposable representations, and the invariants match the calculations of the pre-

vious two examples and those of the Schofield semi-invariants, i.e.

1. det(Ai ) for i = 1,2,3, corresponding to β(4) =β(i ) where i 6= 4.

2. det[Ai |A j ] for i 6= j , and i , j = 1,2,3, corresponding to β(4) =β(i )+β( j ), with i , j 6=4.

3. det(A1|A2|A3), corresponding to all vertices having nonzero vector spaces assigned

to them and β(4) =β(1)+β(2)+β(3).

The Triple Flag Quiver T2,2,2

Example 9.2.5. Now let Q be the quiver T2,2,2,

•1// •2

// •7 •6oo •5

oo

•4

OO

•3

OO

.

Give Q the representation,

Cβ(1)A1

// Cβ(2)A2

// Cβ(7) Cβ(6)A6

oo Cβ(5)A5

oo

Cβ(4)

A4

OO

Cβ(3)

A3

OO

.

From the Cauchy formula we get

C[Rep(Q,β)] = ⊕λ(a):λ∈Q1

⊗a∈Q1


101

which is ⊕λ(ai ): i=1,...,6

Sλ(a1)V (1)⊗Sλ(a1)V (2)∗⊗

Sλ(a2)V (2)⊗Sλ(a2)V (7)∗⊗

Sλ(a3)V (3)⊗Sλ(a3)V (4)∗⊗

Sλ(a4)V (4)⊗Sλ(a4)V (7)∗⊗

Sλ(a5)V (5)⊗Sλ(a5)V (6)∗⊗

Sλ(a6)V (6)⊗Sλ(a6)V (7)∗.

Now, drawing from past experience we list some conditions for semi-invariants to exist,

looking for some summand of the form

⊗x∈Q0

S(σ(x)β(x))V (x)

in the direct sum of irreducibles. For this to happen the following conditions must be

met:

1. Sλ(a1)V (1),Sλ(a3)V (3),Sλ(a5)V (5) must all have rectangular partitions in order for

Sλ(a1)V (1)SL(V (1)) 6= 0,Sλ(a3)V (3)SL(V (3)) 6= 0, and Sλ(a5)V (5)SL(V (5)) 6= 0. They each

have the form (σ(x)β(x)).

2. λ(a1) and λ(a2) must be such that Sλ(a1)V (2)∗ ∼= Sµ(a1)V (2) such that λ(a2) and

µ(a1) fit together to make a rectangle for (Sλ(a2)V (2)⊗Sλ(a1)V (2)∗)SL(V (2)) 6= 0. For

this to happen, λ(a2) must be exactly ((σ(2)+σ(1))β(1),σ(2)β(2)−β(1)), i.e. it is a rect-

angle with dim(V (2)) rows and σ(2) columns, with λ(a1) attached. Similarly for

the pair λ(a3) and λ(a4) we must have that λ(a4) = ((σ(3)+σ(4))β(3),σ(4)β(4)−β(3)),

i.e. λ(a4) is a rectangular partition corresponding to a determinant representa-

tion (σ(4)β(4)), with the rectangular partition σ(3)β(3)) attached. Finally, λ(a6) is a

rectangular partitions (σ(6)β(6)) with the rectangular partition (σ(5)β(5)) attached.

102

3. Now, the space of semi-invariants is isomorphic to

(Sλ(a2)V (7)∗⊗Sλ(a4)V (7)∗⊗Sλ(a6)V (7)∗)SL(V (7))

We have many possible dimension vectors yielding a semi-invariant, one such possibil-

ity would be

β=n 2n 3n 2n n

2nn

.

The weight is then

σ=1 1 −3 1 1

11

.

This gives the semi-invariant

det(A2 A1|A4 A3|A6 A5)

where the Ai are the linear maps corresponding to the arrows ai ∈ Q1. to compute the

dimension of the space of semi-invariants,

dim(Sλ(a2)V (7)∗⊗Sλ(a4)V (7)∗⊗Sλ(a6)V (7)∗)SL(V (7)),

we use the Littlewood-Richardson rule. In particular, for there to be a semi-invariant of

weight σ we need

|λ(a2)|+ |λ(a4)|+ |λ(a6)| = nβ(7), for some n ∈N.

We explain this calculation and its details further in §10.3. We can calculate the other

semi-invariants using methods similar to the work we have done previously with the

quiver

• // • // • •oo •oo

•

OO

We get

103

1. det(A1|A4 A3), det(A3|A6 A5), det(A5|A2 A1)

2. det(A6|A4 A3), det(A2|A6 A5), det(A4|A6 A5)

3. det

(A2 A4 00 A4 A6

).

The quiver Tn,n,n for n ≥ 2 is not of finite representation type, but rather is tame and an

extended Dynkin quiver, i.e. it has a finite number of families of indecomposable repre-

sentations. For Dynkin quivers we can always (relatively) easily find all of the generators

of the ring of semi-invariants since we need only check finitely many indecomposable

representations. For tame and wild quivers we sometimes need more methods. Dimen-

sion vectors of this form for triple flag quivers with equal length arms and ascending

dimensions along arms are what we will need later on to prove the saturation conjec-

ture. This dimension vector is isotropic. A dimension vector β is isotropic if ⟨β,β⟩ = 0,

under the usual inner product given by the Euler form that we have discussed already.

The pattern we have found for T2,2,2 continues for quivers Tn,n,n and isotropic dimen-

sion vectors

β=12...

1 2 · · · n · · · 2 1

.

In other words, the arms of Tn,n,n give semi-invariants when the partitions along the

arms have conjugate partitions of the shape (β(n−1)σ(n−1),β(n−2)σ(n−2), ...,β(2)σ(2),β(1)σ(1)).

104

The Triple Flag Quiver Tp,q,r

Let Q = Tp,q,r be the quiver,

x1// x2

// · · · // xp−1// xp zp−1oo · · ·oo z2

oo z1oo

yp−1

OO

...

OO

y2

OO

y1

OO

.

Further, let yp = zp = xp . Let β be the dimension vector for Tp,q,r and let σ= ⟨α,•⟩ be a

weight with σ(β) = 0, corresponding to some dimension vector α. Further, assume that

the dimensions are weakly increasing along the arms, i.e. that β(xi ) ≤ β(xi+1),β(y j ) ≤β(y j+1), and β(zk ) ≤β(zk+1). Further assume σ(xi ),σ(y j ),σ(zk ) > 0 for all i , j ,k < p.

Theorem 9.2.6. There is an isomorphism,

SI(Q,β)σ ∼= (Sλ(U )⊗Sµ(U )⊗Sν(U ))SL(U )

where U is a vector space of dimension m := β(xp ) = β(yq ) = β(zr ) and the partitions µ,

ν, ω have conjugate partitions

1. λ′ = (β(xp−1)σ(xp−1),β(xp−2)σ(xp−2), ...,β(x2)σ(x2),β(x1)σ(x1)).

2. µ′ = (β(yq−1)σ(yq−1),β(yq−2)σ(yq−2), ...,β(y2)σ(y2),β(y1)σ(y1)).

3. ν′ = (β(zr−1)σ(zr−1),β(zr−2)σ(zr−2), ...,β(z2)σ(r2),β(z1)σ(r1)).

Proof. Define V (xi ) =Cβ(xi ), V (y j ) =Cβ(y j ), and V (zk ) =Cβ(zk ) for all 1 ≤ i , j ,k ≤ p. Fur-

ther, denote the arrows along the x-arm by {ai }, the arrows along the y-arm by {b j } and

105

the arrows along the z-arm by {ck }. Now, from the Cauchy formula we get

SI(Q,β) =C[

p−1⊕i=1

Hom(V (xi ),V (xi+1))q−1⊕j=1

Hom(V (y j ),V (y j+1))r−1⊕k=1

Hom(V (zk ),V (zk+1))

]

=C⊕

ai

Hom(V (t ai ),V (hai ))⊕b j

Hom(V (tb j ),V (hb j ))⊕ck

Hom(V (tck ),V (hck ))

= ⊕λ(ai ),λ(b j ),λ(ck )

⊗ai

Sλ(ai )V (t ai )⊗Sλ(ai )V (hai )∗

⊗b j

Sλ(b j )V (tb j )⊗Sλ(b j )V (hb j )∗

⊗ck

Sλ(ck )V (tck )⊗Sλ(ck )V (hck )∗.

If we define U =V (xp )∗ =V (yq )∗ =V (zr )∗, we have

SI(Q,β)σ ∼= (SµU ⊗SνU ⊗SωU )SL(U )

To see this, let’s write out the details of the above computations,

SI(Q,β) =C[

p−1⊕i=1

Hom(V (xi ),V (xi+1))q−1⊕j=1

Hom(V (y j ),V (y j+1))r−1⊕k=1

Hom(V (zk ),V (zk+1))

]

=C⊕

ai

Hom(V (t ai ),V (hai ))⊕b j

Hom(V (tb j ),V (hb j ))⊕ck

Hom(V (tck ),V (hck ))

=(

p−1⊗i=1

Sym(V (t ai )⊗V (hai )∗)q−1⊗j=1

Sym(V (tb j )⊗V (hb j )∗)r−1⊗k=1

Sym(V (tck )⊗V (hck )∗)

)SL(β)

= ⊕λ(ai ),λ(b j ),λ(ck )

(⊗ai

Sλ(ai )V (t ai )⊗ Sλ(ai )V (hai )∗

⊗b j

Sλ(b j )V (tb j )⊗Sλ(b j )V (hb j )∗⊗ck

Sλ(ck )V (tck )⊗Sλ(ck )V (hck )∗)SL(β)

where for each i , j ,k we have λ(ai ),λ(b j ), and λ(ck ) running through all partitions. Now

106

if we sort the terms in the above expression by vector space as we have in previous ex-

amples, we get the following⊕λ(ai ),λ(b j ),λ(ck )

Sλ(a1)V (x1)SL(V (x1)) ⊗Sλ(b1)V (y1)SL(V (y1)) ⊗Sλ(c1)V (z1)SL(V (z1))⊗

(Sλ(ap−1)V (xp )⊗Sλ(bq−1)V (yq )⊗Sλ(cr−1)V (zr ))SL(V (xp ))×SL(V (yq ))×SL(V (zr ))⊗p−1⊗i=2

(Sλ(ai−1)V (xi )∗⊗Sλ(ai )V (xi ))SL(V (xi ))q−1⊗j=2

(Sλ(b j−1)V (y j )∗⊗Sλ(b j )V (y j ))SL(V (y j ))

r−1⊗k=2

(Sλ(ck−1)V (zk )∗⊗Sλ(ck )V (zk ))SL(V (zk )).

Here V (xp ) =V (yq ) =V (zr ) so we actually have,⊕λ(ai ),λ(b j ),λ(ck )

Sλ(a1)V (x1)SL(V (x1)) ⊗Sλ(b1)V (y1)SL(V (y1)) ⊗Sλ(c1)V (z1)SL(V (z1))⊗

(Sλ(ap−1)V (xp )⊗Sλ(bq−1)V (xp )⊗Sλ(cr−1)V (xp ))SL(V (xp ))⊗p−1⊗i=2

(Sλ(ai−1)V (xi )∗⊗Sλ(ai )V (xi ))SL(V (xi ))q−1⊗j=2

(Sλ(b j−1)V (y j )∗⊗Sλ(b j )V (y j ))SL(V (y j ))

r−1⊗k=2

(Sλ(ck−1)V (zk )∗⊗Sλ(ck )V (zk ))SL(V (zk )).

Now notice the first grouping of the Schur modules Sλ(a1)V (x1),Sλ(b1)V (y1), and Sλ(c1)V (z1).

For each of these to contain invariants of their respective Special Linear Groups, we must

have that λ(a1),λ(b1),λ(c1) are all rectangular partitions of the form

1. λ(a1) =σ(x1)β(x1).

2. λ(b1) =σ(y1)β(y1).

3. λ(c1) =σ(z1)β(z1).

In which case we obtain a determinant representation, and thus a semi-invariant under

the General Linear Groups of each as usual. For the first pair Sλ(a1)V (x2)∗⊗Sλ(a2)V (x2)

107

of the x-arm of the quiver Tp,q,r , we must be a little more clever. In this case we get a

semi-invariant, i.e. a determinant representation of V (x2) exactly when λ(a2) has as its

conjugate partition λ(a2)′ = (β(x2)σ(x2),β(x1)σ(x1)). In other words, λ(a2) is a rectangular

partitionσ(x2)β(x2) with the previous rectangular partitionsλ(a1) attached to it. Now, we

know for the Schur module Sλ(a1)V (x2)∗, that we have an isomorphism

Sλ(a1)V (x2)∗ ∼= Sµ(a1)V (x2)

where on the right hand side we no longer have a dual. In this case we know that λ(a2)

and µ(a1) must fit together to make a rectangular partition giving a determinant repre-

sentation of V (x2), i.e. λ(a2) and µ(a1) must fit together to make a partition of the form

σ(x2)β(x2) if there is to be a semi-invariant of weightσ(x2). We obtainµ(a1) fromλ(a1) by

tensoring with powers of determinant representations (see [8] pg. 231-232, [9] pg. 114).

In this case we get

Sλ(a1)V (x2)∗⊗Sλ(a2)V (x2) ∼= Sµ(a1)V (x2)⊗Sλ(a2)V (x2) = S(σ(1)β(2)−β(1))V (x2)⊗S(σ(2)β(2))V (x2).

In which case we apply the Littlewood Richardson rule to Young diagrams of the form

ν(x2) = ((σ(2)+σ(1))β(2)) and λ(a2) = ((σ(2)+σ(1))β(1),σ(2)β(2)−β(1)).

Example 9.2.7. The following diagram showsλ(a2) = (63,35), whereλ(a1) = (33), labeled

by dots, sitting inside of ν(a2) = (68),

• • • • • •• • • • • •• • • • • •• • •• • •• • •• • •• • • .

108

Then ν(a2)/λ(a2) = (68)/(63,35) looks like

.

We get Sλ(a1)V (x2)∗ = S(33)V (x2)∗ ∼= S(35)V (x2) = Sµ(a1)V (x2). If ν(a2)/λ(a2) is filled with

content µ(a1), i.e. (13,23,33,43,53), we see there is one and only one way to fill the dia-

gram with this content such that the rows are semi-standard and the columns are stan-

dard, and so that we get a lattice permutation word as defined in §9.1, namely

1 1 12 2 23 3 34 4 45 5 5 .

We can continue this reasoning to see that for any pairing Sλ(ai−1)V (xi )∗⊗Sλ(ai )V (xi ), we

must have that λ(ai ) is a rectangular partition corresponding to a power of a determi-

nant representation of V (xi ), with λ(ai−1) attached to it. The same holds for the y-arm

and the z-arm, and the partitions λ(b j ),λ(ck ) for 2 ≤ j ,k < p. This gives us that the

partitions along the arms must have conjugate partitions



3. ν′ = (β(zr−1)σ(zr−1),β(zr−2)σ(zr−2), ...,β(z2)σ(r2),β(z1)σ(r1)).

Taking U =V (xp )∗ =V (yq )∗ =V (zr )∗, we get that

SI(Q,β)σ ∼= (Sλ(U )⊗Sµ(U )⊗Sν(U ))SL(U )

as desired.

109

Theorem 9.2.8. Suppose now, that we reverse the orientation of the arrows on one of the

arms. Then we get the duals on that factor of the tensor product reversed, i.e.

C

[⊕xi

Hom(V (t xi ),V (hxi ))⊕y j

Hom(V (t y j ),V (hy j ))⊕zi

Hom(V (t zk ),V (hzk ))

]

= ⊕λ(ai ),λ(bi ),λ(ci )

⊗ai


⊗b j


⊗ck

Sλ(ck )V (tck )∗⊗Sλ(ck )V (hck )

giving

SI(Q,α)σ = SI(T rp,q ,α) ∼= (SλU ⊗SµU ⊗SνU∗)GL(U ).

Proof. Again, reversing the arrows on one arm, say the z-arm of the quiver will give us

C

[⊕xi

Hom(V (t xi ),V (hxi ))⊕y j

Hom(V (t y j ),V (hy j ))⊕zi

Hom(V (t zk ),V (hzk ))

]

= ⊕λ(ai ),λ(bi ),λ(ci )

⊗ai


⊗b j


⊗ck

Sλ(ck )V (tck )∗⊗Sλ(ck )V (hck ).

Sorting the tensor product by vector space as in the previous theorem we get pairs

Sλ(ck−1)V (zk )∗⊗Sλ(ck )V (zk )

for all 2 ≤ k < r . The partitions λ and µ that we need on the x-arm and y-arm respec-

tively are exactly the same as before, i.e. their conjugate partitions are,


110


But now we need the conjugate ν′ to the partition ν along the z-arm to be

ν′ = (β(zr )−σ(zr ),β(zr−1)−σ(zr−1), ...,β(z2)−σ(z2),β(z1)−σ(z1)).

Recall 9.2.9. Recall how to interpret partitions (−σ(x)β(x)) for weights σ(x) < 0. A neg-

ative weight σ(x) = −σ(x) < 0 is obtained from a dual to a Schur module, i.e. some-

thing of the form S(σ(x)β(x))V (x)∗. In this case we can interpret the weight as a nega-

tive integer σ(x) = −σ(x) < 0, and −σ(x) > 0 gives us a valid partition µ = (−σ(x)β(x)) =

(σ(x)β(x)) such that Sλ(V )∗ = S(σ(x)β(x))(V )∗ ∼= S(−σ(x)β(x))(V ) = Sµ(V ), giving a negative

weight σ(x) < 0; where we have tensored with powers of determinants to get an SL(V )-

isomorphic representation S(−σ(x)β(x))(V ) ∼= S(σ(x)β(x))(V ).

Each weight σ(zi ) < 0, is now a negative weight, as they are the weights corresponding

to determinant representations to negative powers, i.e.

(β(zi )∧

V (zi )∗)⊗−σ(zi )

= detV σ(zi ), for σ(zi ) < 0.

The tensor product corresponding to the z-arm now looks like

Sλ(c1)V (z1)∗⊗(Sλ(c1)V (z2)⊗Sλ(c2)V (z2)∗)⊗·· ·⊗(Sλ(cr−2)V (zr−1)⊗Sλ(cr−1)V (zr−1)∗)⊗Sλ(cr−1)V (zr )

where U = V (xp ) = V (yq ) = V (zr ). For the first representation Sλ(c1)V (z1)∗ to contain a

semi-invariants we need a rectangular partition λ(c1) = (−σ(z1)β(z1)) giving a determi-

nant representation to a negative power,

(β(z1)∧

V (z1)∗)⊗−σ(z1)

= detV σ(z1), for σ(z1) < 0.

111

As before, we need λ(c2) to be a partition with a rectangular shape of size (−σ(z2)β(z2))

with λ(c1) attached so that we can apply the Littlewood-Richardson rule to find a semi-

invariant of weight σ(z2) < 0. Continuing this argument, we get that the partition corre-

sponding to Sλ(cr−1)V (r ) = SνU∗ must have a conjugate partition of the form,

ν′ = (β(zr )−σ(zr ),β(zr−1)−σ(zr−1), ...,β(z2)−σ(z2),β(z1)−σ(z1)).

We can tensor with powers of determinants to get positive weights (see [8] pg. 231-232,

[9] pg. 114). So we have,

SI(Q,β)σ ∼= (SλU ⊗SµU ⊗SνU∗)GL(U ).

Notice, from the formula

Sλ(V )⊗Sµ(V ) = ⊕|ν|=|λ|+|µ|

cνλ,µSν(V )

we get that

dim(Sλ(V )⊗Sµ(V )⊗Sω(V )∗)GL(V ) = dim

( ⊕|ν|=|λ|+|ν|

cνλ,µSν(V )⊗Sω(V )∗)GL(V )

.

Since

(SνV ⊗SωV ∗)GL(V ) 6= 0 ⇐⇒ Hom(SωV ,SνV )GL(V ) 6= 0

⇐⇒ SωV ∼= SνV (as GL(V ) representations, since both are irreducible)

⇐⇒ ν=ω

we get

dim

( ⊕|ν|=|λ|+|ν|

cνλ,µSν(V )⊗Sω(V )∗)GL(V )

= dim(cωλ,µSω(V ))⊗Sω(V )∗)GL(V )

= dim(Hom(SωV ,cωλ,µSωV ))GL(V )

= cωλ,µ.

112

So in particular

dim(SI(Q,α)σ) = cωλ,µ

i.e.

dim(SI(Q,β)σ) = dim(SλU ⊗SµU ⊗SνU∗)GL(U ) = cν∗

λ,µ = dim(SλU ⊗SµU ⊗SνU )SL(U )

where again U =V (xp ) =V (yq ) =V (zr ), and where

ν∗ = (dim(U )−νr−1,dim(U )−νr−2, ...,dim(U )−ν2,dim(U )−ν1)

=β(zr )−ν= (β(zr )−λ(cr−1),β(zr )−λ(cr−2), ...,β(zr )−λ(c2),β(zr )−λ(c1)).

This leads one to believe that for arbitrary dimension vectors with weakly increasing

dimensions along the arms of a general triple flag quiver Tp,q,r , we can associate a triple

of partitions (λ,µ,ν) corresponding to the arms, and we can calculate the dimension

dim(SI(Tp,q,r ,β)σ), for arbitrary σ = ⟨α,•⟩. Further it seems as though we can reverse

the construction and find a quiver Tp,q,r with dimension vectors β and some weight σ

for SI(Tp,q,r ,β)σ. Thus there seems to be a correspondence between triples of partitions

(λ,µ,ν) and weights for triple flag quivers. We will develop this construction further, and

show the correspondence more explicitly in some examples in §10.3.

113

Chapter 10: Application to Littlewood-Richardson Coefficients

10.1 Saturation and Rational Cones

Definition 10.1.1. A rational cone Is the set of all solutions

Ax ≤ 0 : x ∈Zn+

i.e. all positive integer valued vectors inZn , such that the inequality Ax ≤ 0 holds in each

entry, for a fixed m×n matrix A with integer entries. Thus, a rational cone is determined

by m inequalities.

Let Q be a quiver with no oriented cycles with a dimension vector α. As usual SI(Q,α)

denotes the ring of semi-invariant polynomials inC[Rep(Q,α)], where Rep(Q,α) is theα

dimensional representation space of Q.

Definition 10.1.2. Define the set

Σ(Q,α) = {σ : SI(Q,α)σ 6= 0}

to be the set of weightsσ giving nonzero corresponding weight spaces SI(Q,α)σ in SI(Q,α).

It is a rational cone as it is a subset of the space of all weights, and is given by one ho-

mogeneous linear equality and a finite number of homogeneous linear inequalities. The

equality is given byσ(β) = 0, and the inequalities are given byσ(β′) ≤ 0 for all dimension

vectors β′ such that there is a β′ dimensional subrepresentation. This will be discussed

and proven later in Theorem 10.2.3 and Theorem 10.2.7.

We will show that the set Σ(Q,α) is saturated, i.e., for n ∈ N, if nσ ∈ Σ(Q,α) then σ ∈Σ(Q,α). Here nσ indicates that we are multiplying each component of the weight vector

σ by the positive integer n. From the saturation property, and the fact that Schofield

114

semi-invariants span each weight space SI(Q,α)σ in the ring of semi-invariants SI(Q,α),

we show that for a GLn-module SλV ⊗SνV , the module SνV appears in this tensor prod-

uct if and only if the partitions λ,µ, and ν satisfy a certain set of inequalities. Further,

the positive real span R+Σ(Q,α) ⊂R|Q0|, forms a rational real cone in R|Q0|. In the case of

the triple flag quiver Tn,n,n and a particular dimension vector β, the cone of

Σ(Tn,n,n ,β)

turns out to correspond to a rational real cone formed by triples of partitions (λ,µ,ν),

given by

L Rn = {(λ,µ,ν) ∈ (Zn)3 :λ,µ,ν are weakly decreasing sequences and cνλ,µ 6= 0}.

The positive real span of this set, denoted R+L Rn (or some variation of it) is often

referred to as the Klyochko cone. Thus, proving that the weights Σ(Q,β) for triple flag

quivers are saturated in Z|Q0| proves that the Littlewood-Richardson coefficients are sat-

urated as well. In other words, if for n ∈ N we have cnνnλ,nµ 6= 0 then cν

λ,µ 6= 0. Here nλ

denotes multiplication of each component of the partition by the positive integer n.

Recall 10.1.3. We know for dimension vectors β, that β ∈ Γ⊂Z|Q0|, where Γ=N|Q0|. We

can think of σ as being in the dual Γ∗ = Hom(Γ,Z) of the dimension vectors β ∈ Γ. We

can also think of σ as an element of Z|Q0|, or as a function on the vertices σ : Q0 →Z. For

each β we associate a character of GL(Q,β) to the weight σ given by

χ(g ) = ∏x∈Q0

det(gx)σ(ex )

where g = (gx)x∈Q0 ∈ GL(β), gx ∈ GL(β(x)), and ex is the dimension vector corresponding

to the simple representation Ex as described in Example 2.2.4, i.e. ex(y) = δyx , where

x, y ∈Q0 and δyx is the Dirac delta function. For brevity of notation we will still write σ(x)

115

to mean σ(ex), and think of σ as a function on vertices and as a vector in a similar way

to β, i.e.

σ= (σ(x1),σ(x2), ...,σ(xn))

where Q0 = {x1, x2, ..., xn}. We also know that the ring SI(Q,β) has a weight space decom-

position

SI(Q,β) =⊕σ

SI(Q,β)σor equivalently a decomposition

with respect to characters SI(Q,β) =⊕χ

SI(Q,β)χ

where σ runs through all corresponding one-dimensional irreducible characters

χ(g ) = ∏x∈Q0

det(gx)σ(x)

of GL(Q,β), and where each

SI(Q,β)σ = { f ∈C[Rep(Q,β)] : g · f =χ(g ) f ∀g ∈ GL(Q,β)}.

Definition 10.1.4. A generic representation with some property P is a representation

such that the set of representations without property P all lie in a countable union of

subvarieties of Rep(Q,α). For example take the representation

C3 →C4

with some matrix A assigned to the arrow. Then a generic representation V is one such

that A is of full rank, since the set of all matrices of rank r ≤ 3 can be defined by a fi-

nite set of equations. To see this, suppose {[xi1 , ..., xi3 ]} is the set of all 3×3 minors of A

given by choosing three columns. Then {det([xi1 , ..., xi3 ]) 6= 0} give a list of

(43

)equations

defining a variety in Rep(Q, (3,4)). When we speak of representations having a generic

property, this is what we mean. To be precise, the general representation V is the rep-

resentation whose matrix coordinates are indeterminants. A generic representation is

116

an unspecified representation, which refers to a variable point in a Zariski open subset

of Rep(Q,α). We can justify using these terms interchangeably when working over fields

of characteristic zero, as they coincide (see [26] pg. 10).

10.2 Saturation of Weights

Definition 10.2.1. For representations V and W of a quiver Q, define ExtQ (V ,W ) to be

the cokernel of the map

dVW :

⊕x∈Q0

Hom(V (x),W (x)) → ⊕a∈Q1

Hom(V (t a),W (ha))

Further define the generic spaces HomQ (α,β) and ExtQ (α,β) to be the spaces

HomQ (α,β) = {HomQ (V ,W ) : V ∈ Rep(Q,α), W ∈ Rep(Q,β) are generic representations}

ExtQ (α,β) = {ExtQ (V ,W ) : V ∈ Rep(Q,α), W ∈ Rep(Q,β) are generic representations}

Definition 10.2.2. For two dimension vectorsα and βwe say that the space HomQ (α,β)

(respectively ExtQ (α,β)) vanishes generically if and only if for general representations

V and W of dimension α and β respectively, Hom(V ,W ) = 0 (resp. Ext(V ,W ) = 0). If

a general representation of dimension β has an α-dimensional subrepresentation we

write α ,→β.

Theorem 10.2.3. Let α and β be two dimension vectors for the quiver Q.

1. ExtQ (α,β) vanishes generically if and only if α ,→α+β.

2. ExtQ (α,β) does not vanish generically if and only if β′ ,→ β and ⟨α,β−β′⟩ < 0 for

some dimension vector β′.

This result is due to Schofield [22]. The proof uses more advanced tools from algebraic

geometry which we are unable to thoroughly explore and which would require a much

117

lengthier and more technical introduction than was provided in §4. Thus we refer the

reader to [12] and to Schofield’s paper [22].


•1a // •2

Let β= (2,4) be the dimension vector of a general representation W and let α= (2,1) be

the dimension vector of a general representation V . So we have the representations,

V : C2 A // C and W : C2 B // C4 .

Then we have the noncommutative diagram

V : C2 A //

φ(1)��

C

φ(2)��

C2B// C4

where A is a general 1×2 matrix in Rep(Q,α) and B is a general 4×2 matrix in Rep(Q,β),

after a choice of basis. Note that ⟨α,β⟩ = 0 = σ(β) for the weight σ corresponding to α.

There is no subrepresentation of dimension α = (2,1) such that for a general represen-

tation of dimension α+β = (4,5) we have α ,→ α+β. Suppose that we had a general

representation W of dimension (4,5) such that there was a subrepresentation W ′ ⊂ W

of dimension α = (2,1). In this case, the map W ′(a) = W (a)∣∣∣W ′(x1)

restricted to a 2 di-

mensional domain W ′(x1) would be of rank 2, i.e W (a) restricted to a subspace must

always be injective, since the map W (a) in the general representation must be of full

rank. However, a subrepresentation of dimension α = (2,1) would necessarily have at

most a rank 1 map, and thus a nontrivial kernel. Thus, we cannot have a subrepresenta-

tion of dimension α such that α ,→α+β.

118

However, there is a subrepresentation W ′ ⊂ W with dimension vector β′ = (2,2), by the

same argument, so β′ ,→β. Now, β−β′ = (0,2). In this case

⟨α,β−β′⟩ = (2 ·0)+ (−1 ·2) =−2 < 0.

Thus by Theorem 10.2.3, we have that ExtQ (α,β) does not vanish generically. So, dVW has

nontrivial kernel for general representations V and W with dimension vectors α= (2,1)

and β = (2,4) respectively. Thus, the determinant must be zero in general, and there is

no Schofield semi-invariant cVW . This means for σ= ⟨α,•⟩ that Σ(Q,β)σ = 0.

Example 10.2.5. Suppose now that we take α= (1,0) and β= (n,n), for the quiver •→•.

In this case we get the following diagram for general representations V and W ,

V : CA //

φ(1)��

0

φ(2)��

CnB// Cn

.

In this case we see that α+β= (n +1,n). For a general representation

W : Cn+1 M // Cn

of Q, we get a map M of rank n, i.e. M is surjective and has a one dimensional kernel.

Thus there is a subrepresentation

W ′ : C(0) // 0

of dimension α = (1,0), so that α ,→ α+β. This means since ⟨α,β⟩ = 0 we must have a

nontrivial semi-invariant cV .

Example 10.2.6. Now, let Q be the following quiver

•5

a5

��•1 a1// •3 a3

// •6 •4a4oo •2a2

oo

119

with general representation,

V : C

id��

Cid// C

id// C 0

(0)oo 0

(0)oo

.

Now, suppose W is the following general representation,

W : C2n

A′5 ��

CnA′

1

// C2nA′

3

// C3n C2nA′

4

oo CnA′

2

oo

.

We would like to use Theorem 10.2.3 in order to find out if there is a Schofield semi-

invariant cVW . This means that cV

W = det(dVW ) must be nonzero for general representa-

tions V and W , i.e. for general representations dVW has trivial kernel and thus Ext(α,β)

vanishes generically. For the map dVW we have the following noncommutative diagram,

C

id

}}

φ(5)

��

C

φ(1)

��

id // Cid //

φ(3)

��

C

φ(6)

��

0(0)oo

φ(4)

��

0

φ(2)

��

(0)oo

C2n

A′5

}}Cn

A′1

// C2nA′

3

// C3n C2nA′

4

oo CnA′

2

oo

.

120

For the two sub-diagrams,

0

φ(2)��

(0) // 0

φ(4)��

CnA′

2

// C2n

and 0

φ(4)��

(0) // C

φ(6)��

C2nA′

4

// C3n

we can always restrict general representations of dimension

α′+β′ = (0,0)+ (n,2n) =β′,

to trivial subrepresentations. We can also always restrict a general representation of

dimension α′′ +β′′ = (0,1) + (n,2n) = (n,2n + 1) to a subrepresentation of dimension

α′′ = (0,1), since restricting to a trivial subspace in the domain of a representation of di-

mension α′′+β′′ = (n,2n +1) automatically gives a trivial map, and since there always

exists a trivial map into a one dimensional subspace of the codomain of a dimension

α′′ +β′′ representation. Thus we need only worry about the part of the diagram in-

volving subrepresentations, of dimension α′′′ = (1,1,1), of representations of dimension

β′′′ = (n +1,2n +1,3n +1). In other words, we need to focus on the subdiagram

Cid //

φ(1)��

Cid //

φ(3)��

C

φ(6)��

Cidoo

φ(5)��

CnA′

1

// C2nA′

3

// C3n C2nA′

5

oo

.

So, we are looking for a subrepresentation

V : C id // Cid // C C

idoo

inside the general representation

Cn+1A′

1

// C2n+1A′

3

// C3n+1 C2n+1A′

5

oo .

121

Restricting any of the maps A′1, A′

3, A′5 to a one dimensional subspace automatically gives

a rank one map A′i : C→ C, thus there is such a subrepresentation. This means, for a

general representation of dimension

α+β= n +1n +1 2n +1 3n +1 2n +1 n +1

there will always be a subrepresentation of dimension

α= 11 1 1 0 0

,

and so α ,→ α+β. This means by Theorem 10.2.3 that Ext(α,β) vanishes generically

and that there is a nontrivial Schofield semi-invariant cV . We have computed this semi-

invariant already in Example 6.4.1.

Theorem 10.2.7. Let Q be a quiver with no oriented cycles. Letβ be a dimension vector for

Q. The semi-group Σ(Q,β) (under addition of weights σ) is the set of all σ ∈ Γ∗ such that

σ(β) = 0 and σ(β′) ≤ 0 for all β′ such that β′ ,→ β. So, this condition is given by one lin-

ear homogeneous equality and finitely many linear homogeneous inequalities, defining a

rational cone, and in particular Σ(Q,β) is saturated, i.e., if nσ ∈Σ(Q,β) then σ ∈Σ(Q,β).

Proof. Let σ ∈ Γ∗. Then we can write σ = ⟨α,•⟩, for α some dimension vector. Then

α(x) ≥ 0 for all x ∈Q0. Now, we know that SI(Q,β)⟨α,•⟩ 6= 0 ⇐⇒ ∃ V ∈ Rep(Q,α) : cV 6= 0.

Now, the Schofield semi-invariant cV is nonzero if and only if σ(β) = ⟨α,β⟩ = 0 and

ExtQ (α,β) vanishes generically. By the previous theorem of Schofield ExtQ (α,β) van-

ishes generically if and only if for every β′ such that β′ ,→ β we have σ(β′) = ⟨α,β′⟩ ≤ 0.

Thus, we have that SI(Q,β)σ 6= 0 ⇐⇒ σ(β) = 0 andσ(β′) ≤ 0 ∀ β′ ,→β. This linear homo-

geneous equality and the finite number of homogeneous inequalities can be expressed

as a matrix equation

Bσ≤ 0 : σ ∈Zn

122

where B is a matrix with rows given by the row vector β and the row vectors {β′} such

that β′ ,→ β, and σ are the weight vectors giving solutions to the inequalities given by

Bσ≤ 0. Certainly if B(nσ) ≤ 0 then

B(nσ) = n(Bσ) ≤ 0, n ∈N =⇒ Bσ≤ 0.

Thus, if nσ is a solution then so is σ, and Σ(Q,β) must be saturated.

Example 10.2.8. Again, let Q be the quiver

•1a // •2

Also, we again let β = (2,4) be the dimension vector of a general representation W and

we let α= (2,1) be the dimension vector of a general representation V . So we again have

the representations,

V : C2 A // C and W : C2 B // C4 .

Then we have the noncommutative diagram

V : C2 A //

φ(1)��

C

φ(2)��

C2B// C4

where A is a general 1×2 matrix in Rep(Q,α) and B is a general 4×2 matrix in Rep(Q,β),

after a choice of basis. Note that ⟨α,β⟩ = 0 = σ(β) for the weight σ corresponding to

α. We said there is no subrepresentation of dimension α = (2,1) such that α ,→ α+β,

where β = (2,4), since the matrix of such a general representation W , restricted to a 2-

dimensional subspace of W (x1), is a rank 2 matrix. But, we do have a subrepresentation

W ′ with dimension vector β′ = (2,2) such that β′ ,→ β, by the same argument. Comput-

ing the weight vector σ such that σ= ⟨α,•⟩ using the formula

σ(x) =α(x)− ∑y∈Q0−{x}

by,xα(y)

123

where by,x = |{a ∈Q1 : t a = y,ha = x}|, we get that σ= (2,−1). In this case

σ(β′) = (2 ·2)+ (−1 ·2) = 2 > 0.

Thus by Part 2 of Theorem 10.2.7, we have that Σ(Q,β)σ = 0 since there is a subrepresen-

tation W ′ of dimension β′ = (2,2) such that β′ ,→β, and such that σ(β′) 6≤ 0.

Example 10.2.9. Suppose instead we take α= (1,0) and β= (n,n), for the quiver •→ •.

In this case we get the following diagram for general representations V and W ,

V : CA //

φ(1)��

0

φ(2)��

CnB// Cn

.

We calculate σ= ⟨α,•⟩ = (1,−1). So, σ(β) = n −n = 0, and for any subrepresentation W ′

of the representation W , we must have that the map W (a) restricted to the subspace

W ′(x1) ⊆ W (x1) must be of full rank, and thus square, meaning the dimension β′ of W ′

must be of the form β′ = (m,m) for some nonnegative integer m ≤ n. Thus, for any

subrepresentation of dimension β′ such that β′ ,→β, we have that σ(β′) = m−m = 0. By

Theorem 10.2.7 we have that cV exists and is nontrivial.

Example 10.2.10. Now, let Q be the following quiver

•5

a5

��•1 a1// •3 a3

// •6 •4a4oo •2a2

oo

with general representation,

V : C3n

A5��

C2nA1

// CnA3

// C3n CnA4

oo C2nA2

oo

.

124

Now, suppose W is the following general representation,

W : C4n

A′5 ��

C2nA′

1

// C4nA′

3

// C6n C4nA′

4

oo C2nA′

2

oo

.

We would like to use Theorem 10.2.7 or Theorem 10.2.3 to find out if there is a Schofield

semi-invariant cVW . This means that cV

W = det(dVW ) must be nonzero for general repre-

sentations V and W . For the map dVW we have the following noncommutative diagram,

C3n

A5

}}

φ(5)

��

C2n

φ(1)

��

A1 // Cn A3 //

φ(3)

��

C3n

φ(6)

��

CnA4oo

φ(4)

��

C2n

φ(2)

��

A2oo

C4n

A′5

}}C2n

A′1

// C4nA′

3

// C6n C4nA′

4

oo C2nA′

2

oo

For the two sub-diagrams,

C2n

φ(1)��

A1 // Cn

φ(3)��

C2nA′

1

// C4n

and C2n

φ(2)��

A2 // Cn

φ(4)��

C2nA′

2

// C4n

We see there can be no subrepresentations W ′ of dimension α′ = (2n,n) such that α′ ,→α′+β′, where β′ = (2n,4n), by a similar argument to the previous example. In particular,

125

If we restrict A′1 and A′

2 to a 2n-dimensional subspace, we get a rank 2n map, however,

A1 and A2 are both rank n maps. Thus, there can be no

α= 3n2n n 3n n 2n

dimensional subrepresentation of the general representation of dimension

α+β= 7n4n 5n 9n 5n 4n

,

i.e. α 6,→α+β. Computing the weight vector σ such that σ= ⟨α,•⟩ using the formula

σ(x) =α(x)− ∑y∈Q0−{x}

by,xα(y)

where by,x = |{a ∈Q1 : t a = y,ha = x}|, we get that

σ= 3n2n −n −2n −n 2n

.

Further, we can find a subrepresentation of dimension

β′ = 2n2n 2n 2n 2n 2n

such thatβ′ ,→α+β, andσ(β′) = 3n > 0. So, even though ⟨α,β⟩ =σ(β) = 0, we must have

that det(dVW ) = 0 generically, thus there is no Schofield semi-invariant cV

W , and

Σ(Q,β)σ =Σ(Q,β)⟨α,•⟩ = 0

by Theorem 10.2.7.

The equality σ(β) = 0 and the inequalities given by σ(β′) ≤ 0 for all dimension vectors β′

such that β′ ,→ β, translates into a set of equations in terms of partitions. In particular,

conditions for Σ(Q,β)σ 6= 0 translate into conditions cνλ,µ 6= 0 for the set

L Rn = {(λ,µ,ν) ∈ (Zn)3 :λ,µ,ν are partitions of n}.

126

In the next section we describe in more detail a relation between weight spaces for triple

flag quivers, and triples of partitions given by a bijection of sets

ψ :Σ(Q,β)×Z2 →L Rn

defined by H. Derksen and J. Weyman in [6]. In fact, a bijection between the real span

of a rational cone associated to Σ(Q,β)σ and the positive real span of a rational cone

associated to triples of partitions of n ∈N can be constructed

R+Σ(Q,β)×R2 →R+L Rn .

Further, showing that weight vectors are saturated shows Littlewood-Richardson coeffi-

cients are also saturated.

10.3 Saturation of the Littlewood-Richardson Coefficients

Suppose we take the triple flag quiver Tn,n,n with dimension vector

β=

12...

n −11 2 · · · n −1 n n −1 · · · 2 1

.

We know we can view dim(SI(Q,β)σ) as a Littlewood-Richardson coefficient as follows.

Suppose σ is given by

σ=

a1

a2...

an−1

b1 b2 · · · bn−1 cn cn−1 · · · c2 c1

then

dim(SI(Q,β)σ) = cνλ,µ

127

where

λ=λ(σ) = (a1 +·· ·+an−1, a2 +·· ·+an−1, ..., an−1,0)

µ=µ(σ) = (b1 +·· ·+bn−1,b2 +·· ·+bn−1, ...,bn−1,0)

ν= ν(σ) = (−cn ,−(cn + cn−1), ...,−(cn + cn−1 +·· ·+c1))

and conversely if (λ,µ,ν) ∈ (Zn)3 such that λ,µ, and ν are weakly decreasing sequences

of nonnegative integers then,

cνλ,µ = dim(SI(Q,β)σ)

where the arms of σ are given by

σ=σ(λ,µ,ν) =λ1 −λ2 λ2 −λ3 · · · λn−1 −λn

µ1 −µ2 µ2 −µ3 · · · µn−1 −µn λn +µn −ν1

νn−1 −νn νn−2 −νn−1 · · · ν1 −ν2

with λn +µn −ν1 corresponding to the middle vertex. In fact, for

L Rn = {(λ,µ,ν) ∈ (Zn)3 :λ,µ,ν are weakly decreasing sequences of integers },

Derksen and Weyman construct a bijection of the sets

ψ :Σ(Q,β)×Z2 →L Rn

in [6] pg. 40. The bijection is defined as follows,

ψ(σ, a,b) = (λ(σ)+a ·1,µ(σ)+b ·1,ν(σ)+ (a +b) ·1)

where 1 = (1,1, ...,1) ∈Nn . This bijection then extends to a bijection of the cones

R+Σ(Q,β)×R2 →R+L Rn ,

where R+ denote the positive real span of the sets. The inverse is given by

(λ,µ,ν) 7→ (σ(λ,µ,ν),λn ,µn).

Thus we have that the saturation of weights implies the saturation of Littlewood-Richardson

coefficients. Let’s take a look at an example of the construction.

128

Example 10.3.1. Given three partitions (λ,µ,ν) ∈ L Rn , i.e. three partitions such that

cνλ,µ 6= 0, we would like to show that what we compute for cν

λ,µ is actually the dimension

of the space SI(Q,α)σ for appropriate Q and α. Suppose ν= (5,4,2,2), λ= (4,2,2,1) and

µ= (2,1,1,0). Let λ be represented by the dotted boxes in ν,

• • • •• •• ••

and we get ν/λ to be

.

There is only one way of filling this with content (12,21,31,40) such that the filling is a

lattice permutation, and it is

11 2

3 .

So, cνλ,µ = 1. We now would like to find the corresponding quiver given by the rules above.

We have Q = T3,3,3, i.e.

x1// x2

// x3// z10 z9

oo z8oo z7

oo

y6

OO

y5

OO

y4

OO

.

Let λ be the partition corresponding to the arrows on the x arm, µ the partition corre-

sponding to the arrows on the y arm, and ν the partition corresponding to the arrows on

129

the z arm. Let

β=1 2 3 4 3 2 1

321

.

We calculate σ to be

σ(λ,µ,ν) =λ1 −λ2 λ2 −λ3 λ3 −λ4 λ4 +µ4 −ν1 ν1 −ν2 ν2 −ν3 ν3 −ν4

µ3 −µ4

µ2 −µ3

µ1 −µ2

=2 0 1 −4 1 2 0

101

.

Checking that σ(β) = 0 we then proceed to find α such that σ= ⟨α,•⟩, using the formula

α(x) = ∑y∈Q0 py,xσ(y), making sure to count trivial paths. We get for α the following

dimension vector,

α=2 3 3 4 3 2 0

111

and as we proved in our discussion on weights σ(β) = ⟨α,β⟩ = 0. We thus are assured to

have a semi-invariant cV , for a general representation V of dimensionα, and we get that

dim(SI(T3,3,3,β)⟨α,•⟩ = 1.

Let us now discuss the other direction. We can always view dim(SI(Q,β)σ) for triple flag

quivers as a Littlewood-Richardson number cν∗

λ,ν as follows. Let dim(U ) = n, and let

dim(SI(Q,β)σ) = dim(SλU ⊗SµU ⊗SνU )SL(U ) = c(n)λ,µ,ν.

130

For c(n)λ,µ,ν to be nonzero we must have that |λ|+|µ|+|ν| = kn for some positive integer k,

in which case we have

c(n)λ,µ,ν = cν

∗λ,µ

where ν∗ = (k −νn ,k −νn−1, ...,k −ν1).

Definition 10.3.2. Let x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn) be two nondecreasing,

nonnegative integer sequences. Define the partition

P (x, y) = (x yn−yn−1n−1 , x yn−1−yn−2

n−2 , ..., x y2−y11 ,0).

For example, the integer sequences x = (2,3,5,7) and y = (1,2,4,4) result in the partition

P (x, y) = (3,3,2,0).

Now, consider the quiver Tp,q,r . Let α and β be dimension vectors such that the dimen-

sions weakly increase along the arms of Tp,q,r . Label the arms of the quiver by xi , y j , and

zk . Define

α(x) = (α(x1), ...,α(xp ))

and

α(y) = (α(y1), ...,α(yq )).

Define α(z),β(x),β(y), and β(z) in a similar fashion. Further, define

λ= P (α(x),β(x))

µ= P (α(y),β(y))

ν= P (α(z),β(z)).

From this construction, we can get that dim(SI(Tp,q,r ,β))⟨α,•⟩ = cν∗

λ,µ, thus giving the re-

verse construction.

131

Example 10.3.3. Let Q = T8,8,8 be the following quiver

•x1// •x2

// •x3// •x4

// •x5// •x6

// •x7

!!•y1// •y2

// •y3// •y4

// •y5// •y6

// •y7// •x8

•z1// •z2

// •z3// •z4

// •z5// •z6

// •z7

==

taking by convention x8 = y8 = z8. Give T8,8,8 the following dimension vector,

1 2 3 4 5 6 7

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7

choose the weight σ,

1 0 0 0 1 0 0

0 0 1 0 0 1 0 −3

0 0 1 0 0 1 0

.

This corresponds to some dimension vectorα so thatσ= ⟨α,•⟩. We can compute this us-

ing the formulaα(x) =∑y∈Q0 py,xσ(y) where the partitionsλ= (α(x),β(x)),µ= (α(y),β(y)),

and ν= (α(z),β(z)) and get

1. λ= (2,1,1,1,1,0,0,0)

2. µ= (2,2,2,1,1,1,0,0)

3. ν= (2,2,2,1,1,1,0,0),

132

then calculate ν∗. Here k = 3 and n = 8, so that

|λ|+ |µ|+ |ν| = 6+9+9 = 24 = 3 ·8 = kn

Then

(k −νn ,k −νn−1, ...,k −ν1) = (3−ν8,3−ν7,3−ν6,3−ν5,3−ν4,3−ν3,3−ν2,3−ν1)

= (3−0,3−0,3−1,3−1,3−1,3−2,3−2,3−2)

= (3,3,2,2,2,1,1,1)

= ν∗.

Equivalently, we can use

λ=λ(σ) = (a1 +·· ·+an−1, a2 +·· ·+an−1, ..., an−1,0)

µ=µ(σ) = (b1 +·· ·+bn−1,b2 +·· ·bn−1, ...,+bn−1,0)

ν= ν(σ) = (−cn ,−(cn + cn−1), ...,−(cn + cn−1 +·· ·+c1))

to get

1. λ(σ) = (2,1,1,1,1,0,0,0)

2. µ(σ) = (2,2,2,1,1,1,0,0)

3. ν(σ) = (3,3,2,2,2,1,1,1),

then compute the Littlewood-Richardson number via the skew diagram ν/µ,

• •• •• ••••

133

filled with content (12,21,31,41,51). There are two ways to obtain a lattice permutation

word, and they are

12

34

15

and

12

13

45

giving us cνλ,µ = 2.

Example 10.3.4. Now, let Q = T8,8,8 again with the same orientation

•x1// •x2

// •x3// •x4

// •x5// •x6

// •x7

!!•y1// •y2

// •y3// •y4

// •y5// •y6

// •y7// •x8

•z1// •z2

// •z3// •z4

// •z5// •z6

// •z7

==

.

Give T8,8,8 the following dimension vector,

1 2 3 4 5 6 7

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7

134

choose the weight σ,

2 0 0 0 2 0 0

0 0 2 0 0 2 0 −6

0 0 2 0 0 2 0

.

Notice this is 2σ for the weight σ in the previous example. This again corresponds to

some dimension vector α so that σ= ⟨α,•⟩. We can use

λ=λ(σ) = (a1 +·· ·+an−1, a2 +·· ·+an−1, ..., an−1,0)

µ=µ(σ) = (b1 +·· ·+bn−1,b2 +·· ·+bn−1, ...,bn−1,0)

ν= ν(σ) = (−cn ,−(cn + cn−1), ...,−(cn + cn−1 +·· ·+c1))

to get

1. λ(σ) = (4,2,2,2,2,0,0,0)

2. µ(σ) = (4,4,4,2,2,2,0,0)

3. ν(σ) = (6,6,4,4,4,2,2,2).

Notice, these are multiples of the previous partitions we obtained in the last example, i.e

2λ,2µ,2ν, for the λ,µ, and ν of the previous example. Now we compute the Littlewood-

Richardson number via the skew diagram ν/µ,

• • • •• • • •• • • •• •• •• •

135

filled with content (14,22,32,42,52). There are 3 ways to obtain a lattice permutation

word, and they are

1 12 2

1 13 3

4 45 5

and

1 12 2

3 34 4

1 15 5

and

1 12 2

1 33 4

1 44 5

giving us cνλ,µ = 3.

In general, for the quiver T8,8,8 as above with dimension vector β

1 2 3 4 5 6 7

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7

136

and with weight nσ= (nσ(1), ...,nσ(8)) for n ∈N, i.e. nσ is

n 0 0 0 n 0 0

0 0 n 0 0 n 0 −3n

0 0 n 0 0 n 0

we get the corresponding partitions nλ,nµ, and nν where

1. nλ(σ) = (2n,n,n,n,n,0,0,0)

2. nµ(σ) = (2n,2n,2n,n,n,n,0,0)

3. nν(σ) = (3n,3n,2n,2n,2n,n,n,n).

Further, we get that cnνnλ,nµ = n +1 (see [6] pg. 46).

137

Appendix A: The Path Algebra and CQ-modules

Here we will define and briefly discuss the path algebra and some very basic language

and definitions of category theory. We draw attention to some language used in the de-

scription of the irreducible representations of the General Linear Group, and the basic

notion of an equivalence of categories, namely modules over the path algebra of a quiver

Q, and representations of the quiver Q. We provide only enough information for com-

pleteness and refer the reader to other sources for the details.

A.1 The Path Algebra CQ

Definition A.1.1. A path p in a quiver Q is a sequence of arrows am am−1 · · ·a2a1 such

that hai = t ai+1, for i ∈ {1,2, ...,m −1}. The head of the path p, is defined as hp = ham ,

and the tail of the path p is t p = t a1. We also have trivial paths ex for each x ∈Q0, where

hex = tex = x.

Definition A.1.2. For paths p = am · · ·a1 and q = bn · · ·b1, if we have that hp = ham and

t q = tb1 = hp we define the concatenation of p and q as

qp = bn · · ·b1am · · ·a1

Further, for any et a and eha we have that aet a = a = eha a

Remark A.1.3. In the convention used here, we traverse ai then ai+1 in a path p =am am−1 · a1, similar to composition of linear maps acting on a column vector by left

multiplication.

Definition A.1.4. The path algebra, denoted CQ, of a quiver Q is the algebra spanned

138

by all paths in the quiver Q. Multiplication is given by

q ·p ={

qp if t q = hp

0 else

Remark A.1.5. We can also define the path algebra CQ via generators and relations. The

algebra CQ is generated by all {ex : x ∈Q0} and all {a ∈Q1} satisfying the following rela-

tions,

ab = 0 if t a 6= hb, a,b ∈Q1

aex = 0 if t a 6= x, a ∈Q1, x ∈Q0

aet a = a = eha a a ∈Q1

ex a = 0 if ha 6= x

exey = 0 if x 6= y

e2x = ex x ∈Q0

The identity in the path algebra is the sum∑

i ei of all of the trivial paths at each vertex

of the quiver. Notice, the path algebra has an identity if and only if it has a finite vertex

set Q0.

A.2 The Correspondence Between Quiver Representations andCQ-modules

We will sparingly, and casually use the language of categories. Here we introduce only

the very basics and the most essential definitions. For a thorough treatment of cate-

gory theory we refer the reader to [16], and for a more applied treatment and its use in

homological algebra we refer the reader to [18].

Definition A.2.1. A class is a way of talking about collections of objects too large to be

considered as sets. For example, in dealing with Russell’s paradox, i.e. one cannot have

"the set of all sets", thus one speaks of the "class of all sets". A class is called small if it

139

has a cardinal number, and a class is a set if and only if it is small. If a class is not small,

then it is a proper class. So, for example, N,Z,R,C are all sets, whereas the collections of

all sets is a proper class. We run into trouble again when we try to speak of the "class of

all Russell classes". This however will not be an issue for us, thus we leave the reader to

investigate these issues further in [16].

Definition A.2.2. A category is a class of objects Obj(C ), a set of morphisms HomC (A,B)

for every pair of objects (A,B) with A,B ∈ Obj(C ), an identity morphisms idA ∈ HomC (A, A)

for every object A, and a composition map

HomC (A,B)×HomC (B ,C ) → HomC (A,C )

for every triple (A,B ,C ) of objects. We often denoted f ∈ HomC (A,B) by

f : A → B

and when no confusion arises over which category we are working in, we drop the C

and simply write Hom(A,B). For f : A → B and g : B →C , we denote the composition by

g f : A →C . Further, we have the following axioms,

1. h(g f ) = (hg ) f for f : A → B , g : B →C , and h : C → D .

2. idB f = f = f idA for f : A → B .

Example A.2.3. One category which we will work with throughout is the category Rep(Q,α),

of representations of a quiver Q with dimension vector α. The objects are of course rep-

resentations V of Q, with dimension vectorα. The morphisms are quiver representation

morphisms φ : V →V ′ as defined in §2.

Definition A.2.4. A covariant functor F : C → C ′ maps an object A ∈ C to an object

F (A) ∈C ′. Additionally, for any pair (A,B) of objects in C we have

F : HomC (A,B) → HomC ′(F (A),F (B))

140

f 7→F ( f )

with F (idA) = idF (A) for all A ∈ Obj(C ), and F (g f ) =F (g )F ( f ) for all morphisms f and

g in C with a defined composition. A contravariant functor is a functor that reverses

arrows, i.e. given f : A → B we have F ( f ) : F (B) →F (A), and F (g f ) =F ( f )F (g ), for f

and g morphisms with a defined composition in C .

Example A.2.5. For any category C , and for any objects in that category, we have the

functor FA given by,

FA(B) = HomC (A,B)

which is a functor from C to the category of sets, denoted Sets. For morphisms f : B →C

of objects in C , we define

FA( f ) : HomC (A,B) → HomC (A,C )

by

FA( f )g = f g

This functor is covariant, and we denote it by HomC (A,•).

Example A.2.6. For any category C , and for any objects in that category, we also have

the contravariant version of the above functor, F A, which is given by,

F A(B) = HomC (B , A)

which is a functor from C to the category of sets as well. For morphisms f : B → C of

objects in C , we define

F A( f ) : HomC (C , A) → HomC (B , A)

by

F A( f )g = g f

As stated, this functor is contravariant, and we denote it by HomC (•, A).

141

Remark A.2.7. There are various functors from the category of vector spaces to itself,

some of which we will use in the construction of Schur Functors, a special type of functor

from the category of vector spaces to itself. It is what is known as a polynomial functor.

Other examples of polynomial functors from the category of vector spaces to itself are

taking tensor powers, exterior powers, and symmetric powers of vector spaces. For a

more detailed discussion, see [17] pg. 273.

Remark A.2.8. For more details of the category theory in the following statement we

refer the reader to [16]. Denote the category of all left CQ-modules by CQ Mod, the mor-

phisms are module homomorphisms. Denote the category of all representations of the

quiver Q by RepC(Q), the morphisms are quiver representation morphisms. The cate-

gories CQ Mod and RepC(Q) are equivalent, and we can define functors between them

so that the composition of those functors is what is called naturally isomorphic to the

identity functor. We refer the reader to [18] §1.2 for the description of natural trans-

formations, and to [4] for the construction of the equivalence between these two cate-

gories. If the reader is unfamiliar with categories and natural transformations, this just

means when we speak of modules V over the path algebra CQ of some quiver Q, we

are also speaking of quiver representations V of Q, and that the two are equivalent in

a particular sense, so we may often use the language of modules and representations

interchangeably throughout.

142

Appendix B: Auslander-Reiten Quivers

Auslander-Reiten theory is a beautiful theory on the representations of Artinian rings.

We will not go into the details of the theory here, instead we refer the reader to [1] and

[11]. In this section we will simply use a tool from the theory known as the Auslander-

Reiten quiver, a quiver giving a complete list of indecomposable representations of the

algebras we are interested in and the maps between them. Namely, we list the inde-

composables of the path algebras of some of the ADE-Dynkin quivers. We calculate the

Auslander-Reiten quiver for the path algebras of the quivers, that we will use later for

proving other results, using the knitting algorithm. The details of the computations are

not given, but the reader may refer to [1], [11], and [2] for the details of the algorithm

and for computations of Auslander-Reiten quivers similar to those given here, as well as

others. For an alternate method, as well as for details of the knitting algorithm the reader

can also refer to [20]. For the ADE-Dynkin quivers this algorithm is very straightforward.

It is important to note that although this algorithm does not terminate for some quivers,

and cannot be applied successfully for others, it is applicable to many types of quivers,

not just the ADE-Dynkin quivers. It can also be applied to many quivers with relations,

but we do not discuss this here.

Remark B.0.9. Let Q be the following quiver representation,

C3

A2

A1

~~C4 C4

.

In the future we will use a more compact notation for representations in general, sim-

ply giving the dimension vector β in the shape of the quiver. In that case, the above

143

representation would be denoted,

34 4

.

This may seem as though there is a loss of information, as the maps on the arrows are

not given. We introduce the notion of a general object to justify this simplification of

notation. Conceptually, if a representation V ∈ Rep(Q,α) is a general or generic repre-

sentation, it should have a property held by almost every representation in the category

Rep(Q,α). An example of such a generic property is that of full rank matrices. It is well

known that m ×n matrices of full rank form a dense subset in the set of all m ×n ma-

trices. Similarly, since quiver representations are an assignment of finite dimensional

vectors spaces and linear maps (representable by matrices) to the vertices and arrows

of a quiver respectively, we can look at dense subsets of these assignments. This no-

tion is made precise in the main text in §10. In the following examples we use the more

compact notation of dimension vectors to denote general representations of the quiv-

ers. The vertices of the AR-Quiver labeled by these dimension vectors give a complete

list of the dimension vectors of general indecomposable representations of the quivers

in question, and the maps between them.

The Auslander-Reiten quiver Γ(CQ) is a quiver with representations assigned to its ver-

tices and a special type of morphism assigned to the arrows. The modules assigned to

the vertices are all of the indecomposable representations of a specific quiver Q. So in

calculating the Auslander-Reiten quiver Γ(CQ), we obtain all of the indecomposables of

Q, and the maps between them.

B.1 A Quiver with Graph A2

Let Q be the quiver

•1// •2

144

Using the knitting algorithm we compute the Auslander-Reiten quiver Γ(CQ) and get,

01

!!

10

11

== .

B.2 A Quiver with Graph A3

Example B.2.1. Let Q be the quiver

•1// •3 •2oo .

Using the knitting algorithm we construct the Auslander-Reiten quiver Γ(CQ) and get,

010

110 011

111100 001 .

B.3 A Quiver with Graph D4


•2

��•1// •4 •3oo

.

Again, using the knitting algorithm we construct the Auslander-Reiten quiver Γ(CQ),

0010

0110

1010

0011

1121

1011

0111

1110

1111

0100

1000

0001

.

145

B.4 A Quiver with Graph E6


•5

��•1// •3

// •6 •4oo •2

oo

.

Using the knitting algorithm we get the following Auslander-Reiten quiver for the path

algebra,

146

000100

100100

101210

001110

112321

111211

212321

101110

112221

011111

111111

100000

000111

101100

011110

100111

001000

010000

001100

111210

101221

112211

111110

000011

000110

101211

112210

111221

101111

011000

011100

100110

001111

111100

000010

000001

.

147

Appendix C: The Tensor Algebra

Here we give a brief review of the tensor algebra and some of its subalgebras, subspaces,

and quotient spaces. We refer the reader to [8] Appendix B, and to [17] Chapter 5 and

Chapter 9 for a more detailed exposition on multilinear algebra and its relation to the

representation theory of general linear groups. We do not attempt to prove the Cauchy

formulas, we only justify our use of notation. From the Cauchy formulas and identifica-

tions defined in 8.1 we get the following,

C[Hom(V ,W )] ∼=C[V ∗⊗W ]

= Sym(V ⊗W ∗)

= ⊕n≥0

Symn(V ⊗W ∗)

= ⊕n≥0

⊕λ`n

SλV ⊗SλW ∗

=⊕λ

SλV ⊗SλW ∗

In order to justify the use of the equalities we remind the reader of some of the properties

of the tensor algebra over a vector space, giving a way to identify the sum

⊕λ

SλV ⊗SλW ∗

as an algebra of multilinear functions on the vector space V ⊗W ∗, by identifying each

SλV and each SλW ∗ with vectors spaces of multilinear functions on V ∗ and W respec-

tively.

Definition C.0.2. We define the tensor algebra over a finite dimensional complex vector

space V as the space

T (V ) = ⊕n≥0

V ⊗n

148

the direct sum of all tensor powers of V , where we define V ⊗0 =C, along with the multi-

plication

V ⊗p ×V ⊗q →V ⊗p+q

given by

(v, w) 7→ v ⊗w.

Note, this product in T (V ) is not the tensor product, as the tensor product of elements

in V ⊗p with elements in V ⊗q is not defined.

Now, we have a grading of T (V ) by nonnegative integers, and we call the vector space

V ⊗p the homogeneous degree p subspace of T (V ). This makes T (V ) into a N-graded

associative algebra. Now, in an analogous way we define the tensor algebra T (V ∗) on

the dual space of V . Elements of T (V ∗) are tensors of dual vectors, but they are also

multilinear functions on the vector space V given by εi1,...,ip : V →C,

εi1,...,ip (v1, ..., vp ) = εi1 ⊗·· ·⊗εip (v1, ..., vp ) = εi1 (v1) · · ·εip (vp ) ∈C.

where εi is a basis element of V ∗, and vi ∈V . If we choose a basis of V , say

BV = {x1, ..., xk }

then these tensors can be thought of as the free algebra C⟨x1, ..., xk⟩, i.e. an algebra of

noncommutative polynomials in the variables x1, ..., xk . If we take (V ∗)⊗0 = C and the

basis elements {ε1, ...,εk } of (V ∗)⊗1 =V ∗, dual to the basis BV , we have a set of generators

of T (V ∗). We can generate any simple (decomposable) homogeneous element εi1 ⊗·· ·⊗eip ∈ (V ∗)⊗p ⊂ T (V ∗), with the multiplication in T (V ∗) that we have defined, and we

can generate any element of the homogeneous degree p subspace V ⊗p since we can

generate its basis {εi1 ⊗·· ·⊗ εip : i j ∈ {1, ..., p}} by multiplying degree one tensors via the

product in T (V ∗).

149

Example C.0.3. For example if dimV ∗ = 3 and BV ∗ = {ε1,ε2,ε3} then we can generate

the basis of (V ∗)⊗2 in the following way with the multiplication of T (V ∗),

(ε1,ε1) 7→ ε1 ⊗ε1 (ε1,ε2) 7→ ε1 ⊗ε2 (ε1,ε3) 7→ ε1 ⊗ε3

(ε2,ε1) 7→ ε2 ⊗ε1 (ε2,ε2) 7→ ε2 ⊗ε2 (ε2,ε3) 7→ ε2 ⊗ε3

(ε3,ε1) 7→ ε3 ⊗ε1 (ε3,ε2) 7→ ε3 ⊗ε2 (ε3,ε3) 7→ ε3 ⊗ε3.

One can then easily see how to generate a basis {εi1 ⊗ ·· · ⊗ εin : i j ∈ {1, ...,n}} of (V ∗)⊗n ,

and thus any homogeneous degree n component of T (V ∗).

Now, we can identify several subspaces and quotient spaces of T (V ) or T (V ∗) with some

familiar spaces.

Definition C.0.4. An ideal I ⊆ A of an algebra A is a vector subspace such that aI ⊆ I

and I a ⊆ I , for all a ∈ A.

We can define an ideal generated by all

εi ⊗εi

under the multiplication in T (V ∗). Quotienting out by this ideal we then get the relations

εi1 ⊗·· ·⊗εip ∼ sgn(σ)εσ(i1) ⊗·· ·⊗εσ(ip ).

for σ ∈ Sp . This gives us the algebra known as the exterior algebra

∧V ∗ = ⊕

n≥0

n∧V ∗.

Its homogeneous degree p subspace is just the exterior power∧p V ∗. This is the algebra

of alternating (or antisymmetric) multilinear functions on V . If we instead define an

ideal generated by

εi ⊗ε j −ε j ⊗εi

150

under the multiplication of T (V ∗) we can quotient out by this ideal and get the relations

εi1 ⊗·· ·⊗eip ∼ εσ(i1) ⊗·· ·⊗eσ(ip ).

for σ ∈ Sp . This is just the symmetric algebra

Sym(V ∗) = ⊕n≥0

Symn(V ∗).

The symmetric algebra can then be identified with the algebra C[x1, ..., xk ] = C[V ] of all

commuting polynomial functions in the variables x1, ..., xk , i.e. the coordinate ring of V .

This is the algebra of symmetric (commuting) multilinear functions on V . Now, since

Schur modules

Sλ(V ) =λ′

1∧V ⊗·· ·⊗

λ′r∧

V /Qλ(V )

and

Sµ(V )∗ =µ′1∧

V ∗⊗·· ·⊗µ′r∧

V ∗/Qµ(V ∗)

where λ′ and µ′ denote the conjugate partitions, are defined as spaces of tensors sat-

isfying certain conditions, they can be identified as multilinear functions on V ∗ and V

respectively. Further, by Schur-Weyl duality (see [17] pg. 243 and [8] §6.1-6.2), each V ⊗n

decomposes into Schur modules by the formula

V ⊗n = ⊕λ`n

SλV ⊗mλ

where mλ is the dimension of the irreducible representation (Specht module) Vλ, of the

symmetric group Sn (see [8] pg. 77). This gives a decomposition of the tensor algebra

T (V ) in terms of Schur modules

T (V ) = ⊕n≥0

V ⊗n = ⊕n≥0

⊕λ`n

SλV ⊗mλ =⊕λ

SλV ⊗mλ .

151

We can extend this to multilinear functions on the vector space V ⊗W ∗, where V and W

are finite dimensional complex vector spaces by defining the tensor algebra on V ⊗W ∗.

Define the tensor algebra over V ⊗W ∗ as,

T (V ⊗W ∗) = ⊕n≥0

(V ⊗W ∗)⊗n .

The multiplication is given by (x, y) 7→ x ⊗ y ∈ (V ⊗W ∗)⊗p+q , for x ∈ (V ⊗W ∗)⊗p and

y ∈ (V ⊗W ∗)⊗q . There is a natural ismorphism, i.e an isomorphism that does not depend

on a choice of basis,

T (V ⊗W ∗) ∼= T (V )⊗T (W ∗) = ⊕n≥0

⊕a+b=n

V ⊗a ⊗ (W ∗)⊗b .

We can define a multiplication on T (V )⊗T (W ∗) by

((v, w), (v ′, w ′)) 7→ (v, v ′)⊗ (w, w ′) = v ⊗ v ′⊗w ⊗w ′

where v, v ′ ∈ T (V ) and w, w ′ ∈ T (W ∗), and (v, v ′) denotes the multiplication in T (V ),

(w, w ′) denotes the multiplication in T (W ∗). Now we have by the Cauchy formulas,

C[V ∗⊗W ] = Sym(V ⊗W ∗)

= ⊕n≥0

Symn(V ⊗W ∗)

= ⊕n≥0

⊕λ`n

SλV ⊗SλW ∗

=⊕λ

SλV ⊗SλW ∗.

We are summing over all partitions λ, which gives us an infinite dimensional vector

space, just as

T (V ) = ⊕n≥0

V ⊗n and C[V ∗⊗W ] = Sym(V ⊗W ∗) = ⊕n≥0

Symn(V ⊗W ∗)

each give us an infinite dimensional vector space. This should not be surprising consid-

ering we are decomposing the infinite dimensional symmetric algebra (or equivalently

152

the algebra of polynomial functions) Sym(V ⊗W ∗), using the Cauchy formula on each

homogeneous degree n subspace, into a direct sum of tensor products of finite dimen-

sional vector spaces SλV ⊗SλW ∗, by the formula

Symn(V ⊗W ∗) = ⊕λ`n

SλV ⊗SλW ∗

This decomposition is into a direct sum of infinitely many vector spaces that are ten-

sor products of irreducible representations of GL(V ) and GL(W ) (the Schur modules).

According to [27], by the linear reductivity of general linear groups, all irreducible rep-

resentations of GL(V )×GL(W ) are simply tensor products of irreducible GL(V ) repre-

sentations with irreducible GL(W ) representations, thus this also gives a decomposition

of the symmetric algebra Sym(V ⊗W ∗) into irreducible GL(V )×GL(W ) representations.

An explicit description of the tensor elements (multilinear functions) of each SλV , for

arbitrary λ and V , is given in §7. We use this as justification for the use of equalities in

Remark 8.1.7 and 8.1.8, and at various other points where these remarks are applied to

computations. We also note that not only are these equalities referring to equalities of

algebras of multilinear functions, but they are also giving the decomposition of the al-

gebras in terms of direct sums of irreducible GL(V ) and GL(V )×GL(W ) representations,

and thus indicate isomorphisms of representations.

153

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157

Curriculum Vitae Amelie Schreiber

Department of Mathematics email: [email protected] Forest University phone: (336) 809 - 3738Winston Salem, NC 27103 webpage: https://sites.google.com/site/amelieschreiber86/home

Wake Forest University, Winston Salem, NCEducation

M.A. in Mathematics, May 2015Thesis: Semi-invariants of Quivers and Saturation of Littlewood-RichardsonCoefficientsAdvisor: Dr. Ellen Kirman

Ruprecht Karls Universitat Heidelberg, Heidelberg, Germany

Exchange Study in Mathematics, and German as a foreign language,Sept. 2010 -Sept. 2011Studied: Mathematics in German, and German as a foreign language

The University of North Carolina at Greensboro, Greensboro, NC

B.S. in Mathematics, May 2012

The University of North Carolina at Greensboro, Greensboro, NC

B.A. in German, May 2012

Wake Forest UniversityAcademicPositions Graduate Teaching Assistant Aug. 2013 - present

Wake Forest UniversityMath Center Tutor May 2014 - Aug. 2014

Measurement IncorporatedGrader and Evaluator Feb. 2013 -Aug. 2013

The University of North Carolina at GreensboroGraduate Teaching Assistant Aug. 2012 - Dec. 2012

The University of North Carolina at GreensboroTranslator Aug. 2011 - Dec. 2011

UNCG Student Success CenterTutor and Tutor Trainer Aug. 2008 -May 2012

UNCG Department of MathematicsUndergraduate Researcher Jan. 2010 -July 2010

UNCG OrientationsUndergraduate Advisor Aug. 2008 -Dec. 2008

The Carolinian, University NewspaperScience Writer Aug. 2008 -Aug. 2009

representation theory; noncommutative algebra; algebraic geometry; algebraicResearchInterests groups; homological algebra; invariant theory; quiver representations

158

Below are courses for which I was the teaching assistant or tutor and studyTeachingExperience session leader. For each course, I was in charge of preparing and presenting

material and holding tutoring sessions, for all mathematics courses I was alsoresponsible for grading quizzes and homework, holding evening study sessions,and substituting as instructor.

Wake Forest UniversityMath 111: Calculus with Analytic Geometry IMath 112: Calculus with Analytic Geometry IIMath 113: Calculus with Analytic Geometry IIIMath 121: Linear Algebra IMath 321: Abstract Algebra I

The University of North Carolina at GreensboroMath 191: Calculus IMath 292: Calculus IIMath 293: Calculus IIIMath 394: Calculus IVMath 311: Linear Algebra IMath 312: Modern AlgebraGerman 101: Beginning German IGerman 102: Beginning German IIPhysics 291: Physics I with CalculusPhysics 292: Physics II with CalculusPhysics 211: Physics I with trigonometryPhysics 212: Physics II with trigonometry

Summer Research FundingScholarshipsand Awards Wake Forest University Graduate School, Summer-2014 Session I/II

Georgia Algebraic Geometry Symposium (travel funding)Oct. 2014

NSF-REU Grant ($3,500) 2010-2011

Cornelia Strong Scholarship (undergraduate math academic award)University of North Carolina at Greensboro Department of Mathematics,

2010-2012

Baeker Foreign Study AwardUNCG International Programs Center, 2010-2011

King Travel AwardUNCG International Programs Center, 2010-2011

Federal Smart GrantU.S. Department of Education, 2009-2012

159

Pi Mu Epsilon, WFU member since May 2013ProfessionalMemberships American Mathematical Society member since August 2012

Association for Women in Mathematics member since August 2012NOGLSTP member since May 2013German Honors Society, UNCG member since August 2009

English: native fluencyLanguagesGerman: advanced fluencyRussian: Novice Fluency

Georgia Algebraic Geometry Symposium. University of Georgia; Septem-SelectedConference andWorkshopAttendance

ber, 2014.

Maurice Auslander Distinguished Lectures. Woods Hole OceanographicInstitute, Quissett Campus, Massachusetts; 2015 .

ReferencesM.A. AdvisorEllen KirkmanDepartment of MathematicsWake Forest UniversityPO Box 7388Winston-Salem, NC, 27109(336) [email protected]

W. Frank MooreDepartment of MathematicsWake Forest UniversityPO Box 7388Winston-Salem, NC, 27109(336) [email protected]

Andrew ConnerDepartment of Mathematics Computer ScienceSaint Marys College of CaliforniaMoraga, CA 94575(925) [email protected]

Teaching ReferenceJason ParsleyDepartment of MathematicsWake Forest UniversityWinston-Salem, NC 27109(336) [email protected]

Teaching ReferenceHugh HowardsDepartment of MathematicsWake Forest UniversityWinston-Salem, NC 27109(336) [email protected]

160

Amelie Schreiber was born December 19, 1986. She graduated from the University of

North Carolina at Greensboro in 2012 with a B.A. in German, a B.S. in mathematics, and

a minor in physics. She studied German and mathematics at Ruprecht-Karls Univer-

sität Heidelberg, in Heidelberg, Germany from September 2010 to September 2011. She

explored a semester of graduate study in statistics only to be convinced that pure math-

emtics was indeed her passion and the only thing that truly made her happy. She then

began study at Wake Forest University from August 2013 to May 2015, for the degree of

Master of Arts in mathematics. She will begin work towards a Doctor of Philosophy in

Mathematics in August 2015, under the supervision of Dr. Jerzy Weyman at the Univer-

sity of Connecticut. During her time at Wake Forest University she was married on May

17, 2014 to her partner Jae Southerland. They were married in Maryland, and their mar-

riage became legally recognized in North Carolina during fall of that same year. Amelie

Schreiber is a member of the German Honors Society, the mathematical society ΠME,

the AWM, the AMS, NOGLSTP, and the MAA. She is involved in many activities promot-

ing the continued success and empowerment of girls and women in STEM, and she is the

recipient of multiple scholarships, grants, and awards, such as an NSF-REU grant, fed-

eral SMART grants, travel funding and scholarships, departmental funding and scholar-

ships, and research grants funded by the WFU graduate school, allowing her to continue

her passion, mathematics.

161

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