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Semi-Invariants for Gentle String Algebras
A dissertation presented
by
Andrew Thomas Carroll
to
The Department of Mathematics
In partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in the field of
Mathematics
Northeastern University
Boston, Massachusetts
January, 2012
1
Semi-Invariants for Gentle String Algebras
by
Andrew Thomas Carroll
ABSTRACT OF DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Mathematics
in the Graduate School of
Northeastern University, January 2012
2
Abstract
This thesis is devoted to the study of the geometry of representation spaces of string
algebras. For each irreducible component C of a representation space of a gentle string
algebra, we give an algorithm to determine the ring of semi-invariant functions on C. We
show that these rings are semigroup rings (even coordinate rings of toric varieties) whose
generators and relations can be described as walks on a particular graph. In addition, we
determine the canonical decompositions of the modules in C. This decomposition allows
a general discussion of the generating semi-invariants via Schofield’s construction. This
decomposition can be used to describe certain important GIT quotients for particular
choices of C.
3
Acknowledgements
I have been fortunate to find support and encouragement from a myriad of friends, family
and colleagues without whom this journey would have been far more difficult. I cannot
overstate the influence that my advisor, Jerzy Weyman, has had on my professional life. His
devotion to this project and meticulous reading of my work have benefited me immensely,
and I thank him for all of his time. No acknowledgement of accomplishments here would
be complete without recognizing the extraordinary support lent me by Gordana Todorov
in my early education in Auslander-Reiten theory.
I am also thankful to my colleagues in the department of mathematics, whose points of
view have informed my own in tremendously positive ways. In particular, Sachin Gautam
who pushed me to give talks early and often; Kristin Webster, from whom I learned a
number of techniques in the study of semi-invariants; Kavita Sutar, with whom I hope to
solve problems long into the future. A very special thanks is owed to a great friend and
mentor Daniel Labardini-Fragoso.
I don’t quite know how to adequately thank my family. Their support, respect, and
love have kept me grounded and self-confident throughout this experience, and I would not
have made it to this point without them. My partner Jen has truly been my foundation
and the source of my continued strength. I am overjoyed to continue to embrace life with
her, and our tenacious boys Mia and Theo.
4
For Jen
5
Contents
Abstract 2
Acknowledgements 4
Table of contents 7
1 Introduction 8
2 Preliminaries 13
2.1 Quivers and Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Module Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Semi-Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Schofield Semi-invariants . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 GIT Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 String and band modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Gentle String Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Varieties of Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.1 The Coordinate Ring k[Comn(β, r)] . . . . . . . . . . . . . . . . . . 24
6
3 Explicit Description of the Rings of Semi-invariants 30
3.1 Colorings of quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.1 Coordinate rings for algebras admitting a coloring . . . . . . . . . . 32
3.1.2 Application to gentle string algebras . . . . . . . . . . . . . . . . . 34
3.2 Semi-Invariant Functions in k[RepQ,c(β, r)] . . . . . . . . . . . . . . . . . . 36
3.3 Combinatorics: The Semigroup ΛSI(Q, c, β, r) . . . . . . . . . . . . . . . . 42
3.4 Matching Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Degree Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Generic Modules 69
4.1 The Up-and-Down Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.1 Some Combinatorics for Up-and-Down Graphs . . . . . . . . . . . . 74
4.2 Up-and-Down Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.1 Main Theorem and Consequences . . . . . . . . . . . . . . . . . . . 80
4.2.2 Projective resolutions of VQ,c(β, r) and the EXT-graph . . . . . . . 84
4.2.3 Properties of the EXT-graph . . . . . . . . . . . . . . . . . . . . . . 89
4.2.4 Homology and the EXT graph . . . . . . . . . . . . . . . . . . . . . 95
4.3 Higher Extension Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 GIT Quotients 102
5.1 Dimension Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 GIT Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7
Chapter 1
Introduction
Since their inception in the early nineteen-seventies, quivers and their representations have
been the topic of great scrutiny. While quivers were initially developed to study problems
arising in linear algebra, it quickly became apparent that they could be used to study
more general problems in the study of modules over finite-dimensional algebras [20]. In
particular, every finite-dimensional associative algebra over an algebraically closed field
is Morita equivalent (i.e., has an isomorphic module category) to the path algebra of
a quiver modulo relations. Furthermore, modules over path algebras can be viewed as
representations of the quiver.
Drozd [17] showed that the module categories of finite-dimensional algebras can fall
into one of two levels of complexity called tame and wild. Algebras of tame type have
module categories which one could hope to describe; that is, module categories whose in-
decomposable objects could be classified, whereas for algebras of wild type, there is little
hope of such description. (Algebras with only finitely many non-isomorphic indecompos-
able modules are tame in this dichotomy, although we sometimes refer to them as finite
type.)
It was Gabriel [20] who proved that (connected) quivers whose path algebras have only
8
finitely many indecomposable modules are those whose underlying graphs are simply-laced
Dynkin diagrams. One year later, Donovan-Freislich and Nazarova ([15], [31]) indepen-
dently showed that the quivers of tame type are those whose underlying graphs are ex-
tended Dynkin diagrams (otherwise known as Euclidean). In all other cases, the path
algebra is wild.
Gentle string algebras are a generally well-behaved class of algebras, which are special
cases of (special) biserial algebras. They are of tame representation type ([9],[40]), but
exhibit non-polynomial growth, meaning that the number of one-parameter families of
indecomposables increases exponentially with the dimension (in contrast to the situation of
quivers with no relations). The indecomposable modules have been classified ([5], [40]), and
the combinatorial nature of their description makes these algebras an obvious set of algebras
on which to test or disprove conjectures concerning tameness (see [16], for example). The
Auslander-Reiten theory for these algebras is also well-known and very combinatorial ([5]).
More recently, string algebras have appeared in connection with cluster algebras arising
from surfaces ([2], [18], [27], [30]) and in the description of quiver Grassmanians ([6]).
Instead of considering questions pertaining to the module category, it is interesting
to ask about the module varieties (or representation spaces) for these algebras. These
are affine varieties admitting an action of a product of general linear groups. If (Q, I)
is a bound quiver, and β is a dimension vector, we denote by RepQ,I(β) the variety of
representations of Q of dimension β, which is not necessarily irreducible (unless I = 0).
Among many natural questions arising in this context a few are the following:
a. What can be said about singularities of these varieties? Alternatively, what types of
singularities appear in orbit closures? ([1], [4])
b. Can an appropriate geometric quotient be constructed for the group action? If so,
what does its structure imply about representation type? ([7])
9
c. What do the rings of invariant regular functions under the aforementioned product
of general linear group (or some subgroup) imply about representation type? ([38],
[37])
This type of analysis was carried out by Weyman and Skowronski for representation
spaces for quivers without relations ([38]) and for canonical algebras ([37]). The work led
them to conjecture that in general, an algebra is tame if and only if all rings of semi-
invariants are complete intersections (as is true in the cases they considered). In 2010,
Kraskiewicz and Weyman found particular gentle string algebras for which this was not
the case [26].
In this thesis, we work exclusively over an algebraically closed field k of characteristic 0.
We will explicitly determine the irreducible components of representation space RepQ,I(β)
when (Q, I) is a gentle string algebra, and give a combinatorial procedure to construct
generators and relations of the rings of semi-invariant functions. It will turn out that
components will be parameterized by maps r : Q1 → N satisfying certain properties, which
will be called rank maps. The first main theorem concerns the degrees of the generators
and relations on the rings of semi-invariants of RepQ,I(β, r), relative to the grading induced
by the embedding RepQ,I(β, r) ↪→ RepQ(β).
Theorem 1
The ring of semi-invariant functions on RepQ,I(β, r) is a semigroup ring with generators
in degree at most ∑a∈Q1
2
(r(a) + 1
2
)and relations in degree at most ∑
a∈Q1
8
(r(a) + 1
2
).
10
This theorem is shown by defining a more general class of so-called matching semigroups.
The determination of degree bounds for these matching semigroups consists of constructing
a graph in such a way that certain walks on the graph correspond to elements in the
semigroup, and from which relations are clear (see section 3.4). The degree bounds for
these matching semigroups are sharp, however they may not give rise to sharp bounds on
these rings of semi-invariants.
A powerful tool in studying the geometry of these representation spaces is the notion of
the canonical decomposition of a dimension vector β with respect to some irreducible com-
ponent RepQ,I(β, r) of RepQ,I(β) (see section 4). Inherent in the canonical decomposition
is a dense subset whose elements are referred to as generic representations (in contrast to
the notion of generic due to Ringel, for example in [32]).
In chapter 4, we construct a family of modules VQ,I(β, r) for each component of a
representation space RepQ,I(β, r) ⊂ RepQ,I(β). The work of Crawley-Boevey and Schroer
[10] gives homological criteria to determine when a module is generic based on data about
its direct summands. We use these criteria to show the following theorem.
Theorem 2
The union of the orbits of VQ,I(β, r) is the set of generic representations in RepQ,I(β, r).
In particular, it is dense in its irreducible component.
In proving theorem 1, the combinatorics of Schur modules are used heavily. While
this does provide a concrete description of the ring of semi-invariants, there is another
construction of semi-invariants for representations of quivers—the so-called Schofield semi-
invariants—that were extended to representations of bound quivers by Derksen and Wey-
man in [13]. In chapter 5, we will focus on dimension vectors β and rank maps r for which
VQ,I(β, r) consists of representations whose indecomposable direct summands are bands,
11
and determine the GIT quotients for fixed weights. It is conjectured that in tame type,
these should always be products of projective spaces. We show that this holds for for
components whose generic representation is an indecomposable band.
Theorem 3
Suppose that the generic representation in RepQ,I(β, r) is an indecomposable band module.
Let χ be the weight 〈〈β,−〉〉. Then the the GIT-quotient of the set of χ-semi-stable points
of RepQ,I(β, r) (with respect to the group PGLQ(β)) is isomorphic to P1.
12
Chapter 2
Preliminaries
2.1 Quivers and Representations
In this section, we will review basic notions of quivers, bound quivers, and representations.
The primary reference is the text “Elements of the Representation Theory of Associative
Algebras” ([3]).
A quiver Q = (Q0, Q1) is a directed graph with vertices Q0 and arrows Q1. We denote
by ta (resp. ha) the tail (resp. head) of an arrow a ∈ Q1. For each vertex x ∈ Q0, we also
introduce the paths ex of length zero concentrated at x. The path algebra kQ of Q is the
vector space with basis consisting of paths in Q and multiplication given by concatenation
of paths. Namely
p · q :=
pq if h(q) = t(p)
0 otherwise.
(In the above, the head and tail of a path are defined in the obvious way). Notice that kQ
is an associative algebra, which is finite dimensional if and only if Q is finite and without
oriented cycles. Finally, it is a graded algebra with grading given by length of paths.
13
An admissible relation in Q is a k-linear combination of paths of length at least two,
ρ =n∑i=1
λiwi, where λi ∈ k, and w1, w2, . . . , wn have common tail and head, and a zero
relation is a relation with n = λ1 = 1. A quiver Q together with a family {ρi}i∈J of
admissible relations is called a bound quiver or quiver with relations, and the algebra
kQ/〈ρi〉i∈J is called a bound quiver algebra.
2.1.1 Representations
A (finite dimensional) representation of a quiver Q is an assignment of a (finite dimen-
sional) k-vector space Vx to each vertex x ∈ Q0, and a linear map Va : Vta → Vha to
each arrow. All representations will be assumed finite-dimensional. The category of
representations of Q, denoted RepQ, is the category whose objects are representations
of Q with morphisms defined as follows: if V,W are representations of Q, a morphism
ϕ : V → W consists of a Q0-tuple of linear maps ϕx : Vx → Wx such that for every a ∈ Q1,
the following diagram commutes:
VtaVa //
ϕta
��
Vha
ϕha
��Wta Wa
//Wha
If p = am . . . a1 is a path in Q, and V is a representation of Q, then Vρ := Vam · . . . · Va1
is defined to the composition of the maps along the path. If ρ =∑n
i=1 λiwi ∈ kQ is an
admissible relation, then by definition Vρ =∑n
i=1 λiVwi . Notice that since ρ is admissible,
Vρ is a sum of linear maps on a common domain and codomain.
If {ρi}i∈J is a set of admissible relations, let I = 〈ρi〉i∈J be the associated ideal of
relations. Then the category of representations of the bound quiver (Q, I), denoted RepQ,I ,
is the subcategory of RepQ whose objects are representations V of Q such that V (ρi) = 0
for all i ∈ J . The dimension vector of a representation V is the vector dimV ∈ ZQ0
14
with (dimV )x = dimk(Vx).
The category of representations of a bound quiver is equivalent to the category of mod-
ules over the bound quiver algebra. Given this equivalence, we will speak interchangeably
about representations of an algebra and representations of its (bound) quiver.
Being equivalent to a category of modules, the categories of representations of quivers
(or bound quivers) inherit a direct sum operation, and are Krull-Schmidt categories. A
fundamental problem in the representation theory of algebras is to classify all indecompos-
able objects (with respect to this operation). To that end, there are two main classes of
algebras defined below.
Definition 2.1.1. A (bound) quiver algebra A ∼= kQ/I is called tame if for each dimen-
sion vector d ∈ NQ0 , there are finitely many one-parameter families of A-k[t]-bimodules
M1, . . . ,Mh where the Mi are finitely generated free (right) k[t]-modules, such that all
but finitely many d-dimensional indecomposable A-modules M are of the form M ∼=
Mi ⊗k[t] k[t]/(t − λ) for some i and λ ∈ k. In particular, if there are finitely many in-
decomposable modules, then A is tame. The algebra A is called wild if the category of
finitely generated A-modules contains as a subcategory the category of finitely generated
k〈x, y〉-modules.
Drozd showed in [17] that every algebra is of exactly one of the two above types.
The Euler form associated to the algebra kQ/I plays a crucial role both in the repre-
sentation theory of A and the study of semi-invariants. In general, for a finite-dimensional
algebra A (over an algebraically closed field) with finite global dimension, we can identify
the Grothendieck group K0(A) with ZQ0 . This identification takes the class of a module
[V ] to its dimension vector (and is then extended linearly). Now K0(A) can be endowed
15
with a bilinear form defined on Nn × Nn by the following:
〈〈dimV, dimV ′〉〉 =∑i≥0
(−1)i dim ExtiA(V, V ′)
and then extended by linearity to K0(A)×K0(A). Denote the matrix of this bilinear form
by EA. We also write qA to denote the associated quadratic form qA(β) = 〈〈β, β〉〉.
2.2 Module Varieties
We now introduce some geometric objects which will be the primary objects of study. Let
Q be a quiver, and let β ∈ NQ0 be a dimension vector. Define by RepQ(β) the set
RepQ(β) := {V ∈ RepQ | dimV = β}.
Fixing a basis at each vertex, this set can be identified with
∏a∈Q1
Homk(kβta , kβha) =
∏a∈Q1
Matβha×βta(k)
where Matk,l(k) is the set of k × l matrices with entries in k. This latter identification
makes it clear that RepQ(β) is affine space of dimension∑a∈Q1
βtaβha. This variety will be
called the variety of representations of Q of dimension vector β. If (Q, I) is a bound
quiver, then define by RepQ,I(β) ⊂ RepQ(β) the subvariety consisting of representations
of (Q, I) of dimension vector β, i.e.,
RepQ,I(β) = {M ∈∏a∈Q1
Matβha×βta(k) |Mρ = 0 ∀ρ ∈ I}.
Notice that the entries of the matrix Mρ are polynomials in the entries of the matrices Ma,
and recall that I is a finitely generated ideal. Therefore, RepQ,I(β) is an algebraic set in
16
RepQ(β), although may fail to be irreducible.
If β is a dimension vector for some quiverQ, denote by GLQ(β) the product∏x∈Q0
GL(βx),
and SLQ(β) =∏x∈Q0
SL(βx). Both algebraic groups act by simultaneous change of basis on
RepQ(β) (resp. RepQ,I(β)) as follows:
(gx)x∈Q0 .(Ma)a∈Q1 = (ghaMag−1ta )a∈Q1 .
In fact, GLQ(β) preserves irreducible components, so the coordinate ring of each irreducible
component is a GLQ(β)-module.
2.3 Semi-Invariants
Let G be a linear algebraic group, and X be a rational G-variety. Denote by k[X] the
algebra of regular functions on X. Let X(G) be the group of (multiplicative) characters
of G.
Definition 2.3.1. For χ ∈ X(G), the space of semi-invariants of weight χ is the
vector space
SI(G,X)χ = {f ∈ k[X] | g.f = χ(g) · f for all g ∈ G}.
The algebra of semi-invariants is defined to be SI(G,X) :=⊕
χ∈X(G) SI(G, V )χ.
We will work specifically with the group G = GLQ(β), so let us recall pertinent ma-
terial from the representation theory of GL(V ) when V is a finite-dimensional k vector
space of dimension n. Denote by Zn+ the set of dominant integral weights, that is, non-
increasing integer sequences of length n. It is well-known ([19], [41]) that the irreducible
rational representations of GL(V ) are parameterized by Zn+. For λ = (λ1, . . . , λn) ∈ Zn+,
denote by SλV the corresponding representation, called a Schur module. In particu-
lar, S(1,1,...,1,0,...,0)V =∧l V the l-th exterior power (where l is the number of ones) and
17
S(l,0,...,0)V = SlV the l-th symmetric power of V . Therefore, as a GL(V ) module we can
decompose k[X]:
k[X] =⊕λ∈Zn+
m(λ,X)SλV.
Notice that X(GL(V )) = {deta | a ∈ Z} consists of integer powers of the determinant
function, which can be be identified with Z itself. Denote by χa the corresponding character
and λa = (a, a, . . . , a︸ ︷︷ ︸n
). Then SI(GL(V ), X)χa = m(λa, X)SλaV . In particular
SI(GL(V ), X) ∼= k[X]SL(V ),
where the latter is the ring of regular functions invariant under the action of the special
linear subgroup SL(V ).
Notice that each irreducible component Z of RepQ,I(β) is a rational GLQ(β)-variety,
and as such it is a rational GL(βx) variety for each x ∈ Q0. Therefore, the coordinate ring
k[Z] is a rational GLQ(β)-module. Denoting by Λ = {λ ∈∏x∈Q0Zβx+ } the set of dominant
integral weights for GLQ(β), we have
k[Z] =⊕λ∈Λ
m(λ, Z)⊗x∈Q0
Sλ(x)Vx
where Vx = kβx . Furthermore, X(GLQ(β)) can be identified with ZQ0 as products of
integer powers of the determinant functions at each vertex. Denote by χa the character
corresponding to the vector a ∈ ZQ0 . Then SI(GLQ(β), Z)χa = m(λa, X)⊗x∈Q0
SλaxVx and,
in particular,
SI(GLQ(β), Z) ∼= k[Z]SLQ(β).
In the subsequent chapters, we will always work over GLQ(β). For notational simplicity,
we will denote the ring of semi-invariants of the irreducible component Z of RepQ,I(β) by
SIQ,I(β, Z).
18
2.3.1 Schofield Semi-invariants
The difficulty in applying the above techniques is that there may not exist an explicit
description of the coordinate ring of a G variety by Schur modules. An alternate point of
view was developed by Schofield [33] and Derksen-Weyman [13].
Suppose that V is a kQ/I-module of projective dimension one with projective resolu-
tion V P (0)oo P (1)δ0oo . For any kQ/I-module W , we can consider the map dVW :
HomkQ/I(P (0),W )→ HomkQ/I(P (1),W ) obtained by applying the functor HomkQ/I(−,W )
to this presentation. Notice that the kernel of dVW is precisely HomkQ/I(V,W ), and the cok-
ernel is Ext1kQ/I(V,W ). Thus, if 〈〈dimV, dimW 〉〉 = 0, then after choosing bases, dVW is a
square matrix. Furthermore, if either HomkQ/I(V,W ) = 0 or Ext1kQ/I(V,W ) = 0, then dVW
is an isomorphism, so has non-zero determinant.
We can consider cV : RepQ,I(β) → k to be the map with cV (W ) = det dVW . Schofield
constructed these functions in [33] and showed that they are semi-invariants. Furthermore,
we have the following proposition due to Derksen and Weyman.
Proposition 2.3.2. [13] Suppose Z ⊂ RepQ,I(β), and 〈〈dimV, β〉〉 = 0. The function
cV is a semi-invariant of weight 〈〈dimV,−〉〉 and if Z is a faithful component (that is
annkQ(Z) = I), then SIQ,I(β, Z)〈〈α,−〉〉 is spanned by the set of all cV such that
a. V is of projective dimension 1;
b. dimV = α.
2.3.2 GIT Quotients
The study of moduli spaces for finite dimensional algebras was initiated by King [24],
although the general definition of GIT quotients goes back to Mumford [29]. The central
problem is to determine a variety whose points parametrize orbits of a group acting on
19
another variety. In general, such a space is elusive, so one settles for parameterizing only
certain orbits. For this section’s notation, we refer to the quiver notes by Derksen-Weyman
[14]. Let (Q, I) be a bound quiver, β a dimension vector, and Z ⊂ RepQ,I(β) an irreducible
component of the representation space. Suppose that χ is a character for GLQ(β) (which
can be considered as an element in ZQ0). Let GLQ(β)χ be the kernel of the map χ when
viewed as a character.
If χ is not divisible by the characteristic of k, then
Rχ := k[Z]GLQ(β)χ =⊕n≥0
SIQ,I(β, Z)n·χ.
Notice that Rχ ⊂ k[Z], so there is a surjective morphism Ψ : Z → Spec(Rχ). Let Zssχ be
the set of representations V ∈ Z such that there is a function f ∈ Rχ with f(V ) 6= 0.
These are called the χ-semi-stable points of Z. In this case, Ψ−1(0) is the complement
of Zssχ in Z. If Y := Proj(R), then we have the following commuting diagram:
Zssχ
$$
// X \ {0}
��Y
The map π : Zssχ → Y is a geometric quotient called the GIT-quotient of Zss
χ by PGL(β)
the product of projective general linear groups.
2.4 String and band modules
A particularly important class of modules in RepQ,I will be the string and band modules,
which we introduce following the conventions of Crawley-Boevey [8]. If T is an arbitrary
quiver, a bound quiver morphism F = (F0, F1) : T → (Q, I) consists of a set maps F0 :
T0 → Q0, F1 : T1 → Q1 such that
20
• for each a ∈ Q1, h(F1(a)) = F0(h(a)) and t(F1(a)) = F0(t(a));
• if h(a) = h(b) or t(a) = t(b) then F1(a) 6= F1(b);
• for all paths p on T , F (p) is a path in (Q, I) (that is it does not pass through any
relations).
A bound quiver morphism induces a functor F∗ : RepT → RepQ,I so that (F∗V )x =⊕y∈F−1(x) Vy if x ∈ Q0 or x ∈ Q1.
Let T be a tree, that is a finite quiver whose underlying graph is acyclic, and let 1T
be the representation of T consisting of a one-dimensional vector space at each vertex and
with every acting by the identity transformation. The representations F∗(1T ) are called
tree modules, and are indecomposable if T is connected as a result of Gabriel [22]. If T
is a chain (every vertex has valence at most 2), then F∗(1T ) is called a string module.
Let B be any orientation of An for any n. Label the vertices 1, . . . , n, n+ 1, and arrows
a1, . . . , an+1 such that t(a1) = 1 and for each i = 1, . . . , n + 1, the arrows ai, ai+1 share
a vertex. For m ∈ N, and any vector space automorphism ϕ : km → km, let ϕB be the
representation of B with (ϕB)x = km, and
(ϕB)ai =
Idm i 6= n
ϕ i = n.
The representations F∗(ϕB) are called band modules. Let c = cncn−1 . . . c1 be the se-
quence of elements in the letters Q1 ∪Q−11 with
ci =
F (ai) if t(ai) = i
F (ai)−1 otherwise.
21
In this way, the word c can be viewed as a cyclic path on Q. If c cannot be written as the
power of any smaller word (i.e., c 6= (c′)(c′) . . . (c′) for any cyclic word c′), and (km, ϕ) is
an indecomposable k[t, t−1]-module, then F∗(ϕB) is indecomposable.
2.5 Gentle String Algebras
In this section, we collect some results concerning string algebras. Most of the details can be
found in [5]. In addition, Schroer has two articles ([34], [35]) containing nice introductions.
Definition 2.5.1. A bound quiver algebra kQ/I is called a string algebra if I is generated
by paths and the following conditions hold:
a. For each x ∈ Q0, the number of arrows a ∈ Q1 with ha = x (resp. ta = x) is at most
2.
b. Given an arrow β ∈ Q1, there is at most one arrow γ ∈ Q1 with tγ = hβ (resp. at
most one arrow α ∈ Q1 with tβ = hα) such that γβ /∈ I (resp. βα /∈ I).
c. For each β ∈ Q1, there is a bound n(β) such that every path p of length greater than
n(β) whose first or final arrow is β is in I.
A string algebra is called gentle if it satisfies the additional conditions:
d. Given an arrow β ∈ Q1, there is at most one arrow γ ∈ Q1 with tγ = hβ (resp. at
most one arrow α ∈ Q1 with tβ = hα) such that γβ ∈ I (resp. βα ∈ I).
e. I is generated by paths of length two.
Proposition 2.5.2 ([5],[36]). Every indecomposable representation of a string algebra
kQ/I is either a string module or a band module.
22
2.6 Varieties of Complexes
The varieties of complexes were defined and studied by DeConcini-Strickland in [11] and
later extended to cyclic complexes by Mehta-Trivedi [28]. These are varieties whose points
correspond to complexes of vector spaces of some fixed dimension. Here we recall the
related material concerning these varieties including their irreducible components, and a
description of their coordinate rings.
Definition 2.6.1. For n ∈ N and β = (β1, β2, . . . , βn+1) ∈ Nn+1, the variety of complexes
of length n and dimension β is the set
Comn(β) := {(Mi ∈ Matβi×βi+1(k))i=1,...,n |Mi+1 ·Mi = 0 for i = 1, . . . , n− 1}.
Denote by Aeqn+1 the equioriented quiver of type An+1 with vertices 1, 2, . . . , n + 1 and
arrows ai : i → i + 1. Let I =< ai+1ai | i = 1, . . . , n − 1 >, then Comn(β) = RepAeqn ,I(β).
In this way, the varieties of complexes can be viewed as representation spaces for the most
basic gentle string algebras. Their irreducible components can be nicely parameterized
by sequences of maximal ranks in the following way. A function r : {a1, . . . , an} → N
is called a rank map for β if r(ai) + r(ai+1) ≤ βi+1 for i = 0, . . . , n (here we define
r(a0) = r(an+1) = 0). The set of all rank maps for β is a (finite) poset with r ≤ r′ if
and only if r(ai) ≤ r′(ai) for i = 1, . . . , n. Given a fixed rank map r for β, we define the
following subsets of Comn(β):
Com◦n(β, r) = {Mi ∈ Comn(β) | rankkMi = r(ai)}
Comn(β, r) = {Mi ∈ Comn(β) | rankkMi ≤ r(ai)}
Proposition 2.6.2 ([10]). Let r be a maximal rank map for β. Then Comn(β, r) is an
23
irreducible component of Comn(β). Furthermore,
Comn(β) =⋃
r maximal for β
Comn(β, r).
2.6.1 The Coordinate Ring k[Comn(β, r)]
In the same article, DeConcini-Strickland give a basis for k[Comn(β, r)] in terms of multi-
tableau. Their work prescribes a filtration on the coordinate ring whose associated graded
is given by Schur modules. The early portion of this section is a recollection of Young
diagrams. In the last part of this section, we describe a filtration on k[Com(β, r)] and its
associated graded ring. For the remainder of this section, we fix n, a dimension vector β,
and a maximal rank sequence r for β.
A Young diagram λ is a sequence of non-increasing positive integers λ1 ≥ . . . ≥ λm,
m is called the number of parts of λ. We will draw Young diagrams as a table of rows of
left-justified boxes such that the i-th row has λi boxes. For a Young diagram λ, we denote
by λ′ the transpose diagram, where λ′i = {i | λj ≥ i}. If p be a positive integer with p ≥ m.
Denote by [p− λ] the diagram with p parts and [p− λ]j = λ1 − λp−j+1 (in this expression,
if λp−j+1 is not defined, then it is considered to be 0). We will call a filling standard if it is
row increasing and column strictly-increasing. To a filling t of λ, we associated a sequence
of sets I(t) = (I(t)1, . . . , I(t)λ1) where Il = {t1,l, t2,l, . . . , tλ′l,l}.
Definition 2.6.3. Let V be a vector space, and λ a Young diagram with at most dimV
parts. We will denote by∧λ V the product of exterior powers of V prescribed by the
columns of λ. Namelyλ∧V =
λ′1∧V ⊗ . . .⊗
λ′λ1∧V.
For a set I = {i1, . . . , ik}, let eI = ei1 ∧ . . . ∧ eik . If t is a column-increasing filling of
λ with integers from the set {1, . . . , dimV }, then we have the associated sequence of sets
24
(I(t)1, . . . , I(t)λ1) and the associated basis element in∧λ V is eI(t)1 ⊗ . . .⊗ eI(t)λ1
.
Let V be a k vector space and λ a Young diagram with at most dimV parts. Define
the map
opλ :
[dimV−λ]∧V →
λ∧V
as follows: if t is a column-increasing filling of [dimV − λ], and I(t) = (I(t)1, . . . , I(t)λ1)
is the associated sequence of sets, then take t′ the filling of λ with associated sequence of
sets I(t′) = (I(t′)1, . . . , I(t′)λ1) such that I(t′)j = {1, . . . , dimV } \ I(t)λ1−j+1. Then
opλ(t) :=
(λ1∏j=1
sgn(I(t′)j, I(t)λ1−j+1)
)t′.
Here, sgn(I, J) is the sign of the permutation (I, J) with both I, J written in increasing
order.
Suppose now that Vi, Vi+1 are k-vector spaces, and λ is a Young diagram with at most
min(dimVi, dimVi+1) parts. Define a map δ(i)λ :
∧λ Vi⊗∧λ Vi+1 → k[Com(β, r)] as follows:
suppose that ti is a filling of λ from the integers {1, . . . , dimVi} with associated sequence
of sets I(ti), and ti+1 is a filling of λ from the integers {1, . . . , dimVi+1} with associated
sequence of sets I(ti+1). Then
δλ : ti ⊗ ti+1 7→λ1∏j=1
∆(i)I(ti)j ,I(ti+1)j
.
(Recall that ∆(i)I,J is the minor of the matrix Ai with columns given by I and rows given
by J .)
If λ = (λ(1), . . . , λ(n)) is a sequence of Young diagrams such that λ(i) has at most
25
min(dimVi, dimVi+1) parts, then take
δλ :n⊗i=1
λ(i)∧Vi ⊗
[βi+1−λ(i)]∧Vi+1
→ k[Com(β, r)]
To be the composition of the map
n⊗i=1
(id⊗ opλ(i)) :n⊗i=1
λ(i)∧Vi ⊗
[βi+1−λ(i)]∧Vi+1
→ n⊗i=1
λ(i)∧Vi ⊗
λ(i)∧Vi+1
and the map
n⊗i=1
δ(i)λ :
n⊗i=1
λ(i)∧Vi ⊗
λ(i)∧Vi+1
→ k[Com(β, r)].
Remark 2.6.1. If λ(i) has more than r(i) parts for some i, then image δλ = 0 on Com(β, r)
since one factor is the an r(i) + l × r(i) + l minor of Ai, and rankAi ≤ r(i) by definition
of Com(β, r).
Definition 2.6.4. Let Λn(β, r) be the set of sequence of partitions (λ(1), . . . , λ(n)) such
that [βi+1 − λ(i)]λ(i)1 ≤ λ(i+ 1)′1. I.e., the first column of [βi+1 − λ(i)] is shorter than the
last column of λ(i+ 1). If λ ∈ Λn(β, r), denote by [λ(i+ 1) : λ(i)] the Young diagram with
[λ(i+ 1) : λ(i)]j = [βi+1 − λ(i)]j + λ(i+ 1)j.
Diagrammatically, this is simply juxtaposing the diagrams λ(i) and [βi+1 − λ(i)], which is
still a Young diagram by definition of Λn(β, r). We will also write λ(1) = [λ(1) : λ(0)] and
[βn+1 − λ(n)] = [λ(n+ 1) : λ(n)] for the degenerate cases.
A filling of the diagrams [λ(1) : λ(0)], [λ(2) : λ(1)], . . . , [λ(n) : λ(n−1)], [λ(n+1) : λ(n)]]
is the same as a filling of all diagrams λ(i) and [βi+1 − λ(i)] for i = 1, . . . , n and is called
a multitableau.
26
Definition 2.6.5. For two partitions λ, µ, we define λ � µ if (λ′1, . . . , λ′λ1
) ≥ (µ′1, . . . , µ′µ1
).
Extend this to a partial order on Λn(β, r) with λ � µ if
([λ(1) : λ(0)], [λ(2) : λ(1)], . . . , [λ(n+ 1) : λ(n)], [λ(n+ 1) : λ(n)])
� ([µ(1) : µ(0)], [µ(2) : µ(1)], . . . , [µ(n+ 1) : µ(n)], [µ(n+ 1) : µ(n)])
in the lexicographical order.
Definition 2.6.6. Suppose that λ and µ are partitions with r1, r2 parts, respectively,
and V is a vector space of dimension n. Then we denote by S(λ,−µ)V the Schur module
S(λ1,...,λr1 ,0,...,0,−µr2 ,−µr2−1,...,−µ1)V , where we include n−(r1+r2) zeros in the indexing vector.
Furthermore, we will write −µ for the vector (−µr2 , . . . ,−µ2,−µ1).
Proposition 2.6.7 ([11]). Denote by Fλ =∑
µ∈Λn(β,r)µ�λ
image δµ, and F≺λ =∑
µ∈Λn(β,r)µ≺λ
image δµ.
Then Fλ/F≺λ has a basis given by standard fillings of the diagrams [λ(i + 1) : λ(i)] for
i = 0, . . . , n. A collection of fillings of this sequence of diagrams is called a multitableau
of shape λ. Furthermore,
Fλ/F≺λ ∼=n⊗i=1
S(λ(i),−λ(i−1))Vi.
The above proposition is proven by showing that if tλ is a multitableau of shape λ,
then δλ(tλ) can be written, modulo terms in F≺λ, as a linear combination of standard
multitableau of shape λ.
The content of a multitableau t of shape λ is the sequence of vectors (κ1, . . . , κn+1)
where
κij = #{ boxes in [λ(i) : λ(i− 1)] that are filled with the integer j}
27
Corollary 2.6.8 ([11]). Suppose that t is a non-standard multitableau of shape λ. Then
δλ(t) = s(t) + y(t)
where s(t) is a linear combination of standard multitableaux of the same content as t, and
y(t) ∈ F≺λ.
Proposition 2.6.9 ([11]). k[Com(β, r)] =⋃
λ∈Λn(β,r)
Fλ.
So every function in k[Com(β, r)] is in a Fλ. Next we show that the Fλ form a filtration.
For two elements λ, µ ∈ Λn(β, r), let λ+ µ be the sequence of diagrams with
(λ+ µ)(i)j = λ(i)j + µ(i)j. (2.6.1)
Proposition 2.6.10. Suppose that tλ and tµ are multitableaux of shapes λ and µ, respec-
tively. Then
δλ(tλ) · δµ(tµ) ∈ Fλ+µ.
Proof. It suffices to show this when µ consists of a single column, i.e.,
µ(i) = (1, 1, . . . , 1︸ ︷︷ ︸j
, 0, 0, . . . )
for some i and µ(j) = 0 otherwise. Thus, δλ(tµ) = ∆(i)I,J for some sets I ⊂ {1, . . . , βi},
J ⊂ {1, . . . , βi+1}. Therefore, δλ(tλ) · δµ(tµ) = δλ(tλ) ·∆(i)I,J . Now notice that λ + µ is the
sequence diagrams which is the same as λ except for (λ+µ)(i) which has an extra column
of height j. Take multitableau of shape (λ+µ) so that all entries not corresponding to the
extra column are the same as in the filling tλ, and all entries in the columns corresponding to
the extra column are taken from tµ. Denoting by tλ+µ this filling, we have that δλ+µ(tλ+µ) =
δλ(tλ) · δµ(tµ). In short, each column of the sequence µ can be absorbed into λ until the
28
result is the sequence λ+ µ.
Corollary 2.6.11 ([11]). The set {Fλ}λ∈Λ is a filtration of k[Com(β, r)] and the associated
graded algebra is
grΛ (k[Com(β, r)]) =⊕λ∈Λ
n⊗i=1
S(λ(i),−λ(i−1))Vi.
(For the definition of the Schur modules SλV , we refer to [41] Chapter 2.1.)
29
Chapter 3
Explicit Description of the Rings of
Semi-invariants
In the following chapter, we give an explicit description of the rings of semi-invariants for
components of representation spaces of gentle string algebras. This begins by recognizing
that these representation spaces are products of varieties of complexes as discussed in
section 2.6. This allows us to give a filtration on the coordinate rings of these components
whose associated graded algebra is a direct sum of Schur modules. By analyzing this
decomposition, we can in fact exhibit a grading on the rings of semi-invariants. These
rings are then shown to be semigroup rings of a particular sort, which we can investigate
to determine degree bounds on generators and relations.
3.1 Colorings of quivers
As stated in the introduction, the determination of irreducible components for representa-
tion spaces can be difficult. However, for a large class of zero-relation algebras, the problem
can be solved using varieties of complexes. This is achieved by coloring the arrows of the
30
quiver in such a way that the composition of any two same-colored arrows in Q is in the
ideal I.
Definition 3.1.1. Let S be some finite set, whose elements we call colors. A coloring
of a quiver Q is a set map c : Q1 → S satisfying the condition that c−1(s) is a direct
path for each s ∈ S. The ideal Ic ⊂ kQ associated to a coloring is defined to be the ideal
Ic =< ba | a, b ∈ Q1, h(a) = t(b), c(a) = c(b) >. If Q is a quiver with coloring c, we will
write RepQ,c in lieu of RepQ,Ic .
Remark 3.1.1. Notice that the varieties of complexes can be realized as Aeqn+1/Ic where c
is the coloring c : {a1, . . . , an} → {1} on Aeqn+1.
For each color s ∈ S, let Q(s) = (Q(s)0, Q(s)1) be the subquiver of Q consisting of
the arrows of color s and vertices to which they are incident. Suppose that c−1(s) has
n(s) arrows. Notice, then, that Q(s) = Aeqn(s)+1, and let I(s) be the ideal generated by all
length-two paths in Q(s). If β is a dimension vector for Q, let β|s be the restriction of β
to Q(s).
Proposition 3.1.2. Let c be a coloring of Q, and β be a dimension vector. There is an
isomorphism of affine varieties
∏s∈S
Comn(s)+1(β|s) ∼= RepQ,c(β)
Proof. This isomorphism is best viewed as a composition. First, it is clear by definition
that ∏s∈S
Comn(s)+1(β|s) ∼=∏s∈S
RepQ(s),I(s)(β|s)
by remark 3.1.1. Now define the map∏s∈S
RepQ(s),I(s)(β|s) → RepQ,c(β) in the follow-
ing way. Since every arrow a is in a unique Q(s)1 (namely s = c(a)), an element of∏s∈S
RepQ(s),I(s)(β|s) consists of a linear map Va for each a ∈ Q1 such that if c(a) = c(b) = s
31
and h(a) = t(b), then Vb·Va = 0. This is precisely the data of a representation of (Q, Ic).
This isomorphism allows a description of the irreducible components by generalizations
of rank maps. Namely, with the notation as above, a rank map r : Q1 → N is a function
such that for each s ∈ S, r|s : Q(s)1 → N is a rank map for the β|s (on the variety
Comn(s)+1(β|s)). Let ≤ be the partial ordering on rank maps with r ≤ r′ if r|s ≤ r′|s for
each s ∈ S.
For a given rank map r, we have the following sets in RepQ,c(β):
RepQ,c(β, r) = {V ∈ RepQ,c(β) | rankkVa ≤ r(a) for each a ∈ Q1}
Rep◦Q,c(β, r) = {V ∈ RepQ,c(β) | rankkVa = r(a) for each a ∈ Q1}
We finish this section by pointing out two important facts about the representations
spaces of quivers with colored relations. The first important fact gives a parameterization
of irreducible components, and the second is a note about the geometry of these irreducible
components. Both facts rely on the fact that these spaces are viewed as products of varieties
of complexes.
Proposition 3.1.3. The irreducible components of the variety RepQ,c(β) are the sets
RepQ,c(β, r) where r is maximal under the aforementioned partial ordering. Furthermore,
Rep◦Q,c(β, r) is an open (therefore dense) subset of RepQ,c(β, r).
Proposition 3.1.4. For each dimension vector β and each maximal rank sequence r, the
variety RepQ,c(β, r) is normal and Cohen-Macaulay with rational singularities.
3.1.1 Coordinate rings for algebras admitting a coloring
In section we exploit the filtration on the coordinate ring of the varieties of complexes
(section 2.6.1) to construct a filtration on the coordinate ring of RepQ,c(β, r). Denote by
32
X ⊂ Q0×S the set of pairs (x, s) consisting of a vertex and a color incident to that vertex.
For each such pair, we denote by i(x, s) (resp. o(x, s)) the arrow of color s whose head
(resp. tail) is x. If no such arrow exists, write ∅. A vertex will be called lonely if there is
exactly one element (x, s) ∈ X, and coupled if there is more than one.
Definition 3.1.5. Let Λ(Q, c, β, r) be the set of functions λ : Q1 → P (where P is the
set of Young diagrams or partitions) such that λ(a) has at most r(a) non-zero parts. We
write Λ when the parameters are understood. If p(s) = psms . . . ps1 is the full path of color
s, then write λs for the sequence of diagrams λs = (λ(ps1), . . . , λ(psms)).
We can now extend the map δ from section 2.6.1. Suppose λ ∈ Λ(Q, c, β, r). Define by
δλ :⊗s∈S
ms⊗i=1
λ(psi )∧kβt(ps
i) ⊗
[βh(psi)−λ(pi)]∧
kh(psi )
→ k[RepQ,c(β, r)] (3.1.1)
which is simply the product of the maps δλs . It will be convenient to denote the domain
of this map by∧λ V , and let Vx = kβx . Extending the partial ordering from 2.6.5, for
λ, µ ∈ Λ(Q, c, β, r), λ � µ if and only if λs � µs for each s ∈ S. Finally, denote by
Fλ =∑µ�λ
im(δµ) and F≺λ =∑µ≺λ
im(δµ)
Proposition 3.1.6. Let r be a maximal rank map for β. Then {Fλ | λ ∈ Λ(Q, c, β, r)} is
a filtration of RepQ,c(β, r) relative to the partial order just described. Furthermore,
Fλ/∑µ≺λ
Fµ ∼=⊗
(x,s)∈X
Sλ(x,s)Vx
where λ(x, s) = (λ(o(x, s)),−λ(i(x, s))).
33
Corollary 3.1.7. For Q, c, β, r as above,
grΛ,�(k[RepQ,c(β, r)])∼=⊕λ∈Λ
⊗(x,s)∈X
Sλ(x,s)Vx (3.1.2)
As a result of the above remarks, we can give a basis for k[RepQ,c(β, r)] via standard
multitableaux by generalizing the procedure described by DeConcini and Strickland in the
case of the varieties of complexes. The following definitions will introduce the notation
necessary for this generalization.
Definition 3.1.8. Let λ ∈ Λ(Q, c, β, r). For each element (x, s) ∈ X, denote by [λx,s] the
partition with [λx,s]j = λ(o(x, s))j + (βx−λ(i(x, s)))j. This can be viewed as adjoining the
partitions (βx − λ(i(x, s)) and λ(o(x, s)) left-to-right.
This is the natural generalization of the notation [λ(i) : λ(i− 1)] in section 2.6.1, so we
expect to build a basis from fillings of these diagrams.
Definition 3.1.9. Let λ ∈ Λ(Q, c, β, r). A multitableau of shape λ is a column-strictly-
increasing filling of each of the diagrams [λx,s] for (x, s) ∈ X. A multitableau is called stan-
dard if each filling of each diagram is a standard filling. The content κ of a filling of λ is the
collection of vectors κx,s ∈ Nβx with (κx,s)j = #{ occurrences of j in the filling of [λx,s]}.
Using the same conventions as in section 2.6.1, we can see that∧λ V has basis given by
multitableaux of shape λ. In the subsequent section, we will determine explicit elements
of∧λ V whose image under δλ is a semi-invariant function.
3.1.2 Application to gentle string algebras
Of particular interest herein are the colorings that give rise to string algebras. It turns out
that the triangular string algebras that admit colorings are precisely the gentle string alge-
34
bras. Furthermore, all triangular gentle string algebras admit a coloring (see the following
proposition).
Proposition 3.1.10. Suppose that kQ/I is a triangular gentle string algebra. Then there
is a coloring c of Q such that Ic = I.
Proof. Let S be a set of arrows a ∈ Q1 with the property that there is no b ∈ Q1 with
h(b) = t(a) and ab ∈ I (since Q has no cycles, so every arrow incident to a sink is in S). To
avoid confusion, let us denote by sa ∈ S the element corresponding to such an a ∈ Q1. For
each element sa ∈ S, let p(a) = pl(a)(a) . . . p1(a) be the longest path with p1(a) = a and
pi+1(a)pi(a) ∈ I. Notice first that the length is bounded since Q is acyclic. Additionally,
this path is unique and well-defined since for each arrow pi(a) there is at most one arrow
pi+1(a) such that pi+1(a)pi(a) ∈ I. Take c : Q1 → S to be the map with c(pi(a)) = sa for
each i = 1, . . . , l(a). By definition of the gentle string algebras, for each b there is at most
one arrow a with ha = tb and ba /∈ I. Therefore, since I is generated by paths of length 2,
so Ic = I.
Now in a gentle string algebra, by definition, there are at most two colors incident to
each color. Therefore, we can interpret the corollary 3.1.7 in the following way.
Corollary 3.1.11. Let kQ/Ic be a gentle string algebra, β a dimension vector, and r a
maximal rank map. If x is a coupled vertex, let (x, s1(x)) and (x, s2(x)) be the two elements
in X with first coordinate x. Then
grΛ(Q,c,β,r),�(k[RepQ,c(β, r)])∼= (3.1.3)⊕
λ∈Λ(Q,c,β,r)
( ⊗x lonely
Sλ(x,s)Vx
)⊗( ⊗x coupled
Sλ(x,s1(x))Vx ⊗ Sλ(x,s2(x))Vx
)(3.1.4)
In particular, at each vertex is the tensor product of at most two Schur modules. This
restriction allows for a combinatorial description of the rings of semi-invariants.
35
3.2 Semi-Invariant Functions in k[RepQ,c(β, r)]
Fix a gentle string algebra kQ/Ic, a dimension vector β, and a maximal rank sequence r
for β. We denote by Mλ the term Fλ/F≺λ for λ ∈ Λ. With this notation, we may write
gr(k[RepQ,c(β, r)])∼=⊕λ∈Λ
Mλ. We are interested in the ring SIQ,c(β, r) :=
k[RepQ,c(β, r)]SLQ(β). In the forthcoming, we will show that SIQ,c(β, r) is isomorphic to a
semigroup ring. We do so by defining a basis {mλ}λ∈Λ′⊂Λ for SIQ,c(β, r) and then exhibiting
the multiplication on said basis.
Definition 3.2.1. Let ΛSI(Q, c, β, r) be the set of elements λ in Λ(Q, c, β, r) such that
Mλ contains a semi-invariant for GLQ(β). As usual, we write ΛSI if the parameters are
understood.
Proposition 3.2.2. Let λ ∈ Λ. Then λ ∈ ΛSI if and only if there is a vector σ(λ) ∈ ZQ0
such that for each x ∈ Q0, we have
λ(x, s1)i + λ(x, s2)βx+1−i = σ(λ)x i = 1, . . . , βx, (3.2.1)
(here if x is a lonely vertex, then the second summand is suppressed, i.e., λ(x, s)i = σ(λ)x
for i = 1, . . . , βx). Furthermore, if λ ∈ ΛSI , then the space of semi-invariants in Mλ is
one-dimensional.
Proof. The decomposition of the tensor product of two Schur modules is given by the
Littlewood-Richardson rule (see [41] proposition 2.3.1). Applying this to equation 3.1.3,
we see that there is an SLQ(β)-invariant (meaning that the Schur module appearing as a
factor at x is a height-βx rectangle for each x) if and only if the system of equations in the
proposition hold.
Corollary 3.2.3. ΛSI is a semigroup under the + operation as defined on partitions in
equation 2.6.1.
36
Proof. Indeed, if σ(λ) and σ(µ) are the vectors in ZQ0 satisfying proposition 3.2.2 for
the sequences λ, µ ∈ Λ, then σ(λ) + σ(µ) is the vector satisfying the proposition for the
sequence λ+ µ.
Remark 3.2.1. Recall the definition of [λx,s] in 3.1.8. We will collect some useful points:
a. this notation allows us to rewrite the domain of the map 3.1.1 in the form
⊗(x,s)∈X
[λx,s]∧Vx
b. Using this notation, we can restate proposition 3.2.2, namely that λ ∈ ΛSI if and
only if there is a vector σ(λ) ∈ ZQ0 such that
i. For each lonely element (x, s) ∈ X (i.e., with no other color passing through x),
[λx,s]′i = βx, i = 1, . . . , σ(λ)x
ii. For each coupled pair (x, s1), (x, s2) ∈ X, [λx,s1 ]′i + [λx,s2 ]′σx−i+1 = βx for i =
1, . . . , σ(λ)x.
This restatement will be useful for defining a map whose image consists of semi-
invariants.
Definition 3.2.4. For λ ∈ ΛSI , define the following maps:
i. If (x, s) ∈ X is a lonely pair, then let
∆λ,xi :
βx∧Vx →
[λx,s]′i∧Vx
be the identity map for i = 1, . . . , σ(λ)x (since, by the above remark, βx = [λx,s]′i);
ii. If there is a coupled pair (x, s1), (x, s2) ∈ X, then take
∆λ,xi :
βx∧Vx →
[λ(x,s1)]′i∧Vx ⊗
[λ(x,s2)]′σ(λ)x−i+1∧
Vx37
to be the diagonalization map (since, by the above remark, the sum of the two powers
is precisely βx).
We collect these maps into the map ∆λ in the following way:
∆λ :=⊗x∈Q0
σ(λ)x⊗i=1
∆λ,xi :
⊗x∈Q0
(βx∧Vx
)σ(λ)x→ ⊗
(x,s)∈X
[λx,s]∧Vx. (3.2.2)
Notice that ∆λ is a GL(β)-equivariant map, since both identity and diagonalization
are such. Fixing a basis for each space Vx, and let e be the corresponding basis element of⊗x∈Q0
(∧βx Vx
)σ(λ)x(note that this space is one-dimensional).
Definition 3.2.5. Denote by
mλ = δλ∆λ(e).
This is unique up to scalar multiple.
Proposition 3.2.6. For λ ∈ ΛSI , the function mλ is a semi-invariant of weight σ(λ).
Furthermore, mλ 6= 0 ∈ Fλ/F≺λ.
The first statement is evident since both δλ and ∆λ are GL(β)-equivariant homomor-
phisms, and the weight is clear from the action on the domain of the map. We delay
the proof of the second statement for a brief description of the straightening relations in
k[RepQ,c(β, r)] relative to fillings of Young diagrams, since the description of mλ is not
given in terms of standard multitableaux. We will come back to this proof when we can
show that there is a standard multitableau of shape λ whose coefficient is non-zero in mλ.
The following is simply a generalization of the material in section 2.6.1. We record these
statements as corollaries to DeConcini and Strickland.
Corollary 3.2.7 ([11]). If tλ is a filling of λ, then
δλ(tλ) = s(tλ) + y(tλ)
38
where y(tλ) ∈ F≺λ and s(tλ) is a linear combination of standard fillings of the same content
as tλ).
Corollary 3.2.8 ([11]). If tλ and tµ are fillings of shape λ, µ, then
δλ(tλ) · δµ(tµ) ∈ Fλ+µ.
proof of proposition 3.2.6. It remains to be shown that mλ 6= 0 in Fλ/F≺λ. For a filling
tλ of λ, let I(tλ)x,s,i be the set of entries in the i-th column of [λx,s]. Notice that ∆λ(e)
is the sum of all fillings tλ of λ satisfying the property that I(tλ)x,s,i ∪ I(tλ)x,s′,σ(λ)x−i+1 =
{1, . . . , βx}, call this property (∗). Pick one distinguished element from each coupled
pair (x, s), (x, s′) ∈ X. Consider the filling t◦λ of λ with I(t◦λ)x,s,i = {1, 2, . . . , [λx,s]′i}
whenever (x, s) is the distinguished element in the coupled pair and I(t◦λ)x,s′,i = {βx, βx −
1, . . . , [λx,s′ ]′i}. This filling satisfies the property (∗) above so it appears with non-zero
coefficient (namely 1) in mλ. Notice that this filling is standard. We will show that the
content of this filling is unique among fillings appearing with non-zero coefficient in ∆λ(e),
so after straightening the other fillings, this distinguished filling cannot be canceled. Indeed,
the content of this filling is κ(t◦λ)x,s = ([λx,s]1, [λx,s]2, . . . ) if (x, s) is the distinguished pair,
and (κ(t◦λ)x,s)βx−j+1 = [λx,s]j otherwise. This content uniquely determines the filling t◦λ, so
indeed δλ(t◦λ) appears with non-zero coefficient in mλ.
Theorem 3.2.9. The ring of semi-invariants SIQ,c(β, r) is isomorphic to the semigroup
ring k[ΛSI(Q, c, β, r)].
Proof. We have already shown that there is a (vector space) homomorphism
m : k[ΛSI(Q, c, β, r)]→ SIQ,c(β, r)
where m(λ) = mλ.
39
Claim 1: m is injective.
Suppose that y = m(∑
λ∈T aλλ) =∑
λ∈T aλmλ = 0 ∈ SIQ,c(β, r), where T is a finite
subset of ΛSI . Let max(T ) be the set of maximal elements in T under the partial order �
defined on Λ. Then y ∈ ∑λ∈max(T )
Fλ Now for each µ ∈ max(T ) there is a surjection
ϕµ :∑
λ∈max(T )
Fλ →Mµ
given by the quotient of this space by the subspace F≺µ +∑
λ∈max(T )\µFλ. Given that y is a
semi-invariant, its image under this map is aµmµ, since the space of semi-invariants in Mλ
is one dimensional. By assumption, this is 0, and since mµ 6= 0, we must have that aµ = 0
for all µ ∈ max(T ), contradicting the choice of max(T ).
Claim 2: The map m is surjective.
This fact exploits the same methods as the previous claim: we show that the maximal
λ appearing in a semi-invariant must be elements of ΛSI , and subtract the corresponding
semi-invariant mλ and are left with a semi-invariant function with smaller terms. Suppose
that y ∈ SIQ,c(β, r), and write y =∑
λ∈T aλxλ where T ⊂ Λ is a finite subset (recall that
k[RepQ,c(β, r)] has a basis given by standard fillings of all λ ∈ Λ, and take xλ to be the
summands corresponding to λ). Let max(T ) again be the maximal elements in T under
the partial order �. Notice that the collection of empty partitions is indeed an element of
ΛSI , so we will proceed by induction on height(T ) defined to be the length of the longest
chain joining both the empty partition and an element of max(T ). For height(T ) = 0, m
is a constant, which is the image of the same constant under the map m. For µ ∈ max(T ),
notice that ϕµ(y) = aµxµ must be a semi-invariant in grλ,�(RepQ,c(β, r)), so µ ∈ ΛSI .
Therefore, for each µ ∈ max(T ), aµxµ = bµmµ. In particular, aµxµ− bµmµ ∈ F≺µ. Now let
y1 = y −∑
µ∈max(T )
bµmµ.
40
By the above remarks, then, y1 =∑
λ∈T1a′λxλ where T1 = {λ ≺ max(T )}. As the difference
of semi-invariants, y1 is itself a semi-invariant, and height(T1) < height(T ). By induction,
then, y1 =∑
λ∈ΛSIbλmλ, so
y =
∑µ∈max(T )
bµmµ
+
(∑λ∈T1
bλmλ
).
Claim 3: m is a semigroup homomorphism.
This is proven directly. It has already been shown that mλ ·mµ ∈ Fλ+µ. Now ∆λ(e) is
a linear combination of all multitableau of shape λ such that I(tλ)x,s,i∪ I(tλ)x,s′,σ(λ)x−i+1 =
{1, . . . , βx}. The coefficient of each multitableau is the sign of the permutation taking
the sequence (I(tλ)x,s,i, I(tλ)x,s′,σ(λ)x−i+1) into increasing order. Now consider ∆λ+µ(e).
We will simply show a bijection between pairs tλ, tµ, summands in ∆λ(e) and ∆µ(e),
respectively, and summands in ∆λ+µ(e), and show that the signs agree. To this end,
consider [(λ+ µ)x,s]. Recall that this is the shape given by adjoining (βx− (λ+ µ)(i(x, s))
and (λ + µ)(o(x, s)). Notice that by definition of (λ + µ)(o(x, s)), we can choose indices
1 ≤ i1 < i2 < . . . < iλ(o(x,s))1 ≤ (λ+ µ)(o(x, s))1 such that
((λ+ µ)(o(x, s))′i1 , (λ+ µ)(o(x, s))′i2 , . . . , (λ+ µ)(o(x, s))′iλ(o(x,s))1)
= (λ(o(x, s))1, . . . , λ(o(x, s))λ(o(x,s))1).
This is easiest to see in a picture:
+ =
In fact, the entire shape [(λ + µ)x,s] can be partitioned into columns in such a way that
41
the gray columns constitute [λx,s] and those in white constitute [µx,s]. Now for each distin-
guished pair (x, s) ∈ X, choose such a partition of the columns, and partition the columns
of the other shapes [λx,s′ ] accordingly, namely if the column i of [(λ+µ)x,s] is colored gray,
then the σ(λ+µ)− i+1 column of [(λ+µ)x,s′ ] is colored gray as well. Fixing this partition
of the columns, we have that a multitableau of shape (λ+µ) gives rise uniquely to a multi-
tableau of shape λ (given by gray columns), and a multitableau of shape µ, and every pair
of multitableau of shapes λ and µ determine a filling of (λ+µ) by the same partitioning of
the columns. So indeed ∆λ+µ(e) consists of a linear combinations of all products of pairs
of multitableau of shapes λ and µ. Furthermore, since the sign is calculated by taking the
product of the signs given by reordering columns, it is evident that the sign of the product
agrees with the sign in ∆λ+µ(e).
3.3 Combinatorics: The Semigroup ΛSI(Q, c, β, r)
In this section, we determine the structure of the semigroup ΛSI . As we have shown above,
SIQ,c(β, r) ∼= k[ΛSI(Q, c, β, r)].
We will exhibit a grading on k[ΛSI ], and show that k[ΛSI ] is a polynomial ring over a
sub-semigroup ring which we denote by k[U ]. For this section, we fix a quiver Q, a coloring
c, a dimension vector β, and a maximal rank sequence r. For ease of presentation we will
write Λ = Λ(Q, c, β, r) and ΛSI similarly.
Definition 3.3.1. Let {αxi | x ∈ Q0 i = 1, . . . , βx − 1} be the simple roots for the group
SLQ(β). I.e., for λ ∈ Λ, αix(λ(x, s)) := λ(x, s)i − λ(x, s)i+1
Proposition 3.3.2. The element λ ∈ ΛSI if and only if both of the following hold:
42
• For every coupled vertex x with (x, s1), (x, s2) ∈ X, and every i = 1, . . . βx − 1,
αxi (λ(x, s1)) = αxβx−i(λ(x, s2));
• For every lonely vertex x, say (x, s) ∈ X,
αxi (λ(x, s)) = 0.
Proof. Indeed, the equality in the proposition holds if and only if
λ(x, s1(x))i − λ(x, s1(x))i+1 = λ(x, s2(x))βx−i − λ(x, s2(x))βx−i+1
⇔ λ(x, s1(x))i + λ(x, s2(x))βx−i = λ(x, s1(x))i+1 + λ(x, s2(x))βx−i+1
⇔ λ(x, s1(x))i + λ(x, s2(x))βx−i = λ(x, s1(x))j + λ(x, s2(x))βx−j := σx
This is precisely the set of conditions given by proposition 3.2.2.
To organize the equations that arise from proposition 3.3.2, we will set up some notation
and define a graph whose vertices are simple roots, with multiplicity.
Definition 3.3.3.
a. Denote by Σ = Σ(Q, c, β) the set of labeled simple roots {α(x,s)i | (x, s) ∈ X, i =
1, . . . , βx−1} (namely the simple roots from above but with multiplicity for the colors
included).
b. For each λ ∈ Λ, define the function fλ : Σ→ N by
fλ(α(x,s)i ) := α
(x,s)i (λ(x, s)).
c. Define the partition equivalence graph, written PEG(Q, c, β, r) to be the graph with
43
vertices given by the set Σ and the following edges:
i. for each coupled vertex x ∈ Q0, with associated pair (x, s1), (x, s2) ∈ X say, and
each i = 1, . . . , βx − 1, define an edge α(x,s1)i α
(x,s2)βx−i .
ii. for each arrow a : x→ y, and each i = 1, . . . , r(a)− 1, define an edge
α(x,s)i α
(y,s)βy−i .
In words, edges of the first type connect labeled simple roots arising from the same
SL(βx), i.e., from the same vertex, and edges of the second type connect simple roots along
colors. For this reason we may call edges of the second type colored edges.
Proposition 3.3.4. Let λ ∈ Λ. Then λ ∈ ΛSI if and only if fλ(α) = fλ(α′) whenever α
and α′ are in the same connected component of the PEG and fλ(α) = 0 if α corresponds
to a root at a lonely vertex.
Proof. Let a ∈ Q1 be an arrow of color s with ta = x and ha = y. Then λ ∈ Λ implies that
fλ
(α
(x,s)i
)= fλ
(α
(y,s)βy−i
), i.e., fλ(α) = fλ(α
′) whenever α, α′ are connected by a colored
edge. This is so because if λ ∈ Λ, then
fλ
(α
(x,s)i
)= λ(x, s)i − λ(x, s)i+1
= λ(a)i − λ(a)i+1
= (−λ(a)i+1)− (−λ(a)i)
= λ(y, s)βy−i − λ(y, s)βy−i+1
= fλ
(α
(y,s)βy−i
).
But proposition 3.3.2 shows that Mλ contains a semi-invariant if and only if fλ(α) = fλ(α′)
whenever α, α′ are linked by an edge of type (i). Therefore, λ ∈ ΛSI if and only if equality
holds for all roots in the same connected component.
44
Proposition 3.3.5. Let K1, . . . , Kl be the list of connected components in PEG(Q, c, β, r),
and let {α(i)}i=1,...,l be some set of elements in Σ such that the vertex corresponding to α(i)
is in the component Ki for each i. For any vector g = (g1, . . . , gl) ∈ Nl, let Vg be the vector
space with basis {mλ | λ ∈ ΛSI , fλα(i) = gi}. Then
k[ΛSI ] =⊕g∈Nl
Vg
is a graded direct sum decomposition of the semigroup ring k[ΛSI ]. In other words, k[ΛSI ]
has a multigrading by the connected components of PEG(Q, c, β, r).
This follows immediately from the description of the semigroup structure of ΛSI above
and proposition 3.3.4.
Definition 3.3.6. Let E = EQ,c(β, r) be the set of elements in Σ whose corresponding
vertices are endpoints for the PEG associated to (Q, c, β, r). For an element e ∈ E which
is contained in the string, write Θ(e) for the distinct second endpoint contained in this
string (we do not consider an isolated vertex to be a string). Clearly Θ : E → E is an
involution.
In fact, we can explicitly describe E.
Proposition 3.3.7. Each endpoint of the PEG is of one of the following two mutually
exclusive forms:
I. if x is coupled and (x, s) ∈ X, then α(x,s)i is an endpoint for r(o(x, s)) ≤ i ≤ βx −
r(i(x, s));
II. if x is lonely and (x, s) ∈ X, then α(x,s)i is an endpoint for 1 ≤ i ≤ βx.
Proof. This is a consequence of the definition 3.3.3. We will call the edges that connect
roots on the same vertex of the quiver non-colored, and those that connect roots on different
45
vertices of the quiver colored. If x is lonely then there can only possibly be colored edges
containing any of the elements α(x,s)i , and by definition, each vertex can be contained in at
most one such. If, however, x is coupled and (x, s) ∈ X, then each vertex α(x,s)i is incident
to precisely one non-colored edge. Those with i < r(o(x, s)) or i > βx − r(i(x, s)) are also
incident to a colored edge by definition. For r(o(x, s)) ≤ i ≤ βx − r(i(x, s)), there are no
colored edges incident to α(x,s)i .
We will use the endpoints of the strings to find a system of equations so that each
positive integer-valued solution of the system will correspond to an element λ ∈ ΛSI .
Remark 3.3.1. Below lists the endpoints in {αix,s}i=1,...,βx and calculates the values of fλ
on such endpoints. In order to write the system of equations mentioned above in a compact
form, we also label these possibilities:
a. If r(o(x, s))+r(i(x, s)) = βx for some (x, s) ∈ X, then α(x,s)r(o(x,s)) is the unique endpoint
in this set. For this endpoint, we have
fλ
(α
(x,s)r(o(x,s))
)= λ(o(x, s))r(o(x,s)) + λ(i(x, s))r(i(x,s)).
We will denote this endpoint by (o(x, s), i(x, s)).
b. If r(o(x, s)) + r(i(x, s)) < βx for some (x, s) ∈ X, then α(x,s)r(o(x,s)) is an endpoint, and
fλ
(α
(x,s)r(o(x,s))
)= λ(o(x, s))r(o(x,s)).
We will denote this endpoint by the arrow o(x, s).
c. If r(o(x, s)) + r(i(x, s)) < βx for some (x, s) ∈ X, then α(x,s)βx−r(i(x,s)) is an endpoint,
and
fλ
(α
(x,s)βx−r(i(x,s))
)= λ(i(x, s))r(i(x,s)).
46
Such an endpoint will be denoted by the arrow i(x, s).
d. Finally, if r(o(x, s)) < i < βx − r(i(x, s)), or (x, s) has no mirror and i 6= r(o(x, s)),
i 6= βx − r(i(x, s)), then
fλ
(α
(x,s)i
)= 0.
Such endpoints will be denoted by the symbol 0(x,s)i .
Thus, an endpoint can be of type Ia, Ib, Ic, Id, or type IIa, IIb, IIc, IId.
Definition 3.3.8. For any λ ∈ Λ, define uλ : Q1 → N to be the function uλ(a) = λ(a)r(a).
For any function u : Q1 → N, let ϕu : E → N be the function defined as follows:
ϕu(e) =
u(i(x, s)) + u(o(x, s)) if e is of type (Ia) and labeled (o(x, s), i(x, s))
u(o(x, s)) if e is of type (Ib) and labeled o(x, s)
u(i(x, s)) if e is of type (Ic) and labeled i(x, s)
0 if e is of type (Id) or (II).
We call ϕu the companion function to u.
We will denote by U = U(Q, c, β, r) the set of functions u : Q1 → N such that ϕu(e) =
ϕu(Θ(e)) for all e ∈ E. Notice that U is a semigroup with respect to the usual addition of
functions.
Proposition 3.3.9. If λ ∈ ΛSI then uλ ∈ U(Q,C, β, r).
Proof. This is clear from proposition 3.3.4, together with the fact that if λ ∈ ΛSI , and x
is a lonely vertex, (x, s) ∈ X, then fλ(α(x,s)i ) = 0 for i = 1, . . . , βx.
Notice that from uλ one can calculate the values of fλ(α) whenever α ∈ Σ is in a string
of the PEG.
47
Definition 3.3.10. Denote by Y = Y (Q, c, β, r) the set of maps y : {bands in Σ} → N.
For any u ∈ U and y ∈ Y , take λu,y : Q1 → P to be the map defined by the following
conditions:
λu,y(a)r(a) = u(a)
α(λu,y) =
ϕu(e) if e is an endpoint of the string containing α
y(b) if α is contained in the band b.
Remark 3.3.2. Let us summarize the results above:
i. The set U is a semigroup with respect to the usual addition of functions,
ii. λu,y(a) has at most r(a) non-zero parts, so λu,y ∈ Λ,
iii. image((u, y) 7→ λu,y) ⊂ ΛSI .
Proposition 3.3.11. The map (u, y) 7→ λu,y is a semigroup isomorphism between U × Y
and ΛSI .
Proof. We construct an inverse explicitly. For any λ ∈ ΛSI , define (uλ, yλ) as follows:
uλ(a) := λ(a)r(a)
yλ(b) := fλ(α) for any α in the band b.
It is routine that uλ(u,y) = u and yλ(u,y) = y, so this is indeed a bijection, and it is clear
that the composition operation in U × Y is preserved under this map.
Corollary 3.3.12. We have the following ring isomorphism
SIQ,c(β, r) ∼= k[U(Q, c, β, r)][yb]b∈{bands in Σ},
48
that is, SIQ,c(β, r) is a polynomial ring over the semigroup ring k[U ].
Proposition 3.3.13. The semigroup U(Q,C, β, r) is a sub-semigroup of NQ1, satisfying
the following:
a. U(Q,C, β, r) = {(ua)a∈Q1 ∈ NQ1 | ϕu(e) = ϕu(Θ(e)) for e ∈ E},
b. ϕu(e) =∑a∈Q1
ceaua with cea ∈ {0, 1} for each endpoint e ∈ E,
c. ua appears with nonzero coefficient in at most two functions ϕu. I.e., for each a ∈ Q1,
there are at most two endpoints e1, e2 ∈ E with ce1a = ce2a = 1.
Proof. (a) is simply the definition of U(Q,C, β, r), reformulated as a sub-semigroup of NQ1 ,
while (b) is the definition of the control equations. Recall that ϕu(e) = u(a) + u(b), u(a)
or 0 for any endpoint e, and since the quiver is acyclic, a 6= b, so the coefficient on any
summand is at most 1. To show (c), we recall that r is a maximal rank sequence for β. This
implies that if e1 is an endpoint of type (Ia) labeled (a, b) (in which case ϕu(e1) = ub+ua),
then the only other type of endpoint labeled with an a is either another of type (Ia) labeled
(c, a), or one of type (Ic) labeled a. (Similarly the only other type of endpoint labeled with
a b is either another of type (Ia) labeled (b, c), or of type (Ib) labeled b.)
3.4 Matching Semigroups
Fix kQ/Ic a gentle string algebra β a dimension vector vector and rank sequence, together
with its PEG(Q, c, β, r), Σ. We will define a general class of sub-semigroups of Nl, of which
all U(Q, c, β, r) are members. We will then describe a general procedure for determining
generators and relations for these semigroup rings by means of a graph, and show that the
generators of these semigroups occur in multidegree at most 2. First, however, we exhibit
some structure enjoyed by k[U ].
49
Theorem 3.4.1. The semigroup ring k[U ] is the coordinate ring of an affine toric variety.
Proof. Let k[Xa]a∈Q1 be the polynomial ring on the arrows of Q1, and let E be the set of
strings in Σ. Suppose that the PEG has the following endpoints: {e(s)1 , e
(s)2 }s∈E. Then we
define the action of (k∗)E on k[Xa]a∈Q1 as follows: suppose that (ts)s∈E ∈ (k∗)E, then
(ts).∏a∈Q1
Xu(a)a := tϕu(e
(s)1 )−ϕu(e
(s)2 )
s
∏a∈Q1
Xu(a)a .
A polynomial p ∈ k[Xa]a∈Q1 is invariant with respect to this action if and only if its
monomial terms are, so it suffices to assume p is a monomial. Suppose that a monomial∏a∈Q1
Xu(a)a is invariant with respect to each ts. Then for each endpoint pair {e(s)
1 , e(s)2 }, we
have
ts.∏a∈Q1
Xu(a)a = tϕu(e
(s)1 )−ϕu(e
(s)1 )
s
∏a∈Q1
Xu(a)a =
∏a∈Q1
Xu(a)a ,
so ϕu(e(s)1 ) = ϕu(e
(s)2 ) for s ∈ E. Therefore, such a monomial is invariant with respect to
the action if and only if u ∈ U . Then clearly k[U ] = k[Xa](k∗)E is the invariant ring with
respect to this torus action.
Definition 3.4.2. Let {fi : Nl → N}i=1,...,2m be a collection of N-linear functions
fi(x1, . . . , xl) =l∑
j=1
cjixj
satisfying the following properties:
a. cji ∈ {0, 1} for all i = 1, . . . , 2m, j = 1, . . . , l;
b. cji 6= cji+m for i = 1, . . . ,m, j = 1, . . . , l (i.e., the equations
fi(x1, . . . , xl) = fi+m(x1, . . . , xl) are reduced);
c. for j = 1, . . . , l, #{i | cji 6= 0, i = 1, . . . , 2m} ≤ 2 (i.e., each variable xj appears with
non-zero coefficient in at most two functions fi).
50
The semigroup
U({fi}i=1,...,2m) := {u = (u1, . . . , ul) ∈ Nl | fi(u) = fm+i(u), i = 1, . . . ,m}
is called a matching semigroup if the functions fi satisfy the conditions (a)-(c).
The following is the main theorem of this section.
Theorem 3.4.3. Suppose that U = U(f) ⊂ Nl is a matching semigroup with f =
{fi}i=1,...,2m. Then U is generated by vectors u = (u1, . . . , ul) with the property that
fi(u) ≤ 2 for i = 1, . . . , 2m. In particular, ui ≤ 2.
In order to prove this theorem, we construct a graph G(f) and interpret certain walks
on this graph as elements in U .
Definition 3.4.4. Let G(f) be the multigraph with two types of edges, solid and dotted,
on the vertices {1, . . . , 2m}, with a solid edge
i k whenever cji = cjk = 1, i 6= k,
a solid loop
i i whenever i is the unique integer for which cji = 1,
and dotted edges i m+ i for i = 1, . . . ,m. We define a function L : Edges(G(f))→
{1, x1, . . . , xl} with
L(E) =
1 if E is a dotted edge
xj if E is the edge containing i, k arising from the condition cji = cjk = 1.
51
In depicting this graph, we will indicate the labeling as a decoration on the appropriate
edge. Heuristically, each vertex i stands for a function fi. A vertex i is contained in a solid
edge labeled xj if xj appears with non-zero coefficient in fi, and the vertices corresponding
to functions on either side of a defining equation of U(f) are joined by a dotted edge. The
name matching semigroup arises from the fact that the dotted edges form a perfect
matching for the graph G(f). Moreover, while each vertex is contained in exactly one
dotted edge, it can be contained in several solid edges: as many as non-zero coefficients in
the linear function to which it corresponds.
A walk on G(f) is a sequence of vertices and edges w = vnEnvn−1En−1 . . . E1v0 such
that V (Ei) = {vi, vi−1} (i.e., the vertices of Ei are precisely the two surrounding it in the
sequence). To each such walk, associate an integer vector u(w) ∈ Nl with
u(w)j = #{k | the edge Ek is labeled xj}.
A walk will be called alternating if Ek, Ek−1 are of different edge types for k ∈ [n]. Such a
walk will be called a string if both E1 and En are loops, and a band if v0 = vn, E0, En are
different edge types, and none of the Ei are loops. Henceforth, we will refer to “alternating”
strings and bands simply as strings and bands.
Lemma 3.4.1. Suppose that w is a string or band. Then u(w) ∈ U .
Proof. Without loss of generality, assume i ≤ m. Notice that if w is a string, then
fi(u(w)) = #{j ∈ {1, . . . , n− 1} | vj is the vertex i},
while if w is a band, then
fi(u(w)) = #{j ∈ {1, . . . , n} | vj is the vertex i}.
52
But w is alternating, so every occurrence of the vertex i is either immediately preceded or
succeeded by an occurrence of the vertex i+m, so
fi(u(w)) = #{j | vj is the vertex i+m} = fi+m(u(w))
as claimed.
Lemma 3.4.2. G(f) contains no alternating two-cycles.
Proof. If the edge labeled x1 contains two vertices i, i + m which are both contained in a
single dotted edge, then fi(x) = x1 +∑cjixj = x1 +
∑cji+mxj = fi+m(x), contradicting
definition 3.4.2 (b).
Lemma 3.4.3. A matching semigroup U = U(f) is generated by the set
{u(w) | w is either a string or a band on G(f)}.
Proof. Let ≤ be the coordinate-wise partial order on U . We will show that for each
0 6= u ∈ U , there is a non-trivial alternating walk w, which is either a string or a band,
and an element u′ ∈ U such that
i. u′ ≤ u,
ii. u = u(w) + u′.
Case 1: Suppose that uj1 6= 0 for some j1 for which xj1 is a loop. We inductively construct
a sequence of alternating walks tk = v2kE2kv2k−1 . . . v1E1v0 with L(E1) = xj1 satisfying the
following:
(1) 0 < u(tk) < u(tk+1) < u
(2) fv2k−1(u− u(tk)) + 1 = fv2k
(u− u(tk))
53
(3) fi(u− u(tk)) = fi+m(u− u(tk)) whenever {i, i+m} 6= {v2k, v2k−1}.
Let E1 be the edge with L(E1) = xj1 , v0 = v1 the unique vertex contained in this loop, E2
the dotted edge containing v1, and v2 the unique second vertex contained in E2.
Claim 1: t1 satisfies (1)-(3).
Proof. u(t1)j1 = 1, so immediately u(t1) > 0. Furthermore, u(t1)j′ = 0 for j′ 6= j1, and
since cj1v1= 1, cj1v2
= 0, fv2(u(t1)) = 0. On the other hand, fv2(u) = fv1(u) > 0 by
assumption, so u(t1) < u, and (1) is proven.
As for (2) and (3), fv1(u − u(t1)) = fv1(u) − 1 = fv2(u) − 1 = fv2(u − u(t1)) − 1
since u ∈ U . Furthermore, if {i, i + m} 6= {v2, v1}, then cji = cji+m = 0 since fv1 is the
unique function in which xj appears with non-zero coefficient (as E1 is a loop). Therefore,
fi(u− u(t1)) = fi(u) = fi+m(u) = fi+m(u− u(t1)), proving (3).
Claim 2: If tk = v2kE2kv2k−1 . . . v1E1v0 satisfies (1)-(3), and there is no Es for s = 2, . . . , 2k
with Es a loop, then there are two possibilities:
a. There is a loop E2k+1 containing the vertex v2k such that the walk w := v2kE2k+1tk
is an alternating string and u(w) ≤ u;
b. There is a solid edge E2k+1 which is not a loop such that tk+1 =
v2k+2E2k+2v2k+1E2k+1tk is an alternating walk satisfying (1)-(3).
Before proving this dichotomy, we note that this proves the following: if u ∈ U such that
uj 6= 0 with xj a loop, then there is a an alternating string such that u−u(w) ∈ U . Indeed,
u(tk) < u(tk+1) < u by (1), so there must be a tk such that u(tk) < u and for which there
is a loop E2k+1 such that w as defined in (a) is an alternating string and u(w) ≤ u.
Proof. Suppose that tk = v2kE2kv2k−1 . . . v1E1v0 contains no loops other than E1, satisfies
54
(1)-(3), and does not satisfy (a). By property (2),
fv2k−1(u− u(tk)) + 1 = fv2k
(u− u(tk)) =∑
j|cjv2k 6=0
uj − u(tk)j.
Since fv2k−1(u − u(tk)) ≥ 0, there must be a jk such that ujk > u(tk)jk and cjkv2k
= 1. In
terms of the graph, then, there is a solid edge E2k+1 (which is not a loop since tk does not
satisfy (a)) with L(E2k+1) = xjk containing the vertex v2k. Let v2k+1 be the distinct second
vertex contained in E2k+1, E2k+2 the unique dotted edge containing v2k+1, and v2k+2 the
distinct second vertex contained in E2k+2. Let tk+1 = v2k+2E2k+2v2k+1E2k+1tk. We claim
that tk+1 satisfies (1)-(3).
Notice that u(tk)jk+1 = u(tk+1) (as tk+1 has an additional occurrence of the edge labeled
xjk), and u(tk)j′ = u(tk+1)j′ for j′ 6= jk. Therefore, 0 < u(tk) < u(tk+1) and u(tk+1)j′ ≤ uj′
for j′ 6= jk. Furthermore, ujk > u(tk)jk from above, so ujk ≥ u(tk)jk + 1 = u(tk+1)jk , so
u(tk+1) ≤ u. We will show in the course of proving (2) that u(tk+1) /∈ U , implying we
cannot have equality, so u(tk+1) < u as claimed.
Recall that since v2k+1 contains the edge labeled xjk , cjkv2k+1
= 1. By lemma 3.4.2, then,
cjkv2k+2= 0. Furthermore, fv2k+1
(u − u(tk)) = fv2k+2(u − u(tk)) since tk satisfies condition
(3). Therefore
fv2k+1(u− u(tk+1)) = fv2k+1
(u− u(tk))− 1
= fv2k+2(u− u(tk))− 1
= fv2k+2(u− u(tk+1))− 1
proving (2).
55
Finally, if {i, i+m} = {v2k−1, v2k}, then
fv2k−1(u− u(tk+1)) = fv2k−1
(u− u(tk))
= fv2k(u− u(tk))− 1
= fv2k(u− u(tk+1)),
while if {i, i + m} 6⊂ {v2k−1, v2k, v2k+1, v2k+2}, then fi(u − u(tk+1)) = fi(u − u(tk)) =
fi+m(u− u(tk)) = fi+m(u− u(tk+1)), proving (3).
Case 2: Now suppose for all j such that xj is a loop, we have that uj = 0. Take j1 with
uj1 6= 0 (possible since u 6= 0). Let v0, v1 be the vertices (taken in some order) contained in
the edge labeled xj, E1 this edge, E2 the dotted edge containing v1 and v2 the other end of
this edge. Call this walk t1. Notice that v2 6= v0 by lemma 3.4.2. We can again recursively
define alternating walks tk starting with t1 satisfying the following: if v0 6= v2k, then
(1) 0 < u(tk) < u(tk+1) ≤ u
(2) fv2k−1(u− u(tk)) + 1 = fv2k
(u− u(tk))
(3) fi(u− u(tk)) = fi+m(u− u(tk)) whenever {i, i+m} 6⊂ {v2k, v2k−1, v0}.
(4) tk can be extended to an alternating walk tk+1 which is either an alternating band
with u(tk+1) ≤ u or tk satisfies (1)-(3).
Thus, completely analogously to Case 1, there must be a tk that is a band. As the proof
is nearly verbatim of the proof of Case 1, we omit it. Therefore, u =∑u(wi) for wi some
strings or bands.
Notice that it is possible that fi = 0 for some index i ≤ m (say), while fi+m =∑cji+mxj
with some cji+m 6= 0 for some j. It may not be clear why if w is alternating string or band,
then u(w)j = 0, which would be required if u(w) ∈ U . However, if fi = 0, then there are
56
no solid edges containing the vertex i. Any alternating path passing through the solid edge
labeled xj would then pass through the dotted edge between i + m and i. Since the walk
couldn’t finish at that vertex, it would immediately pass back through the dotted edge,
contradicting the alternating property of the walk.
Definition 3.4.5. A string or band w is called irreducible if there does not exist a pair of
non-trivial strings or bands w′, w′′ satisfying u(w) = u(w′) + u(w′′).
Clearly U is generated by {u(w) | w is an irreducible alternating string or band}.
Lemma 3.4.4. If w is an irreducible string or band, then fi(u(w)) ≤ 2 for i = 1, . . . , 2m.
Proof. Suppose that w = vnEn . . . E1v0 is an irreducible string or band, and fi(u(w)) ≥ 3
for some i = 1, . . . ,m (in particular, fi+m(u(w)) ≥ 3). This implies that the vertex i
appears in the set {v1, . . . , vn−1} at least thrice. Let E be the dotted edge containing the
vertices i and i+m. Recall that in an alternating path, each occurrence of the vertex i is
immediately succeeded or immediately preceded by an occurrence of i+m. Let 1 ≤ k1 <
k2 < k3 ≤ n− 1 be the first three integers such that vkj = i, and 1 ≤ l1 < l2 < l3 ≤ n− 1
the first three such that vlj = i + m. Suppose without loss of generality that k1 < l1. We
claim that if k2 < l2 or l3 < k3, then w is not irreducible. In this case, k2 < l2 implies that
w contains a sub-band, namely
w = . . . vl2E(vk2Ek2 . . . vl1Evk1) . . .
In a diagram (although the graph is undirected, the sequence of edges and vertices of the
57
walk will be indicated with arrows):
vk2 vl1El1Ek2
(Here the thinner arc connecting the two bottom vertices represents an alternating walk
that starts and ends with dotted edges.) This contradicts the assumption of irreducibility,
so k2 > l2, and the same contradiction implies that k3 < l2, so we have that k1 < l1 < l2 <
k2 < k3 < l3. But now we have that
w = . . . vl3(Evk3Ek3 . . . Ek2+1vk2Evl2El2 . . . El1+1vl1)Evk1 . . .
which contains the parenthesized band. In diagram form:
E
El1+1
El2Ek2+1
Ek3
again contradicting irreducibility of w.
58
proof of theorem 3.4.3
U(f) is generated by the u(w) for w irreducible strings and bands, and for such walks,
fi(u(w)) ≤ 2 for i = 1, . . . , 2m by lemma 3.4.4. This concludes the proof.
�
The presentation of U(f) using walks on a graph allows us to determine the relations
in the ring k[U(f)] as well. Let W (f) be the free semigroup generated by the irreducible
paths wi on G(f), and extend the function u to W (f) linearly. Let ∼W be the kernel
equivalence of this map, i.e., A ∼W B if and only if u(A) = u(B). The relation ∼W is a
semigroup congruence, so W (f)/ ∼W is a semigroup isomorphic to U(f), and k[U(f)] is
isomorphic to k[W (f)]/IW where IW is generated by all elements tw − tw′ for w ∼W w′.
Remark 3.4.5. Notice that since ∼W is a semigroup congruence, one has cancellation.
That is a + b ∼W a + c if and only if b ∼W c. This can be recognized immediately from
the definition of ∼W .
Definition 3.4.6.
P1
P2 Q2
Q1
E
(a) X-Configuration about E
E
E′
X1X2P1 P2
(b) H-Configuration about E,E′
Figure 3.1: Relations in Graphical Form
59
• A walk P is called a partial string if its first edge is a loop and its last edge is solid;
• Suppose that P1, Q1 are partial strings as in the configuration of figure 3.1a. We will
often abbreviate by Q1P1 the alternating string obtained by joining Q1 and P1 by
the edge E.
• Suppose that P1, X1 are alternating walks as in figure 3.1b. Then we write X1P1 for
the alternating band obtained by joining P1 and X1 along the edges E and E ′.
• Let ∼X be the minimal semigroup equivalence containing the relations:
i. Q1P1 +Q2P2 ∼ Q2P1 +Q1P2 for every collection P1, P2, Q1, Q2 of partial strings
in an X-configuration (figure 3.1a) on G(f);
ii. X1P1+X2P2 ∼ X1X2+P1P2 for every collection of alternating walksX1, X2, P1, P2,
none containing loops, in an H-configuration (figure 3.1b) on G(f).
Remark 3.4.6. Notice that for a given pair P,Q of partial strings as in figure 3.1a (or a
pair of alternating walks X1, P1 as in figure 3.1b), QP (resp. X1P1) may not be irreducible
even while Q,P (resp. X1, P1) contain no sub-bands.
Proposition 3.4.7. The equivalence relations ∼W and ∼X coincide.
Proof. Notice that if two elements are equivalent under ∼X , then they are equivalent under
∼W , as can be seen on the relations that generate the semigroup.
The converse is proven by induction. Suppose that A ∼W B for some A,B ∈ W (f).
We will show that A ∼X B. Notice that the function u : W (f) → U(f) induces a partial
order on W (f) via A′ � A if and only if u(A′) ≤ u(A). Notice that for any A, the set
{0 � A′ � A} is finite, so we can induct on u(A).
For u(A) = 0, the proposition is clear: u(A) = 0 implies u(B) = 0, so A = B = 0,
which are trivially equivalent under ∼X . Now suppose that the implication holds for all
60
A′ ≺ A. We can assume, without loss of generality, that a0 6= 0 while b0 = 0, since
otherwise cancellation would allow us to express the equivalence under ∼W for A′ ≺ A,
which, by induction, would imply equivalence under ∼X . We state the following lemma
and delay the proof in order to show that the proposition follows from it.
Lemma 3.4.7. With all of the above assumptions, B ∼X w0 +B′ for some B′ ∈ W (f).
Assuming that the claim holds, then by the first paragraph of the proof, B ∼W w0 +B′.
By transitivity, then A = w0 +A′ ∼W w0 +B′. But ∼W is a semigroup congruence, so the
aforementioned equivalence holds if and only if A′ ∼W B′. By inductive hypothesis, then,
A′ ∼X B′. Therefore, A = w0 + A′ ∼X w0 +B′ ∼X B as desired.
proof of lemma 3.4.7. For two strings w,w′, choose a longest partial string common to
both w,w′, and denote it by (w||w′). (This may not be unique, but we simply choose one
such for each pair of strings.) Let l(w||w′) be the length of this partial string (notice that
l(w||w′) is odd since the first and last edges are solid and the walk is alternating).
Case 1: Suppose that w0 is a string. Let j be an index such that u(w0)j > 0 and xj is
a loop. Since u(A)j > 0 and A ∼W B, we must have that u(B)j > 0, so there exists a
string wi1 such that u(wi1)j > 0, bi1 6= 0, and such that l(w0||wi1) is maximal. We show the
following: if wi1 6= w0, then B ∼X Φ(B) in such a way that there is a walk wi2 appearing
with non-zero coefficient in Φ(B) such that l(w0||wi2) > l(w0||wi1). Since the length of
w0 if finite, there must be an N > 0 such that w0 appears with non-zero coefficient in
ΦN(B). Since equivalence under ∼X implies equivalence under ∼W , then, we have that
A ∼W ΦN(B), so A ∼W w0 +B′ for some B′, as desired.
Let v be the last vertex in (w0||wi1), E the dotted edge containing said vertex, v′ the
other vertex contained in E, and Q the partial string such that Q(w0||wi1) = wi1 . This is
demonstrated in the diagram below, where the walk w0 is depicted in black, and wi1 is in
61
gray:
E
xj1
(w0||wi1 )
Q
Now xj1 appears in w0, so u(B)j1 = u(A)j1 > 0, implying that there is a walk wl1 with
non-zero coefficient appearing in B with u(wl1) 6= 0. There are three subcases:
(A) wj1 is the (unique) walk appearing in B with this property, then xj1 is an edge in Q;
(B) wl1 is not wj1 , and is an alternating string;
(C) wl1 is an alternating band.
Subcase A: This case impossible, for suppose that wj1 indeed contains xj1 . Said edge
cannot be the first solid edge in Q, or else xj1E(w0||wi1) would be a partial string common
to both w0 and wi1 with length one greater than (w0||wi1), contradicting the definition.
Otherwise, wi1 takes one of the following two forms:
wi1 = . . . Exj1 . . . E(w0||wi1)
wi1 = . . . xj1ECE(w0||wi1),
where C is an alternating walk starting with the vertex v′ and ending with v. In the former
case, the walk wi1 could be written in the form . . . E . . . xj1E(w0||wi1). But xj1E(w0||wi1)
has greater length than (w0||wi1). Contradiction. Finally, in the latter case, wi1 is not
an irreducible walk since EC is a band, so wi1 = . . . xj1E(w0||wi1) + EC, and the first
62
summand is an alternating string with l(w0|| . . . xj1E(w0||wi1)) > l(w0||wi1), contradicting
the choice of wi1 .
Subcase B: Now we have wl1 an alternating string containing the edge xj1 . Let Q′ be
the partial string in wl1 containing xj1 and not E, and P ′ the partial string such that
Q′P ′ = wl1 . This is depicted in the diagram below:
E
xj1
(w0||wi1 )
Q
Q′P ′
I.e., Q′P ′ + Q(w0||wi1) appears in B. Notice that this is an X-configuration about E,
so Q′P ′ + Q(w0||wi1) ∼X Q′(w0||wi1) + QP ′. Take Φ(B) = B − (Q′P ′ + Q(w0||wi1)) +
(Q′(w0||wi1)+QP ′). Then Φ(B) ∼X B and Φ(B) contains a summand, namely Q′(w0||wi1),
with l(w0||(Q′(w0||wi1)) > l(w0||wi1) as claimed.
63
Subcase C: Finally, if wl1 = PExj1 is a band, then we are in the following situation:
Q
E
xj1
(w0||wi1 )
P
In this case, we can define wi2 = QEPxj1E(w0||wi1) (caution: this walk is not irreducible).
Then l(w0||wi2) > l(w0||wi1), as desired.
Case 2: Now suppose that w0 is an alternating band. Notice that we can assume (by
symmetry) that there are no strings appearing as summands in B. Again, for some band
w we will denote by (w0||w) any of the longest alternating paths contained in both w0 and
w. Let y1 be some solid edge contained in w0. Since u(w0)y1 6= 0, there must be a band wi1
appearing in B passing through this edge. This is depicted below, again the black edges
64
form the band w0 and the gray edges are from wi1 .
y1
y2
Fix an orientation on w0, and suppose that y2 is the first edge in w0 (in the chosen orien-
tation) which is not contained in wi1 as in the diagram. But u(B)y2 6= 0, so there must be
a band wl1 containing this edge. By the same reasoning as the proof of case A for strings,
if this band were wi1 (i.e., if wi1 contained y2), then wi1 could be rewritten so as to contain
a longer common subpath with w0. Therefore, this path is distinct from wi1 . There are
two cases:
Subcase A: wl1 contains all other edges in w0 as in the diagram including that labeled y2:
y1
y2
65
then wi1 and wl1 are in an H-configuration.
P2
X2
E
E′
P1
X1
since wi1 = EX1E′P1, and wi2 = EX2E
′P2. Therefore
wi1 + wi2 = EX1E′P1 + EX2E
′P2
= EX1E′P2 + EX2E
′P1
= w0 + EX2E′P1.
As such, B ∼X w0 +B′ with u(B′) = u(B)− u(w0) < u(B).
Subcase B: wl1 does not contain all other edges in w0:
y1
y2
y1
y2X
Let X be the subpath common to both wi1 and wi2 as above, P1 and P2 the paths such
that wi1 = P1X and wi2 = P2X, respectively. Then wi1 +wi2 = XP1XP2 is an alternating
66
band (although clearly not irreducible). Furthermore, l(w0||XP1XP2) > l(w0||wi1). Since
the length of w0 is finite, iteration of this will introduce an H-configuration as in case A
within l(w0) steps.
3.5 Degree Bounds
It is a simple consequence of section 2.6.1 that for λ ∈ ΛSI(Q, c, β, r), the function mλ is
of degree ∑a∈Q1
|λ(a)|
under the usual grading on the polynomial ring. We will use this and the map (u, y) 7→ λu,y
to give degree bounds on the generators and relations for SIQ,c(β, r). Recall that there is a
second grading on SIQ,c(β, r), as in proposition 3.3.5, given by the connected components
of the partition equivalence graph. The first corollary relates to this grading.
Corollary 3.5.1. The generators for SIQ,C(β, r) occur in multi-degrees bounded by ϕuλ(e) ≤
2 and yλ(e) ≤ 1.
As for degree bounds in the polynomial ring, we have the following result:
Corollary 3.5.2. The generators for SIQ,c(β, r) occur in total degrees bounded by
2∑a∈Q0
(r(a) + 1
2
).
67
Proof. Since λ(a)r(a) ≤ 2, and λ(a)i+1 ≤ λ(a)i ≤ λ(a)i+1 + 2, we have
deg(mλ) =∑a∈Q1
|λ(a)| ≤∑a∈Q1
r(a)∑i=1
2i
= 2∑a∈Q1
(r(a) + 1
2
).
Corollary 3.5.3. The relations for SIQ,c(β, r) occur in total degrees bounded by
8∑a∈Q1
(r(a) + 1
2
).
Proof. We may assume that in an X-relation, none of the arms contains a subband, so by
theorem 3.4.3, we have that for each arm u(a) ≤ 2 and ϕu(e) ≤ 2 for any e. Therefore, on
P1P2 ·Q1Q2, the bounds become u(a) ≤ 8 and ϕu(e) ≤ 8. The bound is derived similarly
to the previous corollary. The same technique works for H-relations as well, so the bound
is as desired.
68
Chapter 4
Generic Modules
We call the decomposition β = β(1) + . . . + β(s) the canonical or generic decompo-
sition of β (with respect to a fixed irreducible component Z of RepQ,I(β)) if the generic
representation in Z can be written as a direct sum V (1)⊕ . . .⊕ V (s) of indecomposables
such that V (i) has dimension β(i). Kac ([23] 2.24) points out that such a decomposition
always exists, although in many the explicit description is unknown. In this chapter, we
describe the generic modules in representation spaces for gentle string algebras.
4.1 The Up-and-Down Graph
In this section, we construct a graph for each irreducible component of RepQ,c(β) when
(Q, c) is a gentle string algebra. In section 4.2 we will construct a module from each such
graph.
Denote by X ⊂ Q0 × S the set of pairs (x, s) such that there is an arrow a of color s
incident to the vertex x. We define a sign function, which will dictate how the graph is
constructed.
Definition 4.1.1. A sign function on (Q, c) is a map ε : X→ {±1} such that if (x, s1), (x, s2)
69
are distinct elements in X, then ε(x, s1) = −ε(x, s2).
The following lemma is not used in the remainder of the article, but is recorded here
for completeness.
Lemma 4.1.1. If there are no isolated vertices in Q, then there are 2|Q0| sign functions
on (Q, c).
Proof. Let E be the set of all sign functions on (Q, c). We will define a bijection between
this space and {±1}Q0 . Namely, for each x ∈ Q0, select a color sx ∈ C such that (x, sx) ∈ X.
If ε is a sign function, denote by ε ∈ {±1}Q0 the vector with εx = ε(x, sx). For ε ∈ {±1}Q0 ,
let ε : X→ {±1} be the extension of the map ε by
ε(x, s) =
ε(x, sx) if s = sx
−ε(x, sx) otherwise
These maps are mutual inverses, so indeed |E| = |{±1}Q0| = 2|Q0|.
Definition 4.1.2. Fix a quiver Q with coloring c, a dimension vector β, and a maximal
rank map r. For any sign function ε on (Q, c), denote by ΓQ,c(β, r, ε) the graph with vertices
{vxi | x ∈ Q0 i = 1, . . . , βx} and edges as follows (see figure 4.1 for a visual depiction):
for each arrow a ∈ Q1 and each i = 1, . . . , r(a)
a. vtai vhai if ε(ta, c(a)) = 1, ε(ha, c(a)) = −1,
b. vtai vhaβha−i+1 if ε(ta, c(a)) = ε(ha, c(a)) = 1,
c. vtaβta−i+1 vhai if ε(ta, c(a)) = ε(ha, c(a)) = −1
d. vtaβta−i+1 vhaβha−i+1 if ε(ta, c(a)) = −1, ε(ha, c(a)) = 1.
We will call the graph ΓQ,c(β, r, ε) an up-and-down graph.
70
Such a graph comes equipped with a map w : Edges(ΓQ,c(β, r, ε))→ Q1 where w(e) = a
if e is an edge arising from the arrow a. The vertices vxi will be referred to as the vertices
concentrated at level x. Figure 4.1 depicts the various edge configurations in ΓQ,c(β, r, ε)
for different choices of ε at the tail and head of an arrow.
Figure 4.1: A local picture of edges in ΓQ,c(β, r, ε) with a ∈ Q1, x = ta, y = ha, ands = c(a), and varying choices of ε.
Proposition 4.1.3. Let ΓQ,c(β, r, ε) be an up-and-down graph. Then
a. If a vertex is contained in two edges e, e′, then c(w(e)) 6= c(w(e′));
b. Each vertex in ΓQ,c(β, r, ε) is contained in at most two edges (therefore Γ consists of
string and band components).
c. A connected component of an up-and-down graph is again an up-and-down graph.
71
Proof. For part (a), suppose that vxi is a vertex in Γ incident to two edges e and e′ where
c(w(e)) = c(w(e′)) = s. It is clear from the definition of the edges that w(e) 6= w(e′). Since
there is at most one outgoing and at most one incoming arrow of color s relative to x, it
can be assumed that w(e) = a1 and w(e′) = a2 where h(a1) = t(a2) = x and c(ai) = s.
Suppose that ε(x, s) = 1 (the other case is identical). Then by definition 4.1.2, i ≤ r(a2),
and i ≥ βx− r(a1)+1. But r is a rank map, so βx ≥ r(a1)+ r(a2). Therefore, i ≥ r(a2)+1
and i ≤ r(a2), a contradiction. For part (b), if a vertex vxi in Γ is contained in three
edges, then by part (a) the arrows corresponding to the edges are of three different colors,
and all incident to x, which is false by assumption that kQ/Ic is a gentle string algebra.
Finally, suppose that γ is a connected component of ΓQ,c(β, r, ε). Let us suppose that γ
has β′x vertices at level x for each x ∈ Q0, and has r′(a) edges labeled a for each a ∈ Q1.
Then γ = ΓQ,c(β′, r′, ε) (this is not simply isomorphism of graphs, but one that preserves
the labeling of edges and levels of vertices). Let us label the vertices in ΓQ,c(β′, r′, ε) by
{wxi | x ∈ Q0, i = 1, . . . , β′x}. Let f : ΓQ,c(β′, r′, ε)→ ΓQ,c(β, r, ε) be the homomorphism of
graphs defined as follows: f : wxi 7→ vxγi(x) where γi(x) is the i-th vertex in γ at level x. It
is clear that the image of this map is precisely the graph γ, and that f gives a bijection
between ΓQ,c(β′, r′, ε) and γ.
Remark 4.1.2. It is worth noting that distinct sign functions give rise to a different
numbering on the vertices of the graph Γ, but do not change the graph structure. In fact,
if ε and ε′ differ in only one vertex, x (say), the graphs ΓQ,c(β, r, ε) and ΓQ,c(β, r, ε′) differ
only by applying the permutation i 7→ βx − i + 1 to the vertices {vxi | i = 1, . . . , βx}. We
will soon see that the families of modules arising from different choices of ε coincide.
Here we collect some technical definitions and notations to be used concerning these
graphs. We will extend the terminology of Butler and Ringel ([5]) slightly. Let ΓQ,c(β, r, ε)
be an up and down graph. A vertex vxj′ is said to be above (resp. below) vxj if j > j′
(resp. j < j′). We will depict the graphs of ΓQ,c(β, r, ε) in such a way that above and
72
below are literal.
A vertex vxj in ΓQ,c(β, r, ε) will be referred to as a source (resp. target) if t(w(e)) = x
(resp. h(w(e)) = x) for every edge e containing it. A 2-source (resp. 2-target) is a
source (resp. target) incident to exactly two edges. We will denote the sets of such vertices
by S(Γ), T (Γ), S2(Γ), and T 2(Γ), respectively.
To a path p = vxnin en . . . vx1i1e1v
x0i0
on ΓQ,c(β, r, ε), we will associate a sequence A(p)
of elements in the set alphabet Q1 ∪ Q−11 (that is the formal alphabet with characters
consisting of the arrows and their inverses), with
A(p)i =
w(ei) if t(w(ei)) = xi−1
w(ei)−1 if t(w(ei)) = xi
.
Such a path p will be called direct (resp. inverse) if A(p) is a sequence of elements in Q1
(resp. Q−11 ).
Finally, a path p will be called left positive (resp. left negative if A(p)n ∈ Q1 and
ε(xn, c(en)) = 1 (resp. −1). Analogously the path is called right positive (resp. right
negative) if A(p)1 ∈ Q1 and ε(x0, c(e0)) = 1 (resp. −1).
Example 4.1.3. Consider the quiver below with coloring indicated by type of arrow:
1r1 //
g1
))
2r2 //
p2
))
3
4b1
//p1
55
5
g2
55
b2// 6
Let us say that the color of the arrow ai is a in the above picture. Let β, r be the pair
73
depicted in the following:
3 3 //
2
%%
4 1 //
2
%%
1
2 2 //
2
99
3
1
99
1 // 2
and ε−1(1) = {(1, g), (2, p), (3, g), (4, b), (5, b), (6, p)} (so ε−1(−1) is the complement in X).
Then ΓQ,c(β, r, ε) takes the following form:
v(1)1
r1g1
v(2)1
p2
v(3)1
v(1)2
r1
g1
v(2)2
p2
v(1)3
r1
v(2)3
v(2)4
r2
v(4)1
b1
p1
v(5)1
b2v
(6)1
v(4)2
p1
b1v
(5)2 v
(6)2
v(5)3
g2
i. The vertices v(1)1 , v
(1)2 , v
(1)3 , v
(4)1 , v
(4)2 are sources, and v
(2)3 , v
(5)2 , v
(3)1 , v
(6)1 , v
(6)2 are targets.
ii. The path v(6)2 e2v
(2)1 e1v
(1)3 with w(e1) = r1 and w(e2) = p2 is a direct path that is left
positive (since ε(6, p2) = 1), and right negative; while v(4)1 e2v
(2)3 e1v
(1)1 with w(e1) = p1,
w(e2) = r1 is not a direct path.
4.1.1 Some Combinatorics for Up-and-Down Graphs
The proof of the main theorem requires an explicit description of the projective resolution
of the modules arising from up-and-down graphs. In this section, we collect some technical
lemmas concerning the structure of the graphs ΓQ,c(β, r, ε) to be used in describing the
74
projective resolution.
Lemma 4.1.4. Let vxj be a vertex in ΓQ,c(β, r, ε), and suppose that
p = vxj elvxl−1
il−1. . . vx1
i1e1v
yi
is a left direct path ending in vxj .
A. If p is left negative direct, and vxj′ is above vxj , then there is a left negative direct path
p′ = vxj′e′l−1v
xl−1
i′l−1. . . vx1
i′1e′1v
yi′
with A(p′) = A(p). Furthermore,
A1. vyi′ is above vyi if and only if ε(y, c(w(e′1))) = 1;
A2. vyi′ is below vyi if and only if ε(y, c(w(e′1))) = −1.
B. If p is left positive direct, and vxj′ is below vxj , then there is a left positive direct path
p′ = p′ = vxj′e′l−1v
xl−1
i′l−1. . . vx1
i′1e′1v
yi′
with A(p′) = A(p). Furthermore,
B1. vyi′ is below vxj if and only if ε(y, c(w(e′1))) = −1;
B2. vyi′ is above vxj if and only if ε(y, c(w(e′1))) = 1.
Proof. We will prove this lemma by induction on the length of p. Suppose that p = vxj e1vyi
with A(p) = a. If p is left negative direct, then ε(x, c(a)) = −1. By definition of the graph
Γ, then, j ≤ r(a). But vxj′ is above vxj if and only if j′ < j. By definition 4.1.2 (a), (c),
there is an edge e′1 terminating at vxj′ labeled a, so p′ = vxj′e′1vyi′ . If ε(y, c(a)) = 1, i = j and
i′ = j′, so indeed i′ < i, implying that vyi′ is above vyi . On the other hand, if ε(x, c(a)) = −1,
75
then i = βx − j + 1 and i′ = βx − j′ + 1, so i′ > i, and vyi′ is below vyi . The other direction
is also clear for [A1] and [A2].
Now suppose that p is left positive direct of length one, i.e., ε(x, c(a)) = 1. Write
p = vyi e1vxj . By definition of Γ, then, j ≥ βx − r(a) + 1. Suppose that j = βx − j + 1, and
j′ = βx − j′ + 1. Since vxj′ is below vxj , we have that j′ = βx − j′ + 1 > βx − j + 1 = j,
so j′ < j. Indeed, j′ > βx − r(a) + 1, so by definition 4.1.2 (b) or (d), there is an edge e′1
labeled a terminating at vxj′ . ε(y, c(a)) = −1 if and only if i = βy− j+1 and i′ = βy− j′+1,
i.e., i′ > i, so vyi′ is below vyi . ε(y, c(a)) = 1 if and only if i = j and i′ = j′, i.e., i′ < i, so
vyi′ is above vyi .
Now assume that [A] and [B] are true for all paths of length at most l − 1. Suppose
that
p = vxj elvyl−1
jl−1el−1v
yl−2
jl−2el−2 . . . e1v
yi
is left negative direct, and vxj′ is above vxj . By the first step, there is a path vxj′e′lvyl−1
j′l−1with
w(el) = w(e′l).
Case 1: ε(yl−1, c(w(e′l))) = 1 if and only if vyl−1
j′l−1is above v
yl−1
jl−1. But by proposition 4.1.3,
ε(yl−1, c(w(el))) = −ε(yl−2, c(w(el−1))), so
p = vyl−1
jl−1el−1 . . . e1v
yi
is left negative direct. So by the inductive hypothesis, since p is of length l − 1, we
have a path
p′ = vyl−1
j′l−1e′l−1 . . . e
′1vyi′
with A(p) = A(p′). Taking p′ = e′lp′, we have a left negative direct path p′ terminating
in vxj′ . Again, by the inductive step, vyi′ is above (resp. below) vyi if and only if
ε(y, c(w(e1))) = 1 (resp. −1).
76
Case 2: ε(yl−1, c(w(e′l))) = −1 if and only if vyl−1
j′l−1is below v
yl−1
jl−1. By proposition 4.1.3,
ε(yl−1, c(w(el))) = −ε(yl−2, c(w(el−1))), so
p = vyl−1
jl−1el−1 . . . elv
yi
is left positive direct. By the inductive hypothesis, there is a path
p′ = vyl−1
j′l−1e′l−1 . . . e
′1vyi′
with A(p) = A(p′). Taking p′ = vxj′e′lp′, we have a left negative direct path p′
terminating in vxj′ . By the hypothesis, vyi′ is above (resp. below) vyi if and only if
ε(y, c(w(e1))) = 1 (resp. −1).
The same argument hold if p is left positive direct, interchanging the terms ‘above’ and
‘below’.
Here we collect some properties that determine what types of extremal vertices occur
in which levels.
Lemma 4.1.5. Let ΓQ,c(β, r, ε) be an up-and-down graph, and let a1, a2, b1, b2 ∈ Q1 be
colored arrows as indicated in the figure:
a1** y
a2 44
b2**
b1
44
i. If vyj is a 2-source (resp. 2-target), then r(a1)+r(b1) > βy (resp. r(a2)+r(b2) > βy);
ii. Let m1 = max{r(a1), r(b2)} and m2 = max{r(b1), r(a2)}. Then if vyj is an isolated
vertex, m1 +m2 < βy (in particular, there are neither 2-sources not 2-targets vertices
at level y);
77
iii. If vyi is a 1-target contained in an edge labeled by a1 (resp. b2), then r(b1)+r(b2) < βx
(resp. r(a1) + r(a2) < βx).
Proof. We prove only (iii), since the others are similar. Suppose that vyi is a 1-target
contained in an edge labeled a1. If r(b1) + r(b2) = βy, then each vertex at level y would
be contained in an edge (either labeled b1 or b2), including vyi . But this contradicts the
assumption.
Lemma 4.1.6. Suppose that there is a sequence of arrows a1, a2, a3 ∈ Q1 with c(a1) =
c(a2) = c(a3), h(a1) = t(a2) = x1, and h(a2) = t(a3) = x2. If r is a maximal rank map,
then we have the following:
i. if r(a1) + r(a2) < βx1 then r(a2) + r(a3) = βx2
ii. if r(a2) + r(a3) < βx2 then r(a1) + r(a2) = βx1.
Proof. Suppose that both r(a1) + r(a2) < βx1 and r(a2) + r(a3) < βx2 . Define by r+ the
rank map with r+(a2) = r(a2) + 1 and r+(b) = r(b) otherwise. r+ is a rank map and
r+ > r, contradicting the assumption of maximality of r.
4.2 Up-and-Down Modules
We will now define a module (or family of modules) VQ,c(β, r) based on two additional
parameters, later proving that the isomorphism class of this module (or family of modules)
is independent of these parameters. Fix Q, c, β, r, ε as described above. Recall that propo-
sition 4.1.3 guarantees ΓQ,c(β, r, ε) is comprised of strings and bands. Let B(Γ) be the set
of bands and fix a function Θ : B(Γ)→ Vert(ΓQ,c(β, r, ε)) with Θ(b) a target contained in
the band b.
Definition 4.2.1. For µ ∈ (k∗)B(Γ), denote by Vµ := VQ,c(β, r, ε,Θ)µ the representation of
Q given by the following data. The space (Vµ)x is a βx-dimensional k vector space together
78
with a fixed basis {exj }j=1,...,βx . The linear map (Vµ)a : (Vµ)ta → (Vµ)ha is defined as follows:
if vtaj and vhak are joined by an edge e labeled a, then
(Vµ)a : etaj 7→
µbe
hak if there is a band b with Θ(b) = vhak and ε(ha, c(a)) = 1
ehak otherwise.
If there is no such edge, then (Vµ)a : etaj 7→ 0. If there are no bands in ΓQ,c(β, r, ε), denote
by VQ,c(β, r, ε,Θ) the subset of RepQ(β) containing this module. If there are bands, then
denote by VQ,c(β, r, ε,Θ) the set of all modules VQ,c(β, r, ε,Θ)µ for µ ∈ (k∗)B(Γ).
Example 4.2.1. Continuing with example 4.1.3, let b be the unique band in ΓQ,c(β, r, ε),
and take Θ(b) = v(6)1 . For µ ∈ k∗, the module VQ,c(β, r, ε,Θ)µ is given by the following:
V1
1 0 00 1 00 0 0
��
0 0 10 1 01 0 00 0 0
// V2
0 λ 0 01 0 0 0
��
[0 0 0 1
]
// V3
V4
0 00 01 00 1
??
0 00 11 0
// V5
[0 0 1
]
??
1 0 00 0 0
// V6
Proposition 4.2.2. Every representation in the set VQ,c(β, r, ε,Θ) is a representation of
the gentle string algebra (Q, c).
Proof. If a1, a2 ∈ Q1 are arrows with ha1 = ta2 and c(a1) = c(a2), then by proposition
4.1.3 there is no path v1e1
v2e2
v3 in Γ with w(e1) = a1 and w(e2) = a2. Therefore,
79
a2a1(exi ) = 0 for all x ∈ Q0, i = 1, . . . , βx. Since Ic is generated by precisely these relations,
each module in VQ,c(β, r, ε,Θ) is indeed a kQ/Ic module.
The definition appears highly dependent on both ε and the choice of distinguished
vertices Θ. In the following proposition, we show that the family does not depend on Θ.
Proposition 4.2.3. The family VQ,c(β, r, ε,Θ) does not depend on the choice of vertices
Θ.
Proof. This proof is a simple consequence of [5] (cf. Theorem page 161). Indeed, suppose
that b is a band component of some VQ,c(β, r, ε,Θ)µ. Denote by Vb,µb the submodule
corresponding to this band, and ωb a cyclic word in Q1 ∪ Q−11 which yields this band.
Recall that Butler and Ringel produce, for each such cyclic word, a functor Fωb from the
category of pairs (V, ϕ) with V a k-vector space and ϕ : V → V and automorphism, to
the category RepQ,c. The indecomposable module Vb,µb is isomorphic to the image under
this functor of the pair (k, µb) where µb : x 7→ µb · x. Butler and Ringel show that the
family Vb,µb for µb ∈ k∗ is independent of cyclic permutation of the word ωb. (They show
the image of the functor Fωb itself is independent cyclic permutations of ωb.) Therefore,
any choice of vertices Θ yields the same family VQ,c(β, r, ε).
Henceforth, we drop the argument Θ, and when necessary we make a particular choice
of said function.
4.2.1 Main Theorem and Consequences
The following statement allows us to show that the representations VQ,c(β, r) are generic.
The remainder of the article will be primarily concerned with proving this theorem.
Theorem 4.2.4. Let B(Γ) be the set of bands for the graph ΓQ,c(β, r, ε).
80
a. Suppose that µ, µ′ ∈ (k∗)B(Γ) with µb 6= µ′b′ for all b, b′ ∈ B(Γ). Then
dim Ext1kQ/Ic(VQ,c(β, r, ε)µ, VQ,c(β, r, ε)µ′) = 0.
b. Suppose that ΓQ,c(β, r, ε) consists of a single band component. Then
dim Ext1kQ/Ic(VQ,c(β, r, ε)µ, VQ,c(β, r, ε)µ) = 1.
Corollary 4.2.5. If B(Γ) = ∅, i.e., ΓQ,c(β, r, ε) consists only of strings, then the unique
element V ∈ VQ,c(β, r) has a Zariski open orbit in RepQ,c(β, r).
Proof. If there are indeed no band components in ΓQ,c(β, r, ε), then by theorem 4.2.4 part
(a), we have Ext1kQ/Ic(VQ,c(β, r, ε), VQ,c(β, r, ε)) = 0. The corollary then follows by ([21]
Corollary 1.2, [39]).
If there are band components, then the analogous corollary is more subtle, although
the result is essentially the same. Namely that the union of the orbits of all elements in
VQ,c(β, r) is dense in its irreducible component. The proof relies on some auxiliary results
due to Crawley Boevey-Schroer [10], and so we exhibit those first. Let kQ/I be an arbitrary
quiver with relations. Suppose that Ci ⊂ Rep(Q,I)(β(i)) are GLQ(β(i))-stable subsets for
some collection of dimension vectors β(i), i = 1, . . . , t, and denote by β =∑
i β(i) the sum
of the dimension vectors. Define by C1 ⊕ . . .⊕Ct the GLQ(β)-stable subset of RepkQ/I(β)
given by the set of all GLQ(β) orbits of direct sums M1 ⊕ . . .⊕Mt with Mi ∈ Ci.
Theorem 4.2.6 (Theorem 1.2 in [10]). For an algebra kQ/I, Ci ⊂ Rep(Q,I)(β(i)) irre-
ducible components and t defined as above, the set C1 ⊕ . . .⊕ Ct is an irreducible component
of Rep(Q,I)(β) if and only if
ext1kQ/I(Ci, Cj) = 0
for all i 6= j.
81
Corollary 4.2.7. In general,⋃
µ∈(k∗)B(Γ)
OVµ is dense in RepQ,c(β, r).
Proof. Enumerate the connected components of ΓQ,c(β, r, ε), c1, . . . , ct with ci a band for
i = 1, . . . , l and a string for i = l + 1, . . . , t. Let β|i, r|i be the restrictions of β and
r, respectively, to the i-th connected component. (By proposition 4.1.3, each connected
component is itself an up-and-down graph, so is associated with a dimension vector and
maximal rank map.) Let Ci = RepQ,c(β|i, r|i), which is an irreducible component by
3.1.3. Notice that VQ,c(β|i, r|i) ∈ RepQ,c(β|i, r|i) if ci is a string and VQ,c(β|i, r|i)µi ∈
RepQ,c(β|i, r|i) if ci is a band. Thus, the Ci are irreducible and, assuming theorem 4.2.4 is
true, ext1kQ/Ic
(Ci, Cj) = 0, so RepQ,c(β, r) = C1 ⊕ . . .⊕ Ct.
Thus, all that remains to be shown is that if ci is a band, then the union of the orbits of
all elements in VQ,c(β|i, r|i) contains an open set. Indeed, if this is the case, then denoting
by Si the set GL(β|i) · VQ,c(β|i, r|i) we have
C1 ⊕ . . .⊕ Ct = S1 ⊕ . . .⊕ St.
Suppose that β is a dimension vector and r is a maximal rank map such that ΓQ,c(β, r, ε)
is a single band. Let Vµ = VQ,c(β, r)µ, and denote by OVµ the GL(β)-orbit of Vµ. From
Kraft (2.7 [25]), there is an embedding
TVµ(RepQ,c(β, r))/TVµ(OVµ) ↪→ Ext1(Vµ, Vµ)
where TM(X) denotes the tangent space in X at M . By theorem 4.2.4, then
dimTVµ(RepQ,c(β, r))− dimTVµ(OVµ) ≤ 1.
Claim. Vµ is a non-singular point in RepQ,c(β, r) (and in OVµ).
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Proof. Consider the construction of VQ,c(β, r) as a specific choice of embedding a product
of varieties of complexes into RepQ,c(β, r), i.e.,
∏s∈S
Com(β, r, s) ∼= RepQ,c(β, r)
For each x ∈ Q0, let σ(x) be the matrix of the map (in the distinguished basis of VQ,c(β, r))
corresponding to the permutation (1, βx)(2, βx − 1) . . . . If V ∈ ∏s∈S Com(β, r, s), then
define by ϕ(V ) the element of RepQ,c(β, r) with
ϕ(V )a =
Va if ε(ta, c(a)) = 1 = −ε(ha, c(a))
σ(ha)Va if ε(ta, c(a)) = 1 = ε(ha, c(a))
Vaσ(ta) if ε(ta, c(a)) = −1 = ε(ha, c(a))
σ(ha)Vaσ(ta) if ε(ta, c(a)) = −1 = −ε(ha, c(a))
The map ϕ is an isomorphism, since σ(x) ∈ GL(βx) for each x. Furthermore, rankkVa =
rankkϕ(V )a. Therefore, since rankk(VQ,c(β, r)a) = r(a), there is a V ∈ ∏s∈S
Com(β, r, s) with
ϕ(V ) = VQ,c(β, r), and rankkVa = r(a). Com(β, r, s) has a dense open orbit, given by the
those complexes W such that rankkWa = r(a). Thus,∏s∈S
Com(β, r, s) is smooth at V , and
so RepQ,c(β, r) is smooth at VQ,c(β, r).
Hence, we have the following:
dim(RepQ,c(β, r))− dim(OVµ) = dimTVµ(RepQ,c(β, r))− dimTVµ(OVµ) ≤ 1.
If the difference is 0, then OVµ is a closed set of the same dimension as dim(RepQ,c(β, r)),
so these are equal. On the other hand, if the difference is 1, then X :=⋃µ∈k∗OVµ is a closed
set. For t ∈ k∗, Vµ+t 6∼= Vµ and Vµ+t ∈ X. Therefore, dimTVµX ≥ dimTVµOVµ + 1, and
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therefore dimTVµX = dim(RepQ,c(β, r)). Since X is closed, X = RepQ,c(β, r).
In order to prove theorem 4.2.4, we will explicitly describe the projective resolution of
VQ,c(β, r)µ for any β, r, and then apply the appropriate Hom-functor to the resolution.
4.2.2 Projective resolutions of VQ,c(β, r) and the EXT-graph
The summands in the projective resolutions of VQ,c(β, r) depend on a number of charac-
teristics of the graph ΓQ,c(β, r, ε). We collect the pertinent characteristics in the following
list.
Definition 4.2.8. Let Γ = ΓQ,c(β, r, ε) be a fixed up-and-down graph.
a. Denote by ISO(Γ) the set of isolated vertices in Γ;
b. Denote by S1(Γ) (resp. T1(Γ)) the set of sources (resp. targets) of degree one in Γ.
These will be referred to as 1-sources (resp. 1-targets).
c. For a vertex vxj ∈ T (Γ), we denote by lp+(vxj ) (resp. lp−(vxj )) the longest left posi-
tive (resp. left negative) direct path in Γ terminating in vxj (if such a path exists).
Similarly, for a vertex vxj ∈ S(Γ), denote by rp+(vxj ) (resp. rp−(vxj )) the longest right
positive (resp. right negative) direct path initiating in vxj .
d. For a vertex vxj ∈ T (Γ), we denote by l+(vxj ), (resp. l−(vxj )) the source at the other
end of lp+(vxj ) (resp. lp−(vxj )). Similarly, for a vertex vxj ∈ S(Γ), we denote by r+(vxj )
(resp. r−(vxj )) the target at the other end of rp+(vxj ) (resp. rp−(vxj )).
e. If vxj ∈ S1 ∪T1, let pxj be the direct path of maximal length containing vxj ;
f. If vxj ∈ S1 ∪T1, denote by [vxj ]1 ∈ Q1 be the arrow with the property that t([vxj ]1) = x
and c([vxj ]1) = c(w(e)) where e is the edge in pxj containing vxj ;
84
g. Furthermore, recursively define the arrows [vxj ]l with t([vxj ]l) = h([vxj ]l−1), and c([vxj ]l) =
c([vxj ]1).
h. Suppose vxj ∈ ISO. Denote by [vxj ]+1 (resp. [vxj ]−1 ) the arrow (if such exists) with
t([vxj ]±1 ) = x and ε(x, c([vxj ]δ1)) = δ. Again, recursively define [vxj ]δl with t([vxj ]δl ) =
h([vxj ]δl−1), and c([vxj ]δl ) = c([vxj ]δ1).
i. In case [vxj ]l or [vxj ]±l fails to exist, write h([vxj ]l) := ∅ (or h([vxj ]±l ) := ∅), and let
P∅ be the zero object. (This is nothing more than notation to write the projective
resolution of up-and-down modules in a more compact form.)
Example 4.2.2. Referring again to example 4.1.3, we have the following aspects:
i. ISO(Γ) = ∅;
ii. v(1)3 is a 1-source, and p
(1)3 is the path v
(6)2 e2v
(2)1 e1v
(1)3 where w(e1) = r1 and w(e2) = p2;
iii. r+(v(1)1 ) = v
(6)1 , and r−(v
(1)1 ) = v
(2)3 .
iv. Since ε((6), b2) = −1, we have lp−(v(6)1 ) = b2g1. Similarly, rp−(v
(1)1 ) = b2g1.
To illustrate the situation (e)-(h), consider the dimension vector and rank sequence below:
1 1 //
0
##
2 0 //0
##
0
0 0 //0
;;
00
;;
0 // 0
The associated up-and-down graph is given by
v(1)1
r1v
(2)1
v(2)2
85
In this case, v(1)1 ∈ S1, and [v
(1)1 ]1 = r2 since the longest path containing v
(1)1 is v
(2)1 r1v
(1)1 ,
c(r1) = c(r2), and t(r2) = (2). The vertex v(2)2 is isolated, and in this case, [v
(2)2 ]+1 = p2 and
[v(2)2 ]−1 = r2.
We are now prepared to exhibit the projective resolution in the general case. Notice
that the simple factor modules of VQ,c(β, r, ε,Θ)µ are Sx for vxj ∈ S(Γ).
Proposition 4.2.9. The following is a projective resolution of VQ,c(β, r, ε,Θ)µ is:
. . . // P (Vµ)2
δ(Vµ)1 // P (Vµ)1
δ(Vµ)0 // P (Vµ)0// Vµ // 0
where
P (Vµ)0 =⊕
vxj ∈S(Γ)
Px
P (Vµ)1 =⊕vyi ∈T2
Py ⊕⊕
vxj ∈S1 ∪T1
Ph([vxj ]1) ⊕⊕vxj ∈ISO
Ph([vxj ]+1 ) ⊕ Ph([vxj ]−1 )
P (Vµ)l =⊕
vxj ∈T1 ∪ S1
Ph([vxj ]l) ⊕⊕vxj ∈ISO
Ph([vxj ]+l ) ⊕ Ph([vxj ]−l );
and where the differential is given by the following maps (we write Px,j for the projective
Px arising from vjx):
i. If vyi ∈ T2, vy+
i+ = l+(vyi ) and vy−
i− = l−(vyi ), then the map δ(Vµ)0 restricts to
Py,i
lp+(vyi )
−µblp−(vyi )
// Py+,i+ ⊕ Py−,i−
86
if vyi = Θ(b) for some band b, and
Py,i
lp+(vyi )
−lp−(vyi )
// Py+,i+ ⊕ Py−,i−
otherwise.
ii. If vyi ∈ T1, pyi is the longest direct path terminating at vyi , and vxj is the source at the
other end of pyi , then the restriction of δ(Vµ)0 to Ph([vyi ]1) is given by
Ph([vyi ]1)
[[vyi ]1A(pyi )
]// Px,j .
iii. If vyi ∈ ISO, then restriction of δ(Vµ)0 to Ph([vyi ]+1 ) ⊕ Ph([vyi ]−1 ) is given by
Ph([vyi ]+1 ) ⊕ Ph([vyi ]−1 )
[[vyi ]
+1 [vyi ]
−1
]// Py,i .
iv. If vyi ∈ T1 ∪ S1, then the restriction of δ(Vµ)l to Ph([vyi ]l+1) is
Ph([vyi ]l+1)
[[vyi ]l
]// Ph([vyi ]l) .
v. If vyi ∈ ISO, then δ(Vµ)l restricted to Ph([vyi ]±l+1) is
Ph([vyi ]±l+1)
[[vyi ]
±l
]// Ph([vyi ]±l ) .
We now apply the functor Hom(−, Vν) to the complex P (Vµ)•. Recall that we have a
87
fixed basis for the spaces (Vν)x for each x ∈ Q0, namely {ex1 , . . . , exβx}, relative to which the
arrows act by the description given by the graph ΓQ,c(β, r, ε). So we take {vxi � exj }j=1,...,βx
the basis for Hom(Px,i, Vν), {vxi � eh([vxi ]l)j }j=1,...,βh([vx
i]l)
the basis for Hom(Ph([vxi ]l), Vν) for
vxi ∈ S1 ∪T1, and {vxi � eh([vxi ]tl)j }j=1,...,β
h([vxi
]tl)
for vxi ∈ ISO and t = +,−, relative to the
aforementioned bases.
We will construct a graph EXT whose vertices correspond to a fixed basis for
Hom(P (Vµ)•, Vν) as described above. We will partition the vertices into subsets EXT(i)
for i = 0, 1, . . . called levels. From this graph the homology of the complex can be easily
read.
Definition 4.2.10. Let Vµ be as described above. Let EXT(l) be the sets defined as
follows.
EXT(0) = {vxj � vxj′} vxj ∈S(Γ)
j′=1,...,βx
EXT(1) = {vxj � vxj′} vxj ∈T2
j′=1,...,βx
∪ {vxj � vh([vxj ]1)
j′ } vxj ∈T1 ∪ S1
j′=1,...,βh([vxj
]1)
∪ {vxj � vh([vxj ]t1)
j′ } vxj ∈ISO
j′=1,...,βh([vx
j]t1)
t=+,−
EXT(l) = {vxj � vh([vxj ]l)
j′ } vxj ∈T1 ∪ S1
j′=1,...,h([vxj ]l)
∪ {vxj � vh([vxj ]tl)
j′ } vxj ∈ISO
j′=1,...,βh([vx
j]tl)
t=+,−
and EXT the graph with vertices⋃l≥0
EXT(l) and edges given by
a. vxj � vxj′ vyi � vy′
i′ if
Hom(δ(Vµ)0, Vν)) : vxj � exj′ 7→∑
si,i′,y,y′
j,j′,x vyi � ey′
i′
with si,i′,y,y′
j,j′,x 6= 0 between levels EXT(0) and EXT(1);
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b. vxj � vx′
j′ vyi � vy′
i′ if
Hom(δ(Vµ)l, Vν)) : vxj � ex′
j′ 7→∑
si,i′,y,y′
j,j′,x,x′vyi � ey
′
i′
and si,y,i′,y′ 6= 0 between EXT(l − 1) and EXT(l).
4.2.3 Properties of the EXT-graph
We collect now the properties of the EXT graph that will be used to show exactness of
complex Hom(P (Vµ)•, Vν).
Proposition 4.2.11. Let EXT be the graph given above
E1. There is an edge
EXT(0) 3 vxj � vxj′ vyi � vyi′ ∈ EXT(1)
in the graph EXT if vxj ∈ S2, vyi ∈ T2, vxjp
vyi and vxj′p′
vyi′ are paths in Γ
with A(p) = A(p′).
E2. If vyi ∈ T1, vxj = l±(vyi ) ∈ S(Γ) and p = lp±(vyi ), then there is an edge
EXT(0) 3 vxj � vxj′ vyi � vy′
i′ ∈ EXT(1)
if vxj′p′
vy′
i′ is a path in Γ with A(p′) = [vyi ]1A(p). Furthermore, there is an edge
EXT(l) 3 vyi � vh([vyi ]l)
i′ vyi � vh([vyi ]l+1)
j′ ∈ EXT(l + 1)
in EXT if there is an edge vh([vyi ]l)
i′evh([vyi ]l+1)
j′ in Γ with w(e) = [vyi ]l+1.
89
E3. Similarly, if vyi ∈ S1, then there is an edge
EXT(0) 3 vyi � vyi′ vyi � vh([vyi ]1)
j′ ∈ EXT(1)
in EXT if there is an edge vyi′evh([vyi ]1)
j′ with w(e) = [vyi ]1. Furthermore, there is
an edge EXT(l − 1) 3 vxj � vh([vxj ]l−1)
j′ vxj � vh([vxj ]l)
j′′ ∈ EXT(l) in EXT if there is
an edge vh([vxj ]l−1)
j′e v
h([vxj ]l)
j′′ in Γ with w(e) = [vxj ]l.
E4. Finally, if vyi ∈ ISO, then there is an edge
EXT(0) 3 vyi � vyi′ vyi � vh([vyi ]±1 )j ∈ EXT(1)
in EXT if there is an edge vyi′e v
h([vyi ]±1 )j in Γ with w(e) = [vyi ]
±1 . Furthermore,
there is an edge
EXT(l − 1) 3 vyi � vh([vyi ]±l−1)
i′ vyi � vh([vyi ]±l )
j ∈ EXT(l)
in EXT if there is an edge vh([vyi ]±l−1)
i′e v
h([vyi ]±l )
j in Γ with w(e) = [vyi ]±l .
Lemma 4.2.3. There are no isolated vertices in EXT(1).
Proof. First, suppose vyi � vyi′ ∈ EXT(1) (i.e., vyi ∈ T2). If i′ < i (resp. i′ > i), then by
lemma 4.1.4, there is a path p′ terminating at vyi′ with A(p′) = lp−(vyi ) (resp. A(p′) =
lp+(vyi )). Therefore, there is an edge vxj � vxj′ vyi � vyi′ .
Next, suppose vyi ∈ T1, and [vyi ]1 exists (otherwise, no vertex vyi � vy′
i′ would exist in
Γ). Let p be the path of maximal length terminating at vyi , and vxj the source at which
p starts. Label the edge of p containing y by a1, let b2 := [vxj ]1, and b1 the arrow (if it
exists) with h(b1) = y and c(b1) = c(b2). By lemma 4.1.5, r(b1) + r(b2) < βy. Now denote
by b3 the arrow [vyi ]2. By lemma 4.1.6, r(b2) + r(b3) = βhb2 , so vhb2i′ is contained in an edge
90
with such a label. If said label is b2, then ey′
i′ ∈ imb2A(p), and so vyi � vhb2i′ is contained
in an edge between EXT(1) and EXT(0). Otherwise, b3ehb2i′ = ehb3i′′ 6= 0. In this case,
vyi � vhb2i′ ∈ EXT(1) and vyi � vhb3i′′ ∈ EXT(2) are contained in an edge.
Finally, suppose that vyi ∈ ISO, and let y′ = h([vyi ]+1 ) or h([vyi ]
−1 . We will show that
vyi � vy′
i′ is non-isolated for i = 1, . . . , βy′ . Note first that vy′
i′ is non-isolated in Γ by lemma
4.1.6, for suppose that a0 is the arrow (if it exists) with h(a0) = y, and c(a0) = c([vyi ]±1 ). By
lemma 4.1.5, r(a0) + r([vyi ]±1 ) < βy, so by lemma 4.1.6, r([vyi ]
±1 ) + r([vyi ]
±2 ) = βy′ . Therefore,
there is an edge e incident to vy′
i′ such that w(e) = [vyi ]±1 or [vyi ]
±2 . In the former case,
vyi � vy′
i′ is contained in a common edge with a vertex in EXT(0), and in the latter case it
is contained in a common edge with a vertex in EXT(2).
Lemma 4.2.4. All vertices in EXT are contained in at most two edges, and every vertex
with label vyi �vh([vyi ]li′ for l ≥ 1 is contained in at most one edge. Furthermore, the neighbor
of any vertex vyi �vh([vyi ]l)
i′ in EXT(l) is vyi �vh([vyi ]l−1)
i′′ or vyi �vh([vyi ]l+1)
i′′ for some i′′. Therefore,
the graph EXT splits into string and band components, such that the band components and
strings of length greater than one occur between levels EXT(0) and EXT(1).
Proof. Recall from property E2 that vyi � vh([vyi ]1)
i′ is connected by an edge to vxj � vxj′ ∈
EXT(0) if and only if vyi ∈ T1, vxjp
vyi is the longest left direct path in Γ ending at
vyi , and there is a path vxj′p′
vh([vyi ]1)
i′ with A(p′) = [vyi ]1A(p). It is clear that there is
only one such vertex, if it exists. If such a path does exist, then there is no edge in EXT
between vyi � vh([vyi ]1)
i′ and vyi � vh([vyi ]2)
i′′ , since this would mean that vh([vyi ]1)
i′ and vh([vyi ]2)
i′′ are
contained in an edge e in Γ with w(e) = [vyi ]2. This contradicts proposition 4.1.3, since
vh([vyi ]1)
i′ would be in two edges of the same color. Otherwise, vyi � vh([vyi ]1)
i′ is connected to
the vertex vyi � vh([vyi ]2)
i′′ in EXT if and only if there is an edge vh([vyi ]1)
i′e
vh([vyi ]2)
i′′ with
w(e) = [vyi ]2, by property E3. By definition of the Up and Down graph, this describes a
unique vertex.
As for the other vertices, the lemma is clear from property E1.
91
In terms of the complex Hom(P (Vµ)•, Vν), the above lemma says that the kernel of the
map
Hom(δ2, Vν)) is spanned by the elements {vyi � vyi′ | vyi ∈ T2, i′ = 1, . . . , βy} together with
those of {vyi �vh([vyi ]1)
i′ | vyi ∈ T1 ∪ S1, i′ = 1, . . . , βh([vyi ]1)} which share no edge with vertices
in EXT(2).
Lemma 4.2.5. No string in EXT has both endpoints in EXT(1).
Proof. Suppose that there is a string with one endpoint vy0
j0�vy
′0
j′0∈ EXT(1) and containing
the following substring:
vx1i1
� vx1
i′1vy0
j0� v
y′0j′0
vx2i2
� vx2
i′2vy1
j1� v
y′1j′1
...
vxnin � vxni′n vyn−1
jn−1� v
y′n−1
j′n−1
vynjn � vy′nj′n
with vxtit � vxti′t∈ EXT(0) and vysjs � v
y′sj′s∈ EXT(1). We will show that the string does not
end in the vertex vynjn � vy′nj′n
. Recall by definition of the graph EXT that for such a string
to exist, we must have paths
vx1i1
p0
q1vy0
j0
vx2i2
p1
q2vy1
j1
...
vxninqn
vyn−1
jn−1
vynjn
92
in Γ. A small notational point: if p is a direct path which starts (resp. ends) in the vertex
vzl , with e the edge of p incident to said vertex, then we write ε(z, p) := ε(z, c(w(e))).
Case 1: Assume that vy0
j0, vynin ∈ T2. Let pn be the longest left path terminating in vynin
with pn 6= qn (this is guaranteed since vynin is a 2-target). Similarly, let q0 be the longest
left path terminating in vy0
j0with q0 6= p0.
A: If i′0 < i0, then ε(y0, p0) = −1. If not, then by lemma 4.1.4 there would be a path q′0
terminating at vy0
i0in Γ with A(q′0) = A(q0). By definition of the graph EXT, then,
there would be an other edge terminating at the vertex vy0
i0� vy0
i′0.
A1: if i′n > in, then ε(yn, qn) = −1 by lemma 4.1.4. Thus, by proposition 4.1.3,
ε(yn, pn) = 1. Therefore, again by lemma 4.1.4, there is a path p′n in Γ termi-
nating at vyni′n with A(p′n) = A(pn), so there is another edge in EXT containing
the vertex vynin � vyni′n .
A2: if i′n < in, then ε(yn, qn) = 1 by lemma 4.1.4. Thus, by proposition 4.1.3,
ε(yn, pn) = −1. Therefore, again by lemma 4.1.4, there is a path p′n in Γ termi-
nating at vyni′n with A(p′n) = A(pn), so there is another edge in EXT containing
the vertex vynin � vyni′n .
B: If i′0 > i0, then ε(y0, p0) = 1, by the same reasoning at Subcase A. The subcases B1
and B2 are analogous to A1 and A2.
Case 2: Assume that vy0
j0∈ T2 while vynin ∈ T1. We will show that Let (i′n)− be the integer
such that there is an edge vyn(i′n)−evh([vynin ]1)
i′nin Γ with w(e) = [vynin ]1. This is guaranteed
to exist by the definition of [vynin ]1 (refer to property E2 in proposition 4.2.11).
A: Suppose (i0)′ < i0. Then ε(y0, p0) = −1 by definition of Γ.
A1: If (i′n)− < in, then ε(yn, qn) = 1, and since there is a path eqn in Γ with
w(e) = [vynin ]1, we must have that ε(yn, [vynin
]1) = −1. If this were the case, then
93
by the definition of the edges in Γ, there would be an edge e′ with w(e′) = [vynin ]1
with one end at the vertex vynin . This contradicts the assumption that vynin is a
1-target.
A2: Similarly, if i−n > in, then ε(yn, qn) = 1, and since there is a path eqn in Γ
with w(e) = [vynin ]1, we have that ε(yn, [vynin
]1) = −1. If this were the case, then
there would be an edge e′ with w(e′) = [vynin ]1 with one end at the vertex vynin ,
contradicting the assumption of vynin being a 1-target.
B: Suppose that (i0)′ > i0. Then ε(y0, p0) = 1 by definition of Γ. Subcases b1 and b2
are the same as above with signs of ε flipped.
Case 3: Assume that vy0
i0∈ T1 and vynin ∈ T2. Let pn be the left direct path in Γ of
maximal length with endpoint vynin and pn 6= qn (guaranteed since the vertex is a 2-target).
As above, let (i′0)− be the integer such that there is an edge e with endpoints vy0
(i′0)− and
vh([v
y0i0
]1)
i′0.
A: Suppose that (i′0)− < i0. Then ε(y0, [vy0
i0]1) = 1, so ε(y0, p0) = −1.
A1: If i′n < in, then ε(yn, qn) = 1, so ε(yn, pn) = −1. But then by lemma 4.1.4, there
is an edge p′n with A(pn) = A(p′n) one of whose endpoints is vyni′n .
A2: If i′n > in, then ε(yn, qn) = 1, ε(yn, pn) = 1. By lemma 4.1.4, there is an edge p′n
with A(pn) = A(p′n) one of whose endpoints is vyni′n .
B: If (i′0)− > i0, then the same arguments hold with the values of ε exchanged.
Case 4: Assume that vy0
i0, vynin ∈ T1.
A: Suppose (i′0)− < i0, so ε(y0, [vy0
i0]1) = −1 and ε(y0, p0) = 1.
A1: If (i′n)− < in, then ε(yn, qn) = −1 by lemma 4.1.4. But if this were the case, then
there would be an edge e in Γ with w(e) = [vynin ]1 and one of whose endpoints
was vynin . This contradicts the assumption that said vertex was a 1-target.
94
A2: If (i′n)− > in, then ε(yn, qn) = 1 by lemma 4.1.4. If this were the case, then
there would be an edge e in Γ with w(e) = [vynin ]1 and one of whose endpoints
was vynin . This contradicts the assumption that said vertex was a 1-target.
B: If (i′0)− < i0, then the same argument holds with the values of ε exchanged.
4.2.4 Homology and the EXT graph
Let us pause to interpret the above results into data concerning the maps Hom(δ(Vµ)1, Vν)
and Hom(δ(Vµ)0, Vν). Recall that a vertex vxi � vyj corresponds to the basis element vxi ⊗
eyj . By lemma 4.2.3, there are no isolated vertices in EXT(1), and by lemma 4.2.4, if
Hom(δ(Vµ)1, Vν) : vxi ⊗ eyj 7→ vx′
i′ ⊗ ey′
j′ , then after reordering the chosen basis,
Hom(δ(Vµ)1, Vν) takes the form
1 0 . . . 0
0 ∗ . . . ∗...
.... . .
...
0 ∗ . . . ∗
In particular, ker(Hom(δ(Vµ)1, Vν)) is precisely the span of those vertices in EXT(1) that
have an edge in common with a vertex in EXT(0).
It remains to be shown that every other vertex in EXT(1) corresponds to a basis element
that is in the image of Hom(δ(Vµ)0, Vν). This will show that the image of said map equals
the kernel of Hom(δ(Vµ)1, Vν). Let us denote by C1, C2, . . . , Cm the connected components
of the induced subgraph on the vertices EXT(0) ∪ EXT(1). Then Hom(δ(Vµ)1, Vν) can be
95
written in block form:
δC1 0 . . . 0
0 δC2
. . ....
.... . . . . .
0 . . . 0 δCm 0
Therefore, it suffices to show that each block corresponding to a connected component is
surjective.
Lemma 4.2.6. If vxj � vyi ∈ EXT(1) is contained in a string between levels 0 and 1, then
vxj ⊗ eyi ∈ im(Hom(δ(Vµ)0, Vν)).
Proof. Suppose that the vertex is contained in the connected component Ci, and that Ci
is a string. We have shown in lemma 4.2.5 that if a string is between levels 0 and 1, then
either one endpoint lies in level 0 and the other in level 1, or both endpoints lie in level 0.
In the first case, δCi is strictly upper triangular with nonzero entries on the diagonal which
must be from the set {±1,±µ,±ν}. Therefore, the map is invertible. In the second case,
there is one more vertex in level EXT(0) than in EXT(1), and (δCi)j,j 6= 0 for each j, so
the given map is surjective.
Lemma 4.2.7. If Ci is a band, then δ(Ci) is an isomorphism.
Proof. If a component Ci is cyclic, then it must come from the following cycles on ΓQ,c(β, r, ε):
vx0i0
p1vy0
j0
vx1i1
q1
p2vy1
j1
...
vxnin
qn−1
pnvynjn
vx0
i′0
p1vy0
j′0
vx1
i′1
q1
p2vy1
j′1
...
vxni′n
qn−1
pnvynj′n
96
In particular, by definition of δ(Vµ)1, the matrix of δ(Ci) takes the following form:
1 −µ 0 0 . . . 0
0 ±1 ±1 0 . . . 0
0 0 ±1 0 . . . 0
.... . .
...
±1 0 . . . ±1
where one of the diagonal entries is ν, and in each row there is exactly one positive and
one negative entry. Then it is an elementary exercise (expanding by the first column and
calculating the determinant of upper or lower triangular matrices) to show that det δ(Ci) =
±(µ− ν). Since, by assumption, µ 6= ν, we have that δ(Ci) is nonsingular.
Now that part (a) of the theorem is proved, we move to part (b), recalled here:
Proposition 4.2.12. Suppose that ΓQ,c(β, r, ε) consists of a single band component, and
let µ ∈ (k∗)B(Γ) = k∗. Let Vµ = VQ,c(β, r, ε)µ. Then
Ext1kQ/Ic(Vµ, Vµ) = 1.
Proof. The projective dimension of Vµ is one by the constructions above. Furthermore,
there is exactly one band component in the graph EXT, since there is exactly one pair of
bands b1, b2 in Γ with the A(pi) = A(p′i) and A(qi) = A(q′i) as in the proof of lemma 4.2.7.
Therefore, the image of the restriction of the map Hom(P (Vµ)•, Vµ) to the vectors vxkik ⊗exkik
is in the span of the vectors vykj′k⊗eykj′k . Again, as in the proof of lemma 4.2.7, the restriction
97
of said map to the aforementioned subspaces relative to the basis given above is
C =
−µ 1
±µ ±1
±1 ±1
. . .
±1
±1 1
. . . ±1
±1 ±1
.
Recall that in each row there is exactly one positive and one negative entry. Therefore,
the sum of the last n− 1 columns of this matrix is
[1 ±1 0 . . . 0
]where the sign of
the second entry is opposite of the sign of ±µ. Therefore, the first column is in the span
of the last n− 1 columns. Column reducing gives the matrix
C =
0 1
0 ±1
±1 ±1
. . .
±1 ±1
±1 ±1
.
The lower right n−1×n−1 minor is clearly non-zero, since it is a strictly lower triangular
matrix, so this map has rank n−1, showing that the complex Hom(P (Vµ)•, Vµ) has exactly
one dimensional homology at Hom(P (Vµ)1, Vµ).
Proof of theorem 4.2.4 By lemma 4.2.6, blocks corresponding to strings on EXT are
surjective, and by lemma 4.2.7, blocks corresponding to bands on EXT are surjective, so
98
the homology of the complex
Hom(P (Vµ)0, Vν) // Hom(P (Vµ)1, Vν) // . . .
vanishes in the first degree.
�
Finally, we point out a corollary to the above proof that will be useful for describing
the Schofield semi-invariants.
Corollary 4.2.13. Suppose that the generic module in RepQ,c(β, r) consists of an inde-
composable band module. Then det(Hom(δ0(Vµ), Vν)) = ±µkνl(µ− ν).
Proof. We have already shown that the restriction of the map Hom(δ0(Vµ), Vν) to the cyclic
component of the EXT graph is a multiple of µ − ν. Furthermore, for each of the string
components, the entries on the diagonal are in the set {±1,±µ,±ν}, and in the proof of
lemma 4.2.6, we showed that these restrictions are upper-triangular. Therefore, for some
powers k, l, the determinant is precisely ±µkνl(µ− ν).
4.3 Higher Extension Groups
The graphical representation given above can be used to calculate higher extension groups.
For each vertex vxj ∈ S1 ∪T1, let Xj,x be the complex
Vx[vxj ]1
// Vh([vxj ]1)
[vxj ]2// Vh([vxj ]2)
[vxj ]3// . . . .
and if vxj ∈ ISO, let X+j,x be the complex
Vx[vxj ]+1 // Vh([vxj ]+1 )
[vxj ]+2 // . . . ,
99
and analogously for X−j,x. Let hi(X) be the dimension of the i-th homology space of the
complex X.
Corollary 4.3.1. Let ΓQ,c(β, r, ε) be an up-and-down graph for (Q, c) a gentle string alge-
bra. Then
dim Exti(VQ,c(β, r)µ, VQ,c(β, r)ν) =∑
vxj ∈S1 ∪T1
hi(Xj,x) +∑
vxj ∈ISO
(hi(X+
j,x) + hi(X−j,x)).
4.3.1 Example
We finish by exhibiting the EXT graph for example 4.1.3. Recall that we chose Θ(b) = v(6)1
for the band component. By proposition 4.2.9, the projective resolution of the representa-
tion in the example is given by
Vµ P 31 ⊕ P 2
4oo P2 ⊕ P3 ⊕ P 2
5 ⊕ P6δ0oo P3
δ1oo
where
δ0 =
−r1 0 0 0 −µb2g1
0 0 −g1 0 p2r1
0 0 0 g1 0
p1 −g2b1 0 0 0
0 r2p1 b1 0 0
δ1 =
0
g2
0
0
0
The associated EXT graph is obtained by applying Hom(−, Vν) to the resolution, so we
have the complex:
(Vν)31 ⊕ (Vν)
24
Hom(δ0,Vν) // (Vν)2 ⊕ (Vν)3 ⊕ (Vν)25 ⊕ (Vν)6
Hom(δ1,Vν) // (Vν)3
100
The EXT graph is depicted below, with the vertices lying in a cyclic component of the
graph boxed.
v(1)1 ⊠ v
(1)1
v(1)1 ⊠ v
(1)2
v(1)1 ⊠ v
(1)3
v(1)2 ⊠ v
(1)1
v(1)2 ⊠ v
(1)2
v(1)2 ⊠ v
(1)3
v(1)3 ⊠ v
(1)1
v(1)3 ⊠ v
(1)2
v(1)3 ⊠ v
(1)3
v(4)1 ⊠ v
(4)1
v(4)1 ⊠ v
(4)2
v(4)2 ⊠ v
(4)1
v(4)2 ⊠ v
(4)2
v(2)3 ⊠ v
(2)1
v(2)3 ⊠ v
(2)2
v(2)3 ⊠ v
(2)3
v(2)3 ⊠ v
(2)4
v(3)1 ⊠ v
(3)1
v(5)2 ⊠ v
(5)1
v(5)2 ⊠ v
(5)2
v(5)2 ⊠ v
(5)3
v(1)3 ⊠ v
(5)1
v(1)3 ⊠ v
(5)2
v(1)3 ⊠ v
(5)3
v(6)1 ⊠ v
(6)1
v(6)1 ⊠ v
(6)2
v(1)3 ⊠ v
(3)1
101
Chapter 5
GIT Quotients
5.1 Dimension Combinatorics
In this section, we illustrate conditions on β and r under which the generic module of
RepQ,c(β, r) is a direct sum of band modules. Furthermore, in the case that Q is acyclic,
we consider some combinatorics of the Euler form 〈〈−,−〉〉kQ/I .
We will say that a pair (β, r) consisting of a dimension vector and rank map is called
a band pair if the generic module in RepQ,c(β, r) is a direct sum of band modules, and
called an exact pair if r(i(x, s)) + r(o(x, s)) = βx for every (x, s) ∈ X.
Proposition 5.1.1. The pair (β, r) is a band pair if and only if (β, r) is an exact pair
such that βx = 0 for every lonely vertex x.
Proof. Suppose that (β, r) is a band pair. First note that if βx = 0 for a lonely vertex x,
then there are two possibilities: if (x, s) ∈ X and one of r(i(x, s)), r(o(x, s)) is non-zero,
then there is a vertex vxi incident to exactly one edge (which is of color s), contradicting
the assumption. On the other hand, if (x, s) ∈ X and both r(i(x, s)), r(o(x, s)) are zero.
In this case, there is a vertex vxi which is isolated, so Sx is a direct summand of VQ,c(β, r).
Sx is not a band module, contradicting the assumption. Therefore, βx = 0 whenever x
102
is a lonely vertex. Since (β, r) is a band pair, each vertex vxi in ΓQ,c(β, r, ε) is incident
to precisely two edges, and by proposition 4.1.3, each such vertex is contained in exactly
one edge of color s if (x, s) ∈ X. There are βx vertices in the set {vxi }i=1,...,βx , of which
r(i(x, s))+r(o(x, s)) are incident to edges of color s. Therefore, βx = r(i(x, s))+r(o(x, s)).
On the other hand, suppose that βx = 0 whenever x is a lonely vertex, and βx =
r(i(x, s)) + r(o(x, s)) for all x. This means that for each (x, s) ∈ X, every vertex in
the set {vxi }i=1,...,βx is contained in exactly one edge of color s. Since βx 6= 0, and by
assumption x is not a lonely vertex, there is another element (x, s′) ∈ X with s′ 6= s
and, by assumption, βx = r(i(x, s′)) + r(o(x, s′)), so every vertex in the set {vxi }i=1,...,βx
is contained in exactly one edge of color s′. By proposition 4.1.3, each such vertex is
contained in at most two edges, and we have just shown that it is contained in at least two
edges. Since every vertex is contained in exactly two edges, the graph ΓQ,c(β, r, ε) consists
only of band components.
Proposition 5.1.2. Suppose that (β, r) is a band pair such that the generic module of
RepQ,c(β, r) is an indecomposable band. Then (nβ, nr) is a band pair and the generic
module of RepQ,c(nβ, nr) is a direct sum of n copies of VQ,c(β, r).
Proof. In the course of proving theorem 4.2.4, we showed that Ext1(VQ,c(β, r)µ, VQ,c(β, r)ν) =
0 when µ 6= ν. Therefore, RepQ,c(β, r)⊕ . . .⊕ RepQ,c(β, r) is a generic component in
RepQ,c(β) by Crawley Boevey-Schroer ([10]). But it contains an element V such that
rankkVa = r(a) (specifically V = VQ,c(nβ, nr)), so the generic module of RepQ,c(nβ, nr) is
a direct sum of n copies of VQ,c(β, r) as claimed.
We will now explore the Euler form on dimension vectors of generic band modules (refer
to section 2.1.1 for detailed definitions). We will denote by EA the matrix associated to
this bilinear form, so that for two vertices x, y, (EA)x,y =∑i≥0
(−1)i dim Ext1A(Sx, Sy). We
will show that if (β, r) is a band pair, then 〈〈β, β〉〉 = 0. This will be exploited in section
103
5.2 to determine the structure of some GIT quotients.
Suppose that kQ/Ic is a gentle string algebra. Recall that if s ∈ S is a color, then
Q0(s) is defined to be the set of vertices x ∈ Q0 such that (x, s) ∈ X. If x, y ∈ Q0(s), let
ds(x, y) be the length of the path of color s with endpoints x, y. Such a path clearly exists
since c−1(s) is a direct path, and x, y are vertices in this path. We can define a total order
on Q0(s) by x ≤s y if x appears before y in the path c−1(s). For the remainder of this
section, we fix kQ/Ic, a triangular gentle string algebra (i.e., such that Q has no oriented
cycles), and a sign function ε : X→ {±1}.
Proposition 5.1.3. Let EkQ/Ic = E be the Euler matrix for kQ/Ic. I.e., the matrix
associated to the bilinear form 〈〈−,−〉〉. Then
Ex,y =∑
{s∈S|{x,y}⊂Q0(s)x≤sy}
(−1)ds(x,y)
Proof. Notice that Sx is an up-and-down module, so we have already constructed a pro-
jective resolution of Sx in section 4.2. Namely, we take [x]+0 (resp. [x]−0 ) to be the arrow (if
it exists) with t([x]+0 ) = x and ε(x, c([x]+0 )) = 1 (resp. t([x]−0 ) = x and ε(x, c([x]−0 )) = −1).
Recursively define [x]±1i to be the arrow with t([x]±i ) = h([x]±1
i−1), c([x]±1i ) = c([x]±0 ). Taking
P (i) = Ph([x]+i ) ⊕ Ph([x]−i ) (where, if either doesn’t exist, the summand is suppressed), the
projective resolution of Sx is
Sx P (0)oo P (1)
[x]1i
[x]−11
oo . . .oo P (i− 1)oo P (i)
[x]1i
[x]−1i
oo . . .oo
Applying HomkQ/Ic(−, Sy) to the projective resolution above (which has finite length, since
kQ/I was taken to be of finite global dimension), we have the complex with 0 differential
and which, in degree i, is a vector space of dimension equal to the number of those ver-
104
tices h([x]±1i ) which are precisely y. The Euler characteristic of the complex is precisely∑
{s∈S|{x,y}⊂Q0(s)x≤sy}
(−1)ds(x,y).
Lemma 5.1.1. Suppose that β, r is an exact pair. Then for each s ∈ S and each y ∈ Q0(s),∑x∈Q0(s)
(−1)ds(x,y)βx = 0.
Proof. Suppose that V ∈ RepQ,c(β, r) is a module in the open set (so rankkVa = r(a) for
each a ∈ Q1). Consider the statement in terms of complexes. If x(s)0, x(s)1, . . . , x(s)l(s) are
the vertices incident to an arrow of color s so that x(s)i <s x(s)i+1, and a(s)1, . . . , a(s)l(s)
are the arrows of color s with h(a(s)i) = x(s)i, then the complex
0 // Vx(s)0
a(s)1 // Vx(s)1
a(s)2 // . . . // Vx(s)l(s)// 0
is exact, so has an Euler characteristic of 0. But the Euler characteristic of the above
complex isl(s)∑i=0
(−1)iβx(s)i . Notice that ds(x(s)i, x(s)j) = |j − i|, so indeed, the sum is equal
(up to sign change) to∑
x∈Q0(s)
(−1)ds(x,y)βx, so the latter expression is also zero.
We are now prepared to prove the main proposition.
Proposition 5.1.4. If there is a rank map r such that (β, r) is a band pair, then q(β) = 0.
Proof. Consider the symmetric form (α, β) = 〈〈α, β〉〉 + 〈〈β, α〉〉, and let E = E + ET be
its associated matrix. For any s ∈ S, let E(s) be the Q0 ×Q0 matrix with
E(s)x,y =
(−1)ds(x,y) if {x, y} ⊂ Q0(s), x <s y
0 otherwise
105
Notice that E = I +∑
s∈S E(s), so E = 2I +∑
s∈S E(s), where
E(s) := E(s) + E(s)T =
(−1)ds(x,y) if {x, y} ⊂ Q0(s), x 6= y
0 otherwise
By proposition 5.1.1, (β, r) is an exact pair. From the above description, we have that
(E(s)β)y =∑
x∈Q0(s)\y
(−1)ds(x,y)βx
=
∑x∈Q0(s)
(−1)ds(x,y)βx
− βy= −βy
where the last equality is by lemma 5.1.1. We are now prepared to calculate (β, β):
(β, β) = βT Eβ = 2βT Iβ +∑s∈S
βT E(s)β
= 2∑x∈Q0
β2x +
∑s∈S
βT (E(s)β)
= 2∑x∈Q0
β2x +
∑s∈S
∑y∈Q0(s)
−β2y .
By proposition 5.1.1, if y is a lonely vertex, then βy = 0, so if y is not a lonely vertex, there
are precisely two elements s ∈ S with y ∈ Q0(s). Therefore,
βT Eβ = 2∑x∈Q0
β2x +
∑y∈Q0
−2β2y
= 0.
Since q(β) = 〈〈β, β〉〉 = 12(β, β), this concludes the proof.
106
5.2 GIT Quotients
We now calculate the GIT-quotients for faithful band components of representation spaces.
Recall that a component of RepQ,c(β, r) is called faithful if the annihilator of its generic
module is precisely the ideal Ic. Throughout this section, kQ/Ic will denote a gentle string
algebra that is triangular, that is, Q has no oriented cycles. In particular, kQ/Ic has finite
projective dimension. We will say a pair (β, r) is indecomposable if the generic module in
RepQ,c(β, r) is indecomposable, and faithful if the component itself is.
Proposition 5.2.1. Suppose that kQ/Ic is a triangular gentle string algebra, and (β, r)
an indecomposable, faithful, band pair. Then SIQ,c(β, r)〈〈β,−〉〉 is two-dimensional.
Proof. Let us denote by χ the weight 〈〈β,−〉〉. Since (β, r) is faithful, SIQ,c(β, r)χ is
the span of the functions cV such that dimV = β and is of projective dimension 1 (see
proposition 2.3.2). From a corollary of Derksen-Fei ([12] corollary 2.6) it follows that the set
of modules of projective dimension 1 is irreducible, let us call it Zp.d.1. We have already seen
that the generic module in RepQ,c(β, r) is of projective dimension 1, so VQ,c(β, r) ⊂ Zp.d.1 ⊂
RepQ,r(β, r). In particular VQ,c(β, r) is dense in Zp.d.1. So we can view c? as a map Zp.d.1 →
SIQ,c(β, r)χ, and rephrase proposition 2.3.2 as saying that span{f ∈ image c?} = SIQ,c(β, r).
Since VQ,c(β, r) ⊂ Zp.d.1 is dense, the above span is precisely equal to span{cV | VQ,c(β, r)}.
Let us write Vµ for the indecomposable element in VQ,c(β, r) corresponding to µ ∈ k∗.
Claim. If µ 6= ν ∈ k∗, then cVµ and cVν are independent.
Indeed, recall from corollary 4.2.13 that cVµ(Vx) = µkxl(µ−x) for µ, x ∈ k∗. Therefore,
cVµ(Vµ) = cVν (Vν) = 0 while cVµ(Vν) 6= 0 cVν (Vµ) 6= 0.
Claim. cVγ ∈ span{cVµ , cVν} for every γ ∈ k∗.
107
Let
αµ =
(γ
µ
)k (γ − νµ− ν
)αν =
(γν
)k (γ − µν − µ
).
Then cVγ = αµcVµ + ανc
Vν . Indeed, we need only check this equality on Vz for all z ∈ k∗
since each function is semi-invariant of the same weight, and the union of the orbits of
these modules is dense in RepQ,c(β, r).
αµcVµ(Vz) + ανc
Vν (Vz) =
(γ
µ
)k (γ − νµ− ν
)µkzl(µ− z) +
(γν
)k (γ − µν − µ
)νkzl(ν − z)
=γkyl
µ− ν ((γ − ν)(µ− z)− (γ − µ)(ν − z))
=γkyl
µ− ν (µ− ν)(γ − z)
= γkyl(γ − z)
= cVγ (Vz)
Moreover, these functions are algebraically independent, as shown in the following
proposition.
Proposition 5.2.2. Suppose that f(x, y) is a polynomial function such that f(cVµ , cVν ) = 0.
Then f(x, y) = 0.
Proof. We can also assume that f is homogeneous of minimal degree (since cVµ , cVν are of
the same weight, any relation must be a sum of homogeneous relations), let us call that
108
degree n. Write f(x, y) =∑
i+j=n αi,jxiyj. Notice that
f(cVµ , cVν )(Vµ) =∑i+j=n
αi,j(cVµ(Vµ))i(cVν (Vµ))j
= α0,n(νkµl(ν − µ))n = 0
So α0,n = 0, and symmetrically αn,0 = 0. Therefore,
f(x, y) = xy
(∑i+j=n
αi,jxi−1yj−1
).
Clearly cVµ ·cVν 6= 0 by evaluating on any point Vx with x 6= µ, ν, so f =∑
i+j=n αi,jxi−1yj−1
is a relation f(cVµ , cVν ) = 0. This contradicts minimality of the relation f .
In particular, the GIT-quotient (by PGL(β)) of RepQ,c(β, r)ssχ is a projective space.
Corollary 5.2.3. The ring R =⊕
n≥0 SIQ,c(β, r)n〈〈β,−〉〉 is isomorphic to the polynomial
ring in two variables cVµ , cVν . Therefore, Y = Proj(R) is isomorphic to P1.
109
Bibliography
[1] S. Abeasis, A. Del Fra, and H. Kraft. The geometry of representations of Am. Math.
Ann., 256(3):401–418, 1981.
[2] I. Assem, T. Brustle, G. Charbonneau-Jodoin, and P-G Plamondon. Gentle algebras
arising from surface triangulations. Algebra Number Theory, 4(2):201–229, 2010.
[3] I. Assem, D. Simson, and A. Skowronski. Elements of the representation theory of
associative algebras. Vol. 1, volume 65 of London Mathematical Society Student Texts.
Cambridge University Press, Cambridge, 2006. Techniques of representation theory.
[4] G. Bobinski and G. Zwara. Normality of orbit closures for Dynkin quivers of type An.
Manuscripta Math., 105(1):103–109, 2001.
[5] M. C. R. Butler and C. M. Ringel. Auslander-Reiten sequences with few middle terms
and applications to string algebras. Comm. Algebra, 15(1-2):145–179, 1987.
[6] G. Cerulli Irelli. Quiver Grassmannians associated with string modules. J. Algebraic
Combin., 33(2):259–276, 2011.
[7] C. Chindris. Geometric characterizations of the representation type of hereditary
algebras and of canonical algebras. Adv. Math., 228(3):1405–1434, 2011.
[8] W. Crawley-Boevey. Maps between representations of zero-relation algebras. J. Alge-
bra, 126(2):259–263, 1989.
110
[9] W. Crawley-Boevey. Tameness of biserial algebras. Arch. Math. (Basel), 65(5):399–
407, 1995.
[10] W. Crawley-Boevey and J. Schroer. Irreducible components of varieties of modules.
J. Reine Angew. Math., 553:201–220, 2002.
[11] C. De Concini and E. Strickland. On the variety of complexes. Adv. in Math., 41(1):57–
77, 1981.
[12] H. Derksen and J. Fei. General presentations of algebras. preprint
arxiv:math.RA/0911.4913, 2009.
[13] H. Derksen and J. Weyman. Semi-invariants for quivers with relations. J. Algebra,
258(1):216–227, 2002. Special issue in celebration of Claudio Procesi’s 60th birthday.
[14] H. Derksen and J. Weyman. The combinatorics of quiver representations. preprint
arxiv:math.RT/0608288, 2006.
[15] P. Donovan and M. R. Freislich. The representation theory of finite graphs and as-
sociated algebras. Carleton University, Ottawa, Ont., 1973. Carleton Mathematical
Lecture Notes, No. 5.
[16] P. Dowbor and A. Mroz. On a separation of orbits in the module variety for string
special biserial algebras. J. Pure Appl. Algebra, 213(9):1804–1815, 2009.
[17] J. A. Drozd. Tame and wild matrix problems. In Representation theory, II (Proc.
Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), volume 832 of Lecture
Notes in Math., pages 242–258. Springer, Berlin, 1980.
[18] S. Fomin, M. Shapiro, and D. Thurston. Cluster algebras and triangulated surfaces.
I. Cluster complexes. Acta Math., 201(1):83–146, 2008.
111
[19] W. Fulton. Young tableaux, volume 35 of London Mathematical Society Student Texts.
Cambridge University Press, Cambridge, 1997.
[20] P. Gabriel. Unzerlegbare Darstellungen. I. Manuscripta Math., 6:71–103; correction,
ibid. 6 (1972), 309, 1972.
[21] P. Gabriel. Finite representation type is open. In Proceedings of the International
Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974),
Paper No. 10, Ottawa, Ont., 1974. Carleton Univ.
[22] P. Gabriel. The universal cover of a representation-finite algebra. In Representations of
algebras (Puebla, 1980), volume 903 of Lecture Notes in Math., pages 68–105. Springer,
Berlin, 1981.
[23] V. G. Kac. Infinite root systems, representations of graphs and invariant theory.
Invent. Math., 56(1):57–92, 1980.
[24] A. D. King. Moduli of representations of finite-dimensional algebras. Quart. J. Math.
Oxford Ser. (2), 45(180):515–530, 1994.
[25] H. Kraft. Geometrische methoden in der invariantentheorie. Aspects der Mathematik,
1984.
[26] W. Kraskiewicz and J. Weyman. Generic decompositions and semi-invariants for
string algebras. preprint arxiv:math.RT/1103.5415, 2011.
[27] D. Labardini-Fragoso and G. Cerulli Irelli. Quivers with potentials associated to
triangulated surfaces, part iii: Tagged triangulations and cluster monomials. preprint
arxiv:math.RT/1108.1774, 2011.
[28] V. B. Mehta and V. Trivedi. The variety of circular complexes and F -splitting. Invent.
Math., 137(2):449–460, 1999.
112
[29] D. Mumford. Picard groups of moduli problems. In Arithmetical Algebraic Geometry
(Proc. Conf. Purdue Univ., 1963), pages 33–81. Harper & Row, New York, 1965.
[30] G. Musiker, R. Schiffler, and L. Williams. Positivity for cluster algebras from surfaces.
Adv. Math., 227(6):2241–2308, 2011.
[31] L. A. Nazarova. Representations of quivers of infinite type. Izv. Akad. Nauk SSSR
Ser. Mat., 37:752–791, 1973.
[32] C. M. Ringel. On generic modules for string algebras. Bol. Soc. Mat. Mexicana (3),
7(1):85–97, 2001.
[33] A. Schofield. Semi-invariants of quivers. J. London Math. Soc. (2), 43(3):385–395,
1991.
[34] J. Schroer. On the infinite radical of a module category. Proc. London Math. Soc. (3),
81(3):651–674, 2000.
[35] J. Schroer. On the Krull-Gabriel dimension of an algebra. Math. Z., 233(2):287–303,
2000.
[36] A. Skowronski and J. Waschbusch. Representation-finite biserial algebras. J. Reine
Angew. Math., 345:172–181, 1983.
[37] A. Skowronski and J. Weyman. Semi-invariants of canonical algebras. Manuscripta
Math., 100(3):391–403, 1999.
[38] A. Skowronski and J. Weyman. The algebras of semi-invariants of quivers. Transform.
Groups, 5(4):361–402, 2000.
[39] D. Voigt. Induzierte Darstellungen in der Theorie der endlichen, algebraischen Grup-
pen. Lecture Notes in Mathematics, Vol. 592. Springer-Verlag, Berlin, 1977. Mit einer
englischen Einfuhrung.
113
[40] B. Wald and J. Waschbusch. Tame biserial algebras. J. Algebra, 95(2):480–500, 1985.
[41] J. Weyman. Cohomology of vector bundles and syzygies, volume 149 of Cambridge
Tracts in Mathematics. Cambridge University Press, Cambridge, 2003.
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GRADUATE SCHOOL APPROVAL RECORD
NORTHEASTERN UNIVERSITY
Dissertation Title: Semi-Invariants for Gentle String Algebras
Author: Andrew T. Carroll
Department: Mathematics
Approved for Dissertation Requirements of the Doctor of Philosophy Degree
Dissertation Committee
Jerzy Weyman, Advisor Date
Calin Chindris Date
Donald King Date
Gordana Todorov Date
Head of Department
Richard Porter Date
Graduate School Notified of Acceptance
Director of Graduate Student Services Date
116
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