SEMI-INVARIANTS OF QUIVERS AND SATURATION OF LITTLEWOOD-RICHARDSON COEFFICIENTS 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.
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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.
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,
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.
Example 3.0.16. Let Q be the quiver
•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).
Example 3.0.17. Let Q be the quiver
•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.
Example 3.0.18. Let Q be the quiver
•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
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
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
with representation,
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
with representation,
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)
)
= det(g2)det(g3)det(g6)−1 det
(A4 A2 id3n 0
0 id3n A4
).
This gives us χ2(G) = det(g2)det(g3)det(g6)−1 and σ2 = (0,1,1,0,0,−1).
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)
)
= det(g1)det(g5)det(g6)−1 det
(A3 A1 id3n 0
0 id3n A5
).
This gives us χ3(G) = det(g1)det(g5)det(g6)−1 and σ3 = (1,0,0,0,1,−1).
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)
)
= det(g2)det(g5)det(g6)−1 det
(A4 A2 id3n 0
0 id3n A5
).
This gives us χ4(G) = det(g2)det(g5)det(g6)−1 and σ4 = (0,1,0,0,1,−1).
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(x)β(x)− ∑a∈Q1
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-
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
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]
.
Example 9.2.2. Let Q be the quiver
•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
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
with representation,
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 )
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,
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].
Example 10.2.4. Let Q be the quiver
•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.
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
Example B.3.1. Let Q be the quiver
•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
Example B.4.1. Let Q be the quiver
•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,
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 ∗),
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