ABSTRACT
Title of Dissertation: MUTATION INVARIANT FUNCTIONS
ON CLUSTER ENSEMBLES
ASSOCIATED WITH SURFACES
Dani Kaufman
Doctor of Philosophy, 2021
Dissertation Directed by: Professor Christian Zickert
Department of Mathematics
We define the notion of an invariant function on a cluster ensemble with respect
to a group action of the cluster modular group on its associated function fields. We
realize many examples of previously studied functions as elements of this type of
invariant ring and give many new examples. We construct invariants for cluster
algebras associated with surfaces using hyperbolic geometry, Teichmu?ller theory and
skein algebras of surfaces. We complete a classification of them for surface ensembles
for the action of Dehn twists, and generalize this classification to the non-surface
mutation finite setting. We use this classification to answer some questions about
the structure of affine cluster algebras, to construct a correspondence between A and
X invariants, and to propose an explanation for why many different computations
of canonical bases of cluster algebras agree.
MUTATION INVARIANT FUNCTIONS ON
CLUSTER ENSEMBLES ASSOCIATED WITH SURFACES
by
Dani Kaufman
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2021
Advisory Committee:
Professor Christian Zickert, Chair/Advisor
Professor William Goldman
Professor Niranjan Ramachandran
Professor James Schafer
Professor William Gasarch, Dean?s Representative
? Copyright by
Dani Kaufman
2021
Acknowledgments
I would like to thank all my friends and family who supported me though
this journey. I would especially like to thank my advisor Christian for his constant
encouragement and excitement about my work, my friend Zack who was always
available to talk math and selflessly work out my problems, and I would especially
like to thank my good friend and housemate Katherine, who kept me sane during
the preparation of this document.
ii
Table of Contents
Acknowledgements ii
Table of Contents iii
List of Figures v
List of Mathematical Symbols vii
Chapter 1:Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Canonical basis . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 New ingredients and their inspiration . . . . . . . . . . . . . . 6
1.2 Summary of Main Results . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Structure of this Thesis and Main Results . . . . . . . . . . . . . . . 11
Chapter 2:Preliminaries on Cluster Ensembles 14
2.1 Quivers and Mutations . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Mutation of quivers . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Folding of quivers . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.4 Important mutation classes of quivers . . . . . . . . . . . . . 19
2.2 Positive spaces and Cluster Ensembles . . . . . . . . . . . . . . . . . 21
2.2.1 The A space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 The X space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Cluster Ensembles . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 The Mutation Structures of a Cluster Ensemble . . . . . . . . . . . . 26
2.3.1 The cluster modular group . . . . . . . . . . . . . . . . . . . . 27
2.3.2 Action of the cluster modular group on functions . . . . . . . 32
2.3.3 Folding of cluster ensembles . . . . . . . . . . . . . . . . . . . 34
2.4 Preliminaries on Hyperbolic Geometry and Teichmu?ller spaces . . . . 36
2.4.1 Hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.2 Hyperbolic structures on surfaces . . . . . . . . . . . . . . . . 38
2.4.3 Teichmu?ller space . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Coordinates on Teichmu?ller Space . . . . . . . . . . . . . . . . . . . . 42
2.5.1 Teichmu?ller X space . . . . . . . . . . . . . . . . . . . . . . . 42
iii
2.5.2 Decorated Teichmu?ler space . . . . . . . . . . . . . . . . . . . 45
2.5.3 Representations from points in Teichmu?ller spaces . . . . . . . 46
2.6 Cluster Ensembles associated with Surfaces . . . . . . . . . . . . . . . 48
2.6.1 Quivers from triangulations . . . . . . . . . . . . . . . . . . . 49
2.6.2 Action of the mapping class group . . . . . . . . . . . . . . . . 53
2.6.3 Skein algebras of surfaces . . . . . . . . . . . . . . . . . . . . . 54
Chapter 3:Examples of Mutation Invariant Functions 58
3.1 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Invariants for Doubly Extended Quivers . . . . . . . . . . . . . . . . 64
3.3 Examples from Somos Sequences . . . . . . . . . . . . . . . . . . . . 69
Chapter 4:First Properties and Construction of Invariants 72
4.1 Basic properties of mutation invariant functions . . . . . . . . . . . . 72
4.1.1 Trivial Invariants . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.1.2 Invariants of Subensembles . . . . . . . . . . . . . . . . . . . . 73
4.2 Invariants Of Ensembles Associated With Surfaces . . . . . . . . . . . 78
4.2.1 Calculating trace functions . . . . . . . . . . . . . . . . . . . . 79
4.2.2 Other types of surface invariants . . . . . . . . . . . . . . . . 80
4.3 Sequences of A coordinates from Dehn twists . . . . . . . . . . . . . . 82
Chapter 5:Surface Invariants 86
5.1 Invariants of Surface Cluster Ensembles . . . . . . . . . . . . . . . . . 86
5.2 Invariants on Affine, Doubly Extended and Exotic Ensembles . . . . . 100
5.2.1 Tp,q,r Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.2 Invariants for partial Dehn twists . . . . . . . . . . . . . . . . 103
Chapter 6:Further Problems and Applications 104
6.1 Affine A coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2 Invariants for Cluster Dehn Twists on General Mutation Finite Types
via folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3 Further Geometric structures related to invariants . . . . . . . . . . . 107
6.4 Correspondence between A and X Invariants . . . . . . . . . . . . . . 107
6.5 Laurent Property of Invariants . . . . . . . . . . . . . . . . . . . . . . 108
6.6 Coefficients and Canonical Bases . . . . . . . . . . . . . . . . . . . . 110
Appendix A:Dynkin Diagrams 114
A.1 Finite Dynkin Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 114
A.2 Affine Dynkin Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.3 Doubly-Extended Dynkin Diagrams . . . . . . . . . . . . . . . . . . . 116
Bibliography 117
iv
List of Figures
1.1 An annulus with one marked point on each boundary component. . . 11
2.1 Two quivers of type G?2 . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The exceptional (non surface) quivers. . . . . . . . . . . . . . . . . . 20
2.3 The exotic non-surface finite mutation quivers . . . . . . . . . . . . . 20
2.4 The five A seeds with different cluster variables. . . . . . . . . . . . . 26
2.5 Choices of quiver isomorphism classes in the mutation class of a quiver
of type A3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 The cluster modular groupoid of the A3 cluster ensemble. A selection
of non-identity maps are shown. . . . . . . . . . . . . . . . . . . . . . 32
2.7 The conjugacy classes of non-identity elements in PSL(2,R) . . . . . 37
2.8 The surface S along with its developing image in H2. . . . . . . . . . 41
2.9 The fatgraph for a triangulation of a punctured digon. . . . . . . . . 48
2.10 Untagged vs tagged arcs in a punctured digon. . . . . . . . . . . . . . 50
2.11 Generating the quiver for the surface S and ?. The two middle
arrows are removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.12 The skein relations for arcs and curves on a marked surface. . . . . . 55
2.13 Computing the element ?a1a2 in Sk(S). . . . . . . . . . . . . . . . . . 56
3.1 Quiver of type A?1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 The surface S, along with a choice of triangulation. ? is the generator
of the fundamental group and the mapping class ? corresponds to a
Dehn twist about ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 The quiver associated with a marked annulus. . . . . . . . . . . . . . 61
3.4 Qn?cycle. This quiver is simply an n cycle of nodes and arrows. . . . . 63
3.5 The surface S1,0,1,0 with a choice of triangulation and associated quiver. 64
3.6 Quivers associated with doubly extended Dynkin diagrams with 3
nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
(3,3)
3.7 Quiver of type G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
(1,1)
3.8 Quivers of type D4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.9 Quivers for the Somos 4, 5 and 6 sequences. . . . . . . . . . . . . . . 70
4.1 Quiver of type A2n+1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Quiver with double edge and oriented cycles. . . . . . . . . . . . . . . 78
4.3 The arcs near a boundary with one marked point. . . . . . . . . . . . 79
v
4.4 An ideal hyperbolic triangle with horocycle segments around its three
vertices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 A pair of pants with one marked point before and after performing a
Dehn twist about the bottom boundary . . . . . . . . . . . . . . . . . 83
4.6 The Skein relations obtained from computing a0K. This is a zoomed
in view near the points where the crossings occur . . . . . . . . . . . 84
5.1 The construction of arcs which excise a given closed curve. . . . . . . 87
5.2 The blue closed curve is genus 0, and the red closed curve is genus 1. 88
5.3 The triangulation and associated quiver when g = 1 quiver . . . . . . 90
5.4 The triangulation for when g > 1. Each of the g handles is associated
with 4 arcs of the triangulation as figure 5.3, and each of non-handle
pairs of pants is associated with 2 arcs if the triangulation. . . . . . . 91
5.5 The quiver Q when g > 1 . . . . . . . . . . . . . . . . . . . . . . . . 91
5.6 Two arcs in ? along with the pants that they live in. . . . . . . . . . 93
5.7 The two possibilities for the arrangement of arcs a1 and ai . . . . . . 95
5.8 The collections of arc and closed curves for the calculation of W in
the separate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.9 General form of a Tp,q,r quiver. . . . . . . . . . . . . . . . . . . . . . . 102
6.1 A?1 quiver with principal coefficients before and after applying ?. . . 111
A.1 Simply Laced Finite Dynkin Diagrams . . . . . . . . . . . . . . . . . 114
A.2 Folded Finite Dynkin Diagrams . . . . . . . . . . . . . . . . . . . . . 115
A.3 Simply Laced Affine Dynkin Diagrams . . . . . . . . . . . . . . . . . 115
A.4 Folded Affine Dynkin Diagrams . . . . . . . . . . . . . . . . . . . . . 116
A.5 Simply Laced Doubly Extended Dynkin Diagrams . . . . . . . . . . . 116
vi
List of Mathematical Symbols
F A fixed coefficient field of characteristic 0
(R>0) The positive real numbers
GL(n,Z) The general linear group with elements in Z
PSL(n,Z) The projective linear group
H2 Hyperbolic space
Sk(S) The skein algebra of the surface S
Gr(n, k) The Grassmannian of k planes in n space
A?n The affine root system of type A
(1,1)
A1 The affine extended root system of type A
i The seed of a cluster ensemble
MCG (S) The mapping class group of a surface
PMCG (S) The puncture preserving mapping class group
Dev The developing map
Aut The group of automorphisms of an object
Pos The category of positive varieties
Grp The category of groups
Tr The trace of a matrix
bxc The floor function
dxe The ceiling function
sgn The sign of a real number
Gr(n, k) The Grassmannian of k planes in n space
vii
Chapter 1: Introduction
Cluster algebras were first defined in the early 2000s by S. Fomin and A.
Zelevinsky [1?4] as an explanation of certain functional recurrences called Y-systems
and as a way of organizing particular bases in the theory of quantum groups. They
have since enjoyed many applications to other areas such as Teichmu?ller theory [5,6],
representation theory [7,8], supersymmetric Yang-Mills theory [9], and recently they
were used in a vital way in a proof of Zagier?s conjecture in weight 4 [10].
Essentially, a cluster algebra provides an atlas of coordinate charts on a vari-
ety, called a ?cluster variety?, along with transition maps which satisfy a number
of interesting and important properties. The coordinate variables of each chart are
called the ?cluster variables?, and the transition maps are called ?mutations? be-
tween the clusters. The data that controls the mutations is encoded by a ?quiver?
which underlies each cluster. The cluster algebra is the Z algebra generated by all
of the cluster variables, and the cluster variety is the algebraic variety obtained by
gluing an algebraic torus for each cluster along the mutation maps.
One often finds that there are sequences of mutations which returns you to
a cluster with the same underlying quiver, but with new cluster variables. The
collection of such transformations forms a group called the ?cluster modular group?.
1
When the cluster algebra at hand has finitely many clusters, this group is finite and
one may easily compute all of the information that the cluster algebra encapsulates
directly. However, most cluster algebras are not ?finite type? and have infinitely
many clusters.
This thesis is an attempt to introduce and study a collection of functions
assigned to any cluster variety that are invariant under specified subgroups of the
cluster modular group. It is our hope that these functions will encode important
information about infinite sequences of mutations, and that they will in some ways
behave like cluster variables which live outside of the reach of regular mutation.
1.1 Motivation
Important examples of cluster algebras arise in a natural way from coordinate
rings of double Bruhat Cells, flag varieties and Grassmannians [3, 11]. These typi-
cally give cluster algebras of ?Geometric Type? meaning that the combinatorial data
underlying the algebra has a simple description in terms of quivers and mutations.
In their seminal papers [5, 12, 13], Fock and Goncharov construct algebraic
versions of higher Teichmu?ller spaces consisting of particular types of representations
of surface groups into semi simple split real Lie groups. They explicitly construct all
such representations and show that they may be realized as the positive real valued
points of a pair of cluster varieties called a cluster ensemble. These cluster ensembles
have a number of novel properties, e.g. there is a duality between a cluster ensemble
associated with the surface S and Lie group G and the ensemble associated with
2
the same surface and the Langlands dual group.
Most of these examples are not associated with cluster algebras that have
finitely many clusters.
Example 1.1.1. The coordinate rings of the affine cones over the Grassmannians of
k planes in n space, Gr(k, n), have a cluster algebra structure on them, as shown by
Scott [11]. These algebras only have finitely many clusters when (k?2)(n?k?2) < 4.
One notices from this example, that there is not an obvious change in the
structure of the Grassmannian when we consider one associated with a finite type
cluster algebra and one associated with an infinite type cluster algebra. This leads
us to the problem of determining what information encoded in the structure of finite
type cluster algebras carry over into the infinite type setting, and what information
is lost or modified. Mutation invariant functions are examples of new phenomenon
appearing in the infinite type setting, and we find that they appear naturally as a
part of the solution to this problem.
1.1.1 Canonical basis
An important problem which arises in the study of all forms of cluster algebras,
is the task of finding a ?canonical basis? of a cluster algebra. Such a basis gives
a basis of the cluster algebra as a vector space and satisfies some particularly nice
positivity properties. For cluster algebras with finitely many clusters, this basis
simply consists of the cluster variables [14]. When the cluster algebra has infinitely
many clusters, there are usually basis elements that do not appear in any cluster;
3
this is our primary example of information encoded in finite type cluster algebras
that must be modified in the infinite type setting.
Canonical bases have been studied by Zelevinsky in [15] for an ?affine A?1
type? cluster algebra, by Muskier, Shiffler, and Williams in [16] for cluster algebras
associated with surfaces, and by Geiss, Leclerc and Schro?er in [17] for more general
algebras.
Through their duality map, Fock and Goncharov construct a canonical basis
for the cluster ensemble associated with surface group representations into G =
PSL(n,R). Gross, Hacking, Kontsevich and Keel construct canonical basis for clus-
ter ensembles satisfying some technical conditions in [18]. Their basis, called the
?theta basis?, is constructed using entirely different machinery than that of Fock
and Goncharov.
Our notion of mutation invariant function produces elements of canonical bases
in many cases.
Example 1.1.2. Let C be the cluster algebra associated with an affine A?1 quiver, as
studied in [15]. We describe this algebra in examples 1.2.1 and 3.1.1. We write {ai}
a2+a2+1
for the collection of cluster variables found by mutations. The invariant F = 0 1
a0a1
is an example of an element of a canonical basis of a cluster algebra which is not
a cluster variable. Let Ck(x) be the normalized Chebychev polynomial defined by
Ck(?+ ?
?1) = ?k + ??k. A canonical basis of this cluster algebra is given by
{ }
B = {ai|i ? Z}, {Ck(F )|k ? Z>0} . (1.1)
This same basis is computed in many different ways, and it is not clear why
4
the exact same function F appears each time. This function is realized as a cluster
character in [8, 17], a trace function in [5], a theta function in [18], an element of
the bracelet basis of a surface skein algebra in [16, 19] among others. Viewing the
functions associated with each of these constructions individually, we may see that
they always produce manifestly mutation invariant functions. Our classification
theorem essentially says that they must all be functions of the same invariant. We
conjecture (see 6.6.1) that there is a canonical basis of mutation invariants which
unifies these different constructions.
1.1.2 Other examples
Several other examples of cluster ensemble invariants have already been noted
in the literature, but they have not been studied in a unified way. Of these examples,
the invariant of the Markov quiver, shown in example 3.1.3, is probably the most well
known and studied. This function is essentially the classical Markov Diophantine
equation x2+y2+z2 = 3xyz, and its mutation invariance encodes the transformation
2 2
of Markov triples (x, y, z)? (x, y, x +y ) .
z
Some examples of similar Diophantine equations arising from cluster algebras
were studied in [20], and we recall such an equation in example 3.2.2. The invariants
of the Somos 4 and 5 sequences studied in [21] are further examples, shown in
example 3.3.1. The invariants in each of these examples provide useful tools for
understanding the number theoretical properties of each sequence.
We also generalize the invariant of the Markov quiver in a different way. This
5
quiver can be seen to be related to the Dynkin diagram of an ?affine-extended?
(1,1)
or ?Elliptic? root system of type A1 introduced by Saito [22]. We call quivers
associated to these types of root systems ?doubly extended? type quivers and we
(1,1) (3,3)
show invariants for types D4 and G2 in examples 3.2.4 and 3.2.3.
1.1.3 New ingredients and their inspiration
All of these previously studied functions are invariant under the action of a
single mutation. The primary new ingredient of this paper is the introduction of
an action of cluster modular group on the function fields of the ensemble. We show
that each of the previously known examples can be realized as elements of invariant
rings in regards to this group action. This will allow us to generalize these examples
to invariants associated to more complicated mutation sequences and subgroups of
the cluster modular group.
We hope that this theory provides a method to understand and tame the
infiniteness of certain cluster ensembles. When the cluster modular group is infi-
nite, we naturally have some infinite repeating structures underlying the exchange
complex of the associated ensemble. This is often encoded by the cluster modular
group; when there are cluster modular group elements of infinite order, we have
sequences of mutations which send us to clusters which have the same underlying
quiver, but new cluster variables. Invariants for these elements provide us with ra-
tional functions whose value and formula is the same on each cluster obtained along
this mutation sequence.
6
Often, we may use the invariance of a function to study the limiting behavior
of the cluster variables found along a mutation path. Recently, a cluster algebraic
interpretation of some of the ?symbols? of 8 particle N = 4 SYM scattering am-
plitudes has been studied in [23?25]. These symbols are related to the limiting
structure of an affine cluster algebra and were studied using an invariant of the A?1
affine cluster algebra. We briefly show a similar analysis of this limiting behavior in
example 3.1.1.
This analysis can be rephrased in terms of hyperbolic geometry, as example
3.1.1 shows. The A?1 cluster ensemble is associated with a hyperbolic structure on an
annulus, and the relevant mutation path corresponds to a Dehn twist on the annulus.
We generalize this analysis to a pair of pants in chapter 4, and this becomes a critical
ingredient in the proof of our main theorems.
The key ingredient towards understanding the limiting behavior is the unquan-
tized skein algebra associated with a marked surface studied in the context of cluster
algebras by [19,26]. We find that sequences of cluster variables which appear along
mutation paths corresponding to Dehn twists always satisfy linear recurrences, and
we write these recurrences explicitly in terms of elements of the skein algebra.
1.2 Summary of Main Results
Our primary result is the classification of mutation invariant functions asso-
ciated with some particularly simple mutation sequences on cluster varieties which
can be built from surfaces. The main tools for this classification will come from the
7
geometry and topology underlying these surfaces.
Teichmu?ler spaces of surfaces provide some of the simplest and most important
examples of cluster varieties. For technical reasons the surfaces we consider are
allowed to have boundary components and punctures, but they must have at least
one marked point on each boundary and the total number of marked points and
punctures must be more than 1.
Given such a surface, S, one may consider the Teichmu?ler space T (S), which
we think of as the moduli space of hyperbolic structures on S. These structures are
required to have cusps at each puncture and marked point. Hyperbolic structures
on a surface encode a representation ? : ?1(S)? PSL(2,Z) called the ?monodromy
representation?. Each element ? ? ?1(S) is mapped to a matrix called the mon-
odromy operator of ?.
One may construct a cluster variety associated with S. The main topological
ingredients are ?arcs? on S which are non self intersecting isotopy classes of paths
between marked points and punctures on S. We can construct a cluster variety, AS,
where the cluster variables correspond to arcs, the clusters correspond to triangula-
tions of S and the mutations corresponds to ?flips? of triangulations. The positive
real valued points of AS parameterize points in T (S) along with a ?decoration?
consisting of horocycles around each cusp. The mapping class group is naturally a
subgroup of the cluster modular group of this cluster variety. Moreover, the mon-
odromy representations associated with points of T (S) can be expressed rationally
in terms of the cluster variables, and we call Tr(?(?)) the ?trace function? associated
with ?.
8
We will focus on classifying invariants for particular types of mutation se-
quences, namely those that correspond to Dehn Twists. There is a natural collec-
tion of arcs and closed curves on S which are invariant after applying a Dehn twist
about ?, namely those arcs and closed curves which do not intersect ?. The cluster
variables and trace functions associated with these arcs and curve are examples of
invariant functions on AS. Our main theorem 5.1.1 implies in this case that
Theorem 1.2.1. The ring of invariant rational functions for the action of a Dehn
twist about ? on AS is generated by traces of monodromy operators on S associated
with closed curves that do not intersect ?, and cluster variables associated with arcs
that do not intersect ?.
This theorem is easily rephrased in terms of the skein algebra of the surface.
The primary method for this will arise from the geometric interpretation of many
cluster ensembles coming from Teichmu?ller theory, hyperbolic geometry, and skein
algebras of surfaces.
Corollary 1.2.1. The invariant ring for a Dehn twist about ? on A is exactly the
the subalgebra of the skein algebra of S consisting of elements corresponding to skeins
which do not intersect ?
Example 1.2.1. We will illustrate this theorem with an example. We let S be an
annulus with one marked point on each boundary component, as shown in figure
1.1. There are infinitely many clusters and cluster variables in AS. There are two
cluster variables associated with the two boundary arcs, and these appear in every
cluster. There are infinitely many possible interior arcs, and each cluster contains
9
two of them which do not cross. These arcs may be identified by their winding
number around the closed curve ?.
We write an for the cluster variable with winding number n, and write c, d
for the cluster variables for the outer and inner arcs. Our clusters and mutations
(omitting the variables c, d) look like
. . . (a?1, a0) (a0, a1) (a1, a2) (a2, a3) . . . . (1.2)
The formula for cluster variable mutation relates the new cluster variable obtained
by flipping an arc to the cluster variables in the original cluster. Explicitly, we have
a2n + cdan+1 = (1.3)
an?1
and the cluster algebra is given by
Z[ai]. (1.4)
Every rational function in the cluster variables can be written as a rational function
in {a0, a1, c, d}.
The cluster modular group of this cluster algebra is the group ?S = Z. This
corresponds to the mapping class group of S, and we write ? for the generator that
is a clockwise Dehn twist about the arc ? on S. There is an action of ?S on the
rational functions of AS. This is given by
{ } { a
2 + cd
?( a0, a1, c, d ) = a1,
1 , c, d}. (1.5)
a0
One may compute the trace function associated with ? and find that
a20 + a
2 + cd
Tr(?(?)) = 1 := F (a0, a1, c, d). (1.6)
a0a1
10
Figure 1.1: An annulus with one marked point on each boundary component.
This function is invariant under the action of ?S since
a22 +cd 2 a
2+cd
a1 + (
1 ) + cd (a2
1
a 1 + cd)
1 + 2 2
a2 a
?(F ) = 0 = 0 = 0
+ a1 + cd
2 a = F. (1.7)a +cd 2 1
a 1 (a1 + cd) a a1 a 0 1a 00
Theorem 1.3.1 implies that
F(a0, a1, c, d)?S = F(F, c, d) (1.8)
where F is our fixed coefficient field.
1.3 Structure of this Thesis and Main Results
Chapter 2 reviews the technical details behind cluster ensembles and Te-
ichmu?ler theory, and introduces the notion of a mutation invariant function. Chapter
3 shows many examples of invariant functions in detail. In chapter 4, we prove some
11
foundational results about invariants, show geometric constructions of them, and
use them to compute sequences of cluster variables.
Chapter 5 contains the proof of our main theorem:
Theorem 1.3.1 (Theorem 1). Let S be a marked surface, and let (AS,XS) be the
cluster ensemble associated with S. Let ? be the cluster modular group element
corresponding to a Dehn twist about ?, a simple closed curve on S.
1. The ring F(X )???S is generated by traces of monodromy operators of excised
closed curves on S and invariant X coordinates for an excising triangulation
of ?.
2. The ring F(AS)??? is generated by traces of monodromy operators of excised
closed curves on S and invariant A coordinates of an excising triangulation of
?.
We also extend this to cluster ensembles associated with general mutation
finite quivers and trivial coefficients:
Theorem 1.3.2. Let Q be a Tp,q,r, X6 or X7 Quiver of figures 5.9 or 2.3 and let
(A,X ) be the cluster ensemble associated with Q. Let ? = {1, (12)} ? ?Q be a cluster
Dehn twist in cluster modular group of (A,X ). Let x3, . . . , xm be the X coordinates
associated with nodes that are connected to nodes 1 and 2.
2
1. The invariant ring F(X )???S is generated by the function G(x , x ) = (x2(x1+1)+1)1 2 ,x1x2
x3(x2(x1 + 1) + 1), . . . , xm(x2(x1 + 1) + 1) and the remaining X coordinates
not connected to nodes 1 and 2.
12
?
2. The Invariant ring F(A )???S is generated by ?? G and A coordinates associ-
ated to nodes other than node 1 and 2.
Chapter 6 discusses some corollaries of the main theorems and some conjec-
tures about the general nature of mutation invariants. We prove a conjecture of [27]
on the structure of ?affine A coordinates?
We give strong evidence through an abundance of examples of a deeper theory
underlying the existence and structure of cluster ensemble invariants. We show in
each example a correspondence between invariants on the A and X spaces via de-
nominator vectors. We prove this correspondence in the surface case. We also show
evidence that there should be a basis of A invariants such that the cluster modular
group acts on this basis by positive Laurent polynomials. This is a generalization
of the Laurent phenomenon of a cluster algebra. These conjectures together should
be related to the duality conjectured in [5] and proved in [18].
We strongly suspect that the existence of invariants will always be related to
some other important manifestation of the ensemble, be it geometric or number
theoretic.
This thesis is primarily a continuation and restructuring of the author?s pre-
vious work initiated in [27].
13
Chapter 2: Preliminaries on Cluster Ensembles
We will recall the basic notions of a cluster ensemble introduced by Fock and
Goncharov in [12]. This consists of a pair of positive spaces (A,X ) along with a
map ? from A to X , a notion of seeds and mutations, and a group ? that acts by
automorphisms of the entire structure. We will, however, simplify these definitions
to emphasize a more concrete and computational framework in which to introduce
and study invariants. We introduce an action of ? on the field of rational functions
on each of the A and X spaces. Our notion of an invariant function will be in regards
to this group action. We will also review the notions of a hyperbolic structure on a
surface, the Teichmu?ller space of a surface, and the skein algebra of a surface and
show how these are naturally related to cluster ensembles.
2.1 Quivers and Mutations
Our first simplification is to only consider cluster ensembles of ?geometric?
type, meaning that we can use quivers to define our ensembles.
14
2.1.1 Quivers
Definition 2.1.1. A quiver is a directed and weighted graph with no self loops or
2 cycles. We think of a quiver as a graphical representation of a matrix, M , called
the exchange matrix which has entries [ij] equal to the number of arrows from node
i to node j. We denote the set of nodes of Q by N(Q) and we usually refer to them
by their index in this set.
We will allow non-symmetrically weighted arrows between nodes, to account
for non skew-symmetric exchange matrices. These arrows are labeled with a pair of
weights. In this case we require that the exchange matrix is skew-symmetrizable,
meaning that there is a diagonal matrix, D, such that MD?1 is skew-symmetric.
The matrix D associates to each node a multiplier, di.
In the various diagrams of quivers in the paper, we label the multipliers of our
nodes as superscripts in brackets and label the edges by the weights, where no label
means weight 1, a double arrow means weight 2, a single label means a symmetric
weight and a pair of weights for an arrow from node i to node j is ? 1ji, ij .
Definition 2.1.2. We will call a quiver simply laced, in analogy with Dynkin dia-
grams, if its exchange matrix is skew symmetric and non simply laced otherwise.
Two quivers are isomorphic if there is a map between their nodes that makes
their associated exchange matrices identical. In other words, their exchange matrices
are identical after conjugation by a permutation matrix.
1This ordering is meant to agree with Berhnard Keller?s java applet Quiver Mutation in
JavaScript.
15
2.1.2 Mutation of quivers
Quivers will underlie the coordinate atlases of our cluster ensembles and mu-
tations will give the transition maps between coordinate charts.
Definition 2.1.3. A mutation of a quiver Q at node i, written ?i(Q), generates a
new quiver by the following two operations
a,b c,d
1. For every pair of nodes j, k with weighted arrows j ?? i ?? k, add an arrow
of weight ac, bd from j to k.
2. Swap the direction and weights of all the arrows coming into and out of node
i.
Definition 2.1.4. The set of all quivers up to quiver isomorphism obtained by all
possible sequences of mutations of a particular quiver is called the mutation class
of the quiver, denoted by ?(Q).
Definition 2.1.5. It is useful to include nodes in a quiver that we do not allow
mutations at. These nodes are called frozen and we write Q? for the subquiver of Q
consisting of non frozen nodes and the arrows between them. We call this subquiver
the mutable portion of Q. The rank of a quiver is equal to #N(Q?).
Remark 2.1.1. When considering an isomorphism of quivers, we generally ignore
their frozen nodes and only consider maps between their mutable portions. In
this way, we have that two quivers with the same mutable portions have the same
mutation class. We will call an isomorphism including a map on frozen nodes a
frozen isomorphism or quivers.
16
1 1,3 3[3]
1 1,3 2[3] 3[3] 3,1
2[3]
(a) Q .
G?2
(b) ?2(Q ).G?2
Figure 2.1: Two quivers of type G?2
Example 2.1.1. Let QG? be the quiver shown in figure 2.1a. We read the pair of2
a?rrow weight?s as ?1 in , 3 out? meaning that the matrix associated to this quiver is
???? ?? 0 3 0?? ?
??
?
? ???1 0 1?????. We may take D = diag(1, 3, 3) so that nodes 2 and 3 get multipliers??? ???
0 ?1 0
of 3. This quiver is associated to a root system of affine type G?2, where node
1 is associated with the larger root and nodes 2 and 3 are associated with the
?smaller roots. ??2(QG? ) is shown in figure 2.1b. The exchange matrix changes to2
???? ?? 0 ?3 3 ?????
??
?? 1 0 ?1?
????? after mutating.?? ???
?1 1 0
2.1.3 Folding of quivers
The relationship between the classical ?folding? of the simply laced Dynkin
diagrams to form the non simply laced diagrams can be extended to quivers. We
17
refer to [28] for full details about folding of quivers.
Essentially, to fold a quiver we will group its nodes into disjoint sets and
perform mutations by requiring that we mutate all the nodes of a given set together.
We call the operation of mutating each of the elements of a set of nodes in turn a
?group mutation?. This does not depend on the order of mutations when there are
no arrows between any two nodes in the set, i.e. each of the mutations of nodes in
the set commute with each other.
With this in mind, we have the following definition:
Definition 2.1.6. A folding of a quiver, Q, with n nodes is a choice of k non empty
and disjoint sets of nodes whose union contains all of the nodes of Q satisfying the
following conditions:
1. The nodes contained in a given set have no arrows between themselves.
2. Condition 1 is satisfied after any number of group mutations of these fixed
sets.
These properties will always be satisfied when Q is the quiver which is an
orientation of a simply laced finite, affine or doubly extended Dynkin diagram and
the sets are given by the orbits of nodes under an automorphism of Q.
In this situation, all of the mutation structures we will be interested in studying
will be the same for the folding of Q and for a particular skew-symmetrizable quiver.
Let K1, . . . , Kk be the groups of nodes in this folding and let mij be the total
number of arrows from nodes in Ki to nodes in Kj. Then we can construct a skew-
m m
symmetrizable quiver Qfold with k nodes respectively and arrows of weight
ij , ij|Ki| |Kj |
18
from node Ki to node Kj.
Example 2.1.2. We may obtain the quiver QG? of example 2.1.1 by folding a quiver2
associated with an E?6 Dynkin diagram in the following way:
3? 3[3]
2? Fold 2[3] (2.1)
1,3
3 2 1 2?? 3?? 1
2.1.4 Important mutation classes of quivers
The combinatorial properties of a cluster ensemble are controlled by the mu-
tation class of the quiver, ?(Q), underlying its seeds. Before we give the definitions
of a cluster ensemble, we will briefly discuss some important mutation classes of
quivers.
Quivers which have a finite mutation class are called ?finite mutation type?.
For the most part, all of the examples and situations we will consider will be with
finite mutation type quivers. These quivers are classified in [28, 29]. Importantly,
all of the non simply laced finite mutation class quivers can be obtained by folding
a simply laced one.
The classification of finite mutation type quivers can be split in two ways.
First, we find that these quivers are either associated with Dynkin diagrams, called
Dynkin-type, or are of non-Dynkin type. The Dynkin-type quivers are then split
into finite, affine or doubly extended types. These mutation classes contain quivers
which are orientations of the Dynkin diagrams of finite, affine, or affine-extended
19
Finite Affine Doubly Extended Exotic
(1,1)
E6 E?6 E6 X6
(1,1)
E7 E?7 E7 X7
(1,1)
E8 E?8 E8
Figure 2.2: The exceptional (non surface) quivers.
2 4 2 4
1 3 5 1 3 5
6 6 7
(a) X6 (b) X7
Figure 2.3: The exotic non-surface finite mutation quivers
root systems respectively.
We refer to [22] for a background on our notations used in regards to affine
extended root systems. These Dynkin diagrams are shown in Appendix A.
Secondly, we may see that every simply laced finite mutation class quiver can
either be obtained as the mutation class of a quiver associated with a triangulation
of a marked surface (see section 2.4) or is one of 11 exceptional examples listed in
figure 2.2. The non-simply laced finite mutation class quivers can all be obtained
by folding a simply laced one, as shown in [28].
20
2.2 Positive spaces and Cluster Ensembles
We will simplify the notion of positive space of [12] in hopes of making com-
putations more concrete.
Definition 2.2.1. A positive chart will simply mean an algebraic torus, (Gm)n,
along with a distinguished coordinate chart consisting of its n characters. A map
between positive charts is called a positive map and satisfies the property that
pullback of the distinguished coordinate variables of the target is a rational function
with positive integral coefficients.
A positive variety is a variety obtained by gluing positive charts together with
positive maps. We write Pos for the category of positive varieties.
Remark 2.2.1. Each positive variety, X, has a well defined set of positive real
valued points denoted by X(R>0). This has a structure of a real manifold. Positive
maps ? : X ? Y induce homeomorphisms X(R>0)? Y (R>0).
Definition 2.2.2. A positive space is a collection of positive varieties along with
invertible positive maps between them. Equivalently, A positive space is a functor
from a groupoid to the category of positive varieties.
Let Q be a quiver with mutable nodes N1, ? ? ? , Nm. Let ? = {?1, ? ? ? , ?m}
be algebraically independent generators of the function field F(?1, ? ? ? , ?m). We
associate these variables variables to the nodes of Q in an obvious way.
Definition 2.2.3. The pair of a quiver along with variables associated to its nodes
is called a seed, denoted by i = (Q,?). The field F(?) is the function field of the
21
seed. The positive chart with characters given by the seed variables is called the
Seed Torus of i.
Definition 2.2.4. Two seeds are isomorphic if there is a quiver isomorphism which
induces an isomorphism of the associated function fields. When there are frozen
node, we also require that the fields generated by the frozen variables are isomorphic.
Definition 2.2.5. We can mutate seeds as follows: Given a seed i = (Q,?), we
construct a seed ?i(i) = (?
?
i(Q), ? ) where the new quiver is given by mutating Q at
node i and we associated a new collection of variables ?? = {?? , ? ? ? , ??1 m}.
Equipped with the notions of quivers, mutations and seeds, we can define the
pair (AQ,XQ) of positive spaces associated to an initial seed with quiver Q. Both
of these spaces will be defined by generating a collection of seeds by mutating an
initial seed generated from Q, but in each case we define different positive maps
between the seed torus of i and ?(i).
2.2.1 The A space
LetQ be a quiver with mutable nodesN1, ? ? ? , Nn and frozen nodes Fn+1, ? ? ? , Fm.
Let a = {a1, ? ? ? , an, fn+1, ? ? ? , fm} be algebraically independent generators of the
function field F(a1, ? ? ? , an, fn+1, ? ? ? , fm) called the initial variables. We associate
these variables to the nodes of Q in the obvious way to make a seed i = (Q, a).
Definition 2.2.6. We call these particular seed variables A coordinates. The A
coordinates associated to the frozen nodes are called coefficients. We write Ai for
22
the positive chart with coordinate variables given by the seed i = (Q, a). This seed
is called the initial seed associated with Q.
Mutation of Q at a node i produces a new seed consisting of a new quiver
?i(Q) and a new collection o??f variables a
?. We define a map ?i : Ai ? A? (i) byi
???? 1 ? ?( a ik + a ik) i = j
? ? ???
k k
a
? (a ) = i ik>0  <0i j ??
ik . (2.2)
aj i =6 j
We can thereby write all of the new A coordinates obtained by mutations in
terms of the initial A coordinates.
Definition 2.2.7. Each collection of A coordinates generated in this way is called a
cluster, and the F-subalgebra of F(a1, ? ? ? , an)[fn+1, ? ? ? , fm] generated by the clus-
ters of A coordinates obtained from all possible sequences of mutations at non-frozen
nodes is the cluster algebra associated to the quiver Q, denoted by C(Q).
Definition 2.2.8. AQ is the positive space obtained from the positive charts asso-
ciated to every seed generated by mutation of i and maps given by the mutation
isomorphism for each mutation. We describe this space as a functor in section 2.3.
2.2.2 The X space
The X space will have a definition similar to that of the A space, but with
a different exchange rule. In the case that Q has frozen nodes, we will not include
extra variables, and we may essentially ignore these nodes in the definition.
Given a quiver, Q, with mutable nodes N1, ? ? ? , Nn, we associate independent
variables x = {x1, ? ? ? , xn} called X coordinates. This pair is called an X seed, also
23
denoted by i. Mutation of Q at node i produces a new X seed, i? consisting of a
quiver ?i(Q) and a new collection of X coordinates, x?. We define a map of positive
varieties ?i : Xi ? X?i(i) by ?????x?1 i = j
?? ?
i
i (xj) = ???? . (2.3)? sgn x (1 + x jij i )?ji i =6 j
We note that we do not assign variables for the frozen nodes in Q.
Definition 2.2.9. XQ is the positive space consisting of the positive varieties asso-
ciated to every seed generated by mutation and maps given by the above mutation
isomorphism for each mutation.
2.2.3 Cluster Ensembles
It is worth viewing these positive spaces as two different functors from the
same groupoid.
Definition 2.2.10. The Seed groupoid Gi is the groupoid with one object for each
seed obtained from mutations of i and maps given by mutation paths. We do not
include any notion of seed isomorphism in this definition.
Remark 2.2.2. This notion of seed groupoid is not the same as that of Fock and
Goncharov in [5]. Their notion is the same as our notion of cluster modular groupoid
in section 2.3.
We may view theA and X spaces associated with the same seed as two different
functors A,X : Gi ? Pos, where we assign positive charts to each seed and maps
according to the appropriate mutation rules.
24
Definition 2.2.11. Given an initial quiver, Q, we write (AQ,XQ) for the pair of
positive spaces generated by using Q along with independent variables for each node
as a seed for each space, where XQ = XQ? since we do not assign X coordinates to
the frozen nodes. This pair of spaces is the cluster ensemble associated to Q. The
rank of this ensemble is #N(Q?). We often drop the subscripts when Q is implied.
Remark 2.2.3. The positive varieties Ai and Xi associated to the initial seed Q
provides a base point of each space. Changing Q to any other quiver in ?(Q) gives
the same positive space, just with a different base point.
The final important property of a cluster ensemble is a natural transformation
? : AQ ? XQ. ? has a simple definition in terms of the initial coordinates. If
(a1, ? ? ? , an, an+1, ? ? ? , an+f ) and (x1, . . . , xn) are the initial A an?d X coordinates and

the exchange matrix of Q is [ij], then we have that ?
?(x iji) = ai . ? commutes
i
with mutations and provides a map of positive spaces.
Example 2.2.1. We illustrate these definitions with a well known example. Let Q
be a quiver with two mutable nodes, 1 and 2, with a single arrow from node 1 to
node 2. Each mutation of Q gives an isomorphic quiver, and so all of the seeds we
obtain by mutation are isomorphic.
We compute cluster variables using the A mutation rule. This is shown in
figure 2.4. We see that after 5 mutations we return to the original cluster variables.
Thus this seed is not identical to the original seed since the variables have switched
places. After 5 more mutations we return to a seed which is identical to the initial
seed.
25
( )
a2+1 ? a
a 21
( ) ?1 ?2( )
a ? a a2+1 ? a2+1+a11 2 a1 a1a2
( ) ( ?1 )
a ? a a1+1 ? a2+1+a12 1 a2 a1a2
?1 ( ) ?2
a1+1 ? a
a 12
Figure 2.4: The five A seeds with different cluster variables.
The cluster algebra C(Q) is the algebra
C a2 + 1 a1 + 1 a + 1 + aF 2 1(Q) = [a1, a2, , , ] (2.4)
a1 a2 a1a2
The positive spaces AQ and XQ each consist of 10 distinct positive charts with
positive maps given by the mutation rules. For example, if i is the initial X seed
with variables x1, x2 and ?1(i) = i
? with associated variables x?1, x
?
2 then we have the
positive map given by
?i : Xi ? X ? ? ?i? , ?i (x1, x2) = (x?11 , x2(1 + x1)). (2.5)
The map ? : AQ ? XQ is given in terms of the initial coordinates by
??(x1, x2) = (a2, a
?1
1 ) (2.6)
2.3 The Mutation Structures of a Cluster Ensemble
The mutation class of a quiver underlying a seed of a cluster ensemble controls
many of that ensembles properties. The most well known example of this is the fact
26
that a cluster ensemble has finitely many clusters if and only if its quiver is of finite
type, see [2]. We call these cluster ensembles ?finite type? and refer to them by the
name of their Dynkin diagram.
We will be generally interested in ?mutation finite? cluster ensembles, i.e.
ensembles for which their underlying quiver has a finite mutation class. When this
quiver is of Dynkin type, we will refer to that ensemble as being that particular
type.
Mutation finite cluster ensembles which are not of finite type have infinitely
many clusters, but only finitely many quivers up to isomorphism underlying their
seeds. A natural way to study such a situation is to study the group of transfor-
mations which act by permuting different clusters associated with isomorphic seeds.
Such a transformation is determined entirely by where it sends a single cluster, since
the other clusters may all be obtained by mutations from a given one. We call this
type of transformation an automorphism of the mutation structure of our cluster
ensemble.
2.3.1 The cluster modular group
Our current notion of seed groupoid has different objects for all seeds obtained
by mutations. It is natural to define a groupoid with objects corresponding to seeds
up to seed isomorphism; This will allow us to define the cluster modular group.
Definition 2.3.1. A seed i? obtained by a sequence finite of mutations, P = ?i1 ?
?i2 ? . . . , from an initial seed i is cluster equivalent to i if the map P : Ai ? Ai? is
27
simply a permutation of the A coordinate variables. In other words, the seeds i and
i? have the same unordered collection of cluster variables.
Remark 2.3.1. We defined this notion in terms of the A mutation rule, but it is
equivalent to the condition that P : Xi ? Xi? is a permutation of the X coordinate
variables. It is always the case that two cluster equivalent seeds are isomorphic.
A seed isomorphism ? : i? i? induces a map on the varieties associated with
each seed by ??(???(i)) = ?i.
Definition 2.3.2. A pair {P, ?} of a mutation path and seed isomorphism is called
a trivial cluster transformation if the composition of the mutation path and seed
isomorphism induces the identity map on the A and X varieties.
A trivial cluster transformation must consist of a path to a cluster equivalent
seed along with a seed isomorphism which induces the identity permutation on the
cluster variables.
Definition 2.3.3. The cluster modular groupoid, GQ, is the groupoid with objects
given by seeds obtained from mutating the seed i up to seed isomorphism. The
maps are given by pairs of mutation sequences and seed isomorphisms up to those
which are trivial cluster transformations. The group associated with this groupoid
is called the Cluster Modular Group.
This group can be considered as the group of symmetries of the ?exchange
complex? of the cluster ensemble, see section 2.4 of [12], but we will not need this
notion here.
28
We will give a concrete description of the cluster modular group in terms of
sequences of mutations and isomorphisms of qu{ivers. }
Fix a seed i with quiver Q. Let ??i = {P, ?} where P is a sequence of
mutations and ? is an isomorphism Q? ? P (Q)?. Because P transforms Q into
an isomorphic quiver, we may compose these mutation paths. We write paths of
mutations as a sequence of the node names read from left to right. The action of ?
as an element of the symmetric group is written with the standard left action.
Since each mutation is an involution, we have that ??i is a subgroup of (Z/2Z?k)o
S where Z/2Z?kk is a free product of cyclic groups and the right hand side of the
semidirect product acts by permuting the factors.
Remark 2.3.2. ??i acts on the elements of the seeds of A and X coordinates whose
underlying quiver is isomorphic to Q. P provides a path to a new seed and ?
provides a map between the initial seed variables and the final variables. We note
that ? is only an isomorphism of quivers, not of seeds, which is why this action is
nontrivial.
Definition 2.3.4. The cluster modular group based at i, ?i is defined to be ??i
modulo the subgroup generated by trivial cluster transformations.
Our definition currently depends on the choice of initial seed i. If i? is any
other seed mutation equivalent to i, then ?i? ' ?i, with the isomorphism given by
conjugation of paths by a mutation path between i and i?.
Definition 2.3.5. The cluster modular group, ?Q : GQ ? Grp, is defined to be
the functor from the cluster modular groupoid which assigns the group ?i for each
29
seed i along with group isomorphisms given by conjugation by mutation paths. An
element of the cluster modular group is a collection of elements ?i ? ?i which are
mapped to each other by the group isomorphisms defined for each mutation path.
Remark 2.3.3. We will often simply refer to the group ?i as the cluster modular
group, knowing that we really have a collection of isomorphic groups for each seed
isomorphism class. We will also simply refer to elements of the abstract group ?i as
elements of the cluster modular group, knowing that these elements take on different
presentations depending on which seed we are interested in.
Example 2.3.1. Let?s examine the cluster modular groupoid and group of a type
A2 cluster ensemble, continuing example 2.2.1. The quiver of a A2 cluster ensemble
consists of two nodes, 1 and 2, with a single arrow between them. As we saw, each
of the new seeds we obtained by mutations are isomorphic to the original seed. Thus
the cluster modular groupoid only has one object. Therefore, each map is a cluster
modular group element.
Mutation at node 1 produces a quiver that is isomorphic to the starting quiver
(after permuting the nodes). We can represent this group element by ? = {1, (12)}.
This element clearly generates the cluster modular group. The only relation is that
?5 = e. This relation comes from the fact that 5 consecutive applications of ?
reproduces the original cluster variables. Thus the cluster modular group is Z/(5Z).
We can see the distinction between the seed groupoid and cluster modular
groupoid quite clearly in this example. The seed groupoid has many more objects
than we really need.
30
1 1
3 2 3 2
(a) Q1 (b) Q2
1 1
3 2 3 2
(c) Q3 (d) Q4
Figure 2.5: Choices of quiver isomorphism classes in the mutation class of a quiver
of type A3.
Example 2.3.2. If we take a quiver of type A3 as our initial seed, it is easy to
compute the cluster modular groupoid and cluster modular group. We can do this
by taking a representative for each of the quivers in the mutation class of our seed
and then writing all of the possible pairs of mutation paths and quiver isomorphisms
between them. A choice of possible representatives for the quiver isomorphism
classes is shown in figure 2.5 and a diagram of the groupoid is shown in figure 2.6.
It is not too difficult to compute the cluster modular group by checking that every
map from Q1 to itself is generated by the two maps shown in the figure. These two
elements commute and the top has order 2 and the bottom has order 3. Thus the
cluster modular group is Z/6Z.
31
Q3
{3,(23)}
{1231,(23)}
{2,()}
Q1 {1,()} Q2 {1,()}
{3,()}
{<>,(123)}
Q4
{<>,(23)}
Figure 2.6: The cluster modular groupoid of the A3 cluster ensemble. A selection
of non-identity maps are shown.
2.3.2 Action of the cluster modular group on functions
Given (A,X ) a cluster ensemble, we wish to define a notion of a field of rational
functions on the A and X spaces.
There is a natural functor from the category of positive varieties to the category
of function fields over F obtained by associating a positive variety its function field.
We write F(A) for the composition of the functor realizing our positive space A with
this functor. This will give us a notion of rational functions on a positive space.
Concretely, F(A) provides us with a function field, F(Ai), for each seed i.
When we change to another quiver, Q? ? ?(Q) we can pull back functions along
P : Ai ? Ai? to obtain an isomorphism
F(Ai) ' F(Ai?). (2.7)
We will occasionally write F(A,X ) for the functions on a cluster ensemble.
32
Again fix a starting seed i with quiver Q. We can now define an action of ?i
on the rational functions on the AQ or XQ space.
Let ? ? ?i. Given f ? R(Ai) we can define ?(f)(a1, . . . , an) = P (?(f)), where
? acts as a map from the cluster variables on the initial seed i to the seed P (i) and
P acts by pullback along P : Ai ? AP (i). We define an action on elements of F(Xi)
in an analogous way.
Elements in ?Q act via natural transformations F(A) ? F(A) and F(X ) ?
F(X ) where we act on each function field as above.
Definition 2.3.6. Given f ? F(Ai) or f ? F(Xi) we call the set ?i(f) the exchange
class of f . ?Q(f) provides a set of exchange classes for each quiver isomorphism
class in the mutation class of Q.
Example 2.3.3. Let Q = Q1 of figure 2.5. Let f ? F(AQ) be given by
a1 + a2 + a3
f(a1, a2, a3) = . (2.8)
a1a2a3
Then we may compute that {1231, (23)}(f) = f?1. Lets write this out in detail.
We write ai, bi, ci, di, ei for the variables on each of the five seeds found along the
path. The important exchange relations are
? a2 + a3 1 + b1?1(b1) = ?
?
a 2
(c1) = (2.9)
1 b2
? 1 + c1 ? d2 + d3?3(d3) = ?1(e1) = (2.10)c2 d1
and all other exchange relations are trivial.
The isomorphism (23) provides the map {a1, a3, a2} ?
(?2?3) {e1, e2, e3} and the
mutation path acts on (23)(f) by pullback. Writing the value of f by abuse of
33
notation, we compute that ( )
{ a1 + a2 + a31231, (23)}(f) =?1 ? ?2 ? ?3 ? ?1((23)( ) =a1a2)a3 ( )
? ? ? e1 + e3 + e2 d1 + 1?1 ?2(?3) ?1 ( ) = ?1 ? ?2 ? ?3 =e1e3e2 d2d3
? c2 b2b3 a1a2a3?1 ?2 = ? = = f?11 .
c3 b1 + 1 a1 + a2 + a3
We clearly have that {<>, (123)}(f) = f . Therefore, the cluster modular
group ?Q = Z/6Z acts on f via the quotient group Z/2Z.
Definition 2.3.7. Let ?? ? ?Q be a functor which assigns a subgroup ??i ? ?i for
each seed i. The invariant field for ?? on AQ or XQ is a collection of invariant fields
F(A ?i)?i or F(X )?
?
i i respectively for each seed i.
It is not clear from this definition that this collection of invariant rings is non
empty. It is most likely not true that there are invariants for any cluster ensemble
with non-trivial cluster modular group, but we will not discuss this here.
Remark 2.3.4. We generally write elements of these rings in terms of the initial
seed, i, associated with Q.
2.3.3 Folding of cluster ensembles
When our cluster ensemble is associated with a non-skew symmetric quiver
which can be obtained by folding, there is a natural relationship between the A and
X coordinate variables which gives a map between the functions defined on each
space. Let R be a quiver obtained by folding Q and let j, i be seeds associated with
R,Q respectively. Let Ki be the sets of nodes of Q which are folded to obtain R.
34
Picking an order to mutate the nodes in each set realizes the seed groupoid
associated with R as a subgroupoid of the seed groupoid of Q. Pulling the functors
AQ,XQ : Gi ? Pos back along this inclusion gives functors A?Q, X?Q : Gj ? Pos.
There is a natural transformation, ? : AR,XR ? A?Q, X?Q which maps the positive
charts (Aj,Xj) to (Ai,Xi) defined on coordinates by assigning all of the cluster
variables associated to nodes in Ki to the cluster variable associated to the folded
node in j. We can think of this map as being similar to a diagonal embedding of
the varieties (Aj,Xj) in (Ai,Xi).
fold
Pull back along this embedding gives a map between the functions F(Ai,Xi) ???
F(Aj,Xj) for each seed i which folds to a seed for the cluster ensemble associated
with j. This map does not depend on the order of the nodes chosen for each set.
Since elements of F(AQ,XQ) are collections of functions on each positive variety
which agree along mutation paths, we can define a map
F f(AQ,XQ) ??
o?ld F(AR,XR) (2.11)
by folding the functions on the positive varieties coming from seeds in Gi appearing
in the subgroupoid associated with j. This map is clearly surjective.
We can also realize the cluster modular group ?j as a sub-quotient of the
cluster modular group ?i, where the ?sub? comes from only considering sequences
which mutate the folding sets together, and ?quotient? being that we ignore auto-
morphisms which map the folding sets into themselves. Thus, if ? ? ?i is an element
which can be mapped into ?j, then we get a map
F(A foQ,X ???Q) ???
ld F(A ,X )???R R (2.12)
35
on the invariant rings.
2.4 Preliminaries on Hyperbolic Geometry and Teichmu?ller spaces
We will recall some background about hyperbolic structures on surfaces and
their moduli spaces
2.4.1 Hyperbolic space
Recall that the hyperbolic place, H2, has two important models, the Poincare?
disk, {z ? C, |z| < 1} and upper half plane , {z ? C,=(z) > 0}. These two models
are related by a Mobius transformation mapping the upper half plane into the unit
disk.
The boundary of the hyperbolic plane, ?H2, is identified with RP1 = R?{?}.
This identification is quite natural in the upper half plane model, but we may also
label points on the boundary in the disk model with elements of R ? {?}. Points
on the boundary are called cusps. There is a natural cyclic order on the points of
?H2. There are two choices for this, which we may fix by deciding if a given triple
of points is positively or negatively orientated. As a convention we will fix (?, 1, 0)
as a positive triple.
Each of these models comes equipped with a complete Reimannian metric, g
with constant curvature equal to ?1. The group of isometries of H2 is identified
with the group of Mobius transformations which fix the upper half plane, and is
isomorphic to PSL(2,R).
36
Name Example Trace Fixed Points
? ?
???? ?? cos(?) ? sin(?) ??Elliptic ?? ???? Tr < 2 One point in H
2
sin(??) co?s(?)
??? ?1 a ??
Parabolic ???? ???? Tr = 2 One point on ?H
2
? 0 1? ??? ?? 0 ?
Hyperbolic ???? ?
?
??? Tr > 2 Two points on ?H
2
0 ??1
Figure 2.7: The conjugacy classes of non-identity elements in PSL(2,R)
The elements of PSL(2,R) are in 3 distinct non identity conjugacy classes.
They are shown in figure 2.7
We will also need to recall the notion of a ?horocycle? about a boundary point
on H2. These are curves on H2 with constant curvature and which only touch touch
the boundary at a single point. In the disk model, they are given by circles which
are tangent to a boundary point, and in the upper half plane model, they are given
by tangent circles or horizontal lines (these lines are the horocycles tangent to the
point? ? ?H2). Each parabolic element of PSL(2,R) fixes a single boundary point
and translates points in H2 along the horocycles touching this point.
37
2.4.2 Hyperbolic structures on surfaces
Let S be a surface of genus g with b boundary components, p punctures, and
n marked points on the boundary. Let B be the set of boundary curves, P be the
set of punctures and M be the set of marked points on the boundary. Let S? be
S ? {B ? P} = ?S, the interior of S.
Definition 2.4.1. We call S a marked surface if it satisfies that each boundary
component has at least 1 marked point on it and that n?3?(S) = 6g?6+3b+3p+n >
0.
Definition 2.4.2. An arc on S is an isotopy class of paths between two marked
points or punctures on S. A closed curve on S is an isotopy class of embeddings of
the circle on S. A closed curve is called simple if it has a representative isotopy class
which does not intersect itself. An Ideal Triangulation of S is a maximal collection
of pairwise non-intersecting arcs on S.
Each marked surface admits an ideal triangulation, and every such triangula-
tion has the same cardinality, N = 6g ? 6 + 3b+ 3p+ 2n.
Definition 2.4.3. A hyperbolic structure on S is a Riemannian metric, g, on S
satisfying the following properties:
1. g has constant curvature equal to ?1.
2. g has finite area.
3. g takes punctures and marked points to cusps.
38
4. the arcs of B ?M are geodesics of g.
There are several important equivalent ways of defining a hyperbolic structure.
The hyperbolic metric on S locally defines a map S ? H2. In other words, for any
point q on S? there exists U ? S , q ? U with U isometric to an open set of H2.
Definition 2.4.4. Let S? be the universal covering space of S and let ? be the
covering map S? ? S. This local isometry can be extended to an immersion
Dev : S? ? D ? H2 (2.13)
where D is some subset of H2, called the developing map. Any element ? ? ?1(S)
corresponds to a deck transformation of S? over S, and the developing map takes
such transformations to hyperbolic isometries of H2. Thus, the developing map gives
a representation
? : ?1(S)? Isom(H2) = PSL(2,R) (2.14)
called the monodromy representation.
Let ? be the image of ?1(S) under ?. Then ? is a discrete subgroup of
PSL(2,Z), and ? takes loops homotopic to punctures to parabolic elements in ?.
? acts on D by isometries, so D/? naturally has a hyperbolic structure which is
isometric to the structure on S.
This situation is encompassed in the following commutative diagram:
S? Dev D
? . (2.15)
S ' D/?
39
Essentially, the developing map allows us to view a hyperbolic structure on a surface
as directly coming from the canonical hyperbolic metric on H2.
Example 2.4.1. The surface S = S0,1,1,2, a once punctured digon is shown in figure
2.8. The monodromy representation takes the element of ?1(S) corresponding to ?
to the parabolic element ? ? PSL(2,R) which fixes p ? H2 and moves n1 to ?n1.
2.4.3 Teichmu?ller space
The notion of Teichmu?ller spaces comes from several different perspectives.
For us, we will consider the Teichmu?ller space of a surface, T (S) as the moduli
space of hyperbolic structures on S up to an equivalence relation. To aid with our
definitions, we need the notion of a marked hyperbolic structure on S.
Definition 2.4.5. A marked hyperbolic structure on S is a triple (M, g, ?) such that
? : S ?M is a homeomorphism and (M, g) is a hyperbolic structure on the surface
M .
Definition 2.4.6. Two marked hyperbolic structures on S, (M, g, ?) and (M ?, g?, ??)
are equivalent if there is a homeomorphism, ? : M ? M ? such that the pullback
of g? along ? is g and ???1 ? ? ? ? : S ? S is isotopic to the identity map. The
Teichmu?ller space of S, T (S) is defined to be the set of marked hyperbolic structures
on S up to equivalence.
There are several important equivalent ways of describing T (S). In view of
developing maps and monodromy representations, we can see that two equivalent
marked hyperbolic structures gives rise to an isometry, ?? of H2 such that Dev(M?) =
40
(a) S
(b) Dev(S)
Figure 2.8: The surface S along with its developing image in H2.
41
??(Dev(M? ?) which descends to the map ? on the quotients. Furthermore, the two
groups ? = ?(?1(M)) and ?
? = ??(?1(M
?)) are conjugate in PSL(2,R) by the
element ??. Thus, we have the following proposition:
Proposition 2.4.1. In the absence of marked points, the Teichmu?ller space T (S) is
equivalent to the set of representations, ? : ?1(S)? PSL(2,R), up to conjugation by
PSL(2,R) such that ? is discrete, faithfull, and sends loops homotopic to punctures
to parabolic elements.
2.5 Coordinates on Teichmu?ller Space
We will review three different coordinate systems on several variations of Te-
ichmu?ller space.
Let S be a marked surface. For the first two coordinate systems, we will fix
an ideal triangulation ? of S.
2.5.1 Teichmu?ller X space
Let N? = 6g ? 6 + 3b+ 3p+ n be the number of interior arcs of ?.
Definition 2.5.1. The Teichmu?ller X space of S, TX (S), is defined to be the real
?
manifold homeomorphic to (R>0)N with a positive real valued coordinate for each
interior arc of ?. We call these coordinates the X coordinates of TX (S).
The main ingredient of our first set of coordinates will be the cross ratio of
four points on RP1.
42
Definition 2.5.2. Let (z1, z2, z3, z4) be four points in R ? {?}. The cross ratio,
r(z1, z2, z3, z4) is defined to be
(z1 ? z2)(z3 ? z4)
r(z1, z2, z3, z4) = (2.16)
(z2 ? z3)(z1 ? z4)
This particular choice of cross ratio has the property that it is a positive real
number for any positively oriented quadruple of points.
Identifying ?H2 with RP1, we see that the cross ratio is a PSL(2,R) invariant
for four points on ?H2. There is a natural way to assign a cross ratio to each interior
(non boundary) edge in our ideal triangulation ?.
We may lift ? to a triangulation, ?? of S? and embed this in H2 via the de-
veloping map. Thus each interior edge, e?, in ?? is identified with the four points
in ?H2 of the square containing the edge after developing. Picking z1 to be one of
the endpoints of e? and the other three points z2, z3, z4 to cyclically oriented from
z1 allows us to assign the cross ratio r(z1, z2, z3, z4) to the edge e?. The particular
choice of endpoint for z1 does not change the assigned cross ratio.
We assign cross ratios to the interior edges, denoted xe of ? by choosing a
lift of them to ?? and assigning the cross ratio of the lift as before. This does not
depend on the choice of lift since the difference of lifts is described by an isometry
of H2.
Definition 2.5.3. An opening, S? of a marked surface, S, is a hyperbolic surface
where the punctures on S have been replaced with geodesic boundary components,
along with a choice of orientation of each new boundary.
An ideal triangulation of S can be lifted to a triangulation of S? where the
43
arcs which touch the punctures are lifted to arcs which spiral around the boundaries
in the direction which agrees with their chosen orientation. The collection of sur-
faces which are openings of S are parameterized by signed lengths of their geodesic
boundaries, {lp} for p a puncture, where the sign indicates whether the orientation
of the boundary component agrees with the orientation of S. The situation when
the boundary has length 0 is equivalent to there being a puncture on the opened
surface
Proposition 2.5.1 (Thurston-Fock). Any collection of 6g?g+3b+3p+n positive
real numbers can be realized as the cross ratio coordinates of an opening of S. In
other words TX (S) parameterizes hyperbolic surfaces which are openings of S.
Part of the proof of this proposition is an explicit construction of a represen-
tation ?1(S) :? PSL(2,R) leading to a hyperbolic structure on an opened surface.
We recall the basic principals of this construction
For each puncture p on S there is an important associated product of X
coordinates given by ?
pX = xe = e
lp (2.17)
e|p??e
which is equal to the exponential of the signed boundary length.
Proposition 2.5.2. The Teichmu?ler space T (S) embeds in TX (S) as the submani-
fold where pX = 1 for each puncture of S.
44
2.5.2 Decorated Teichmu?ler space
Decorated Teichmu?ller space, denoted TA(S), introduced by Penner in [30]
parameterizes hyperbolic structures on S along with a ?decoration? of S, consisting
of a choice of horocycle around each puncture and marked point. Choices of horo-
cycles on S correspond via the developing map to collections of horocycles around
the cusps of D ? H2 which are fixed by the group ?.
Again we fix ? an ideal triangulation of S. Let N = 6g? g + 3b+ 3p+ 2n be
the number of arcs in ?.
Definition 2.5.4. The decorated Teichmu?ller space, TA(S), is defined to be the real
manifold homeomorphic to (R>0)N with a positive real valued coordinate for each
arc of ?. We call these coordinates the A coordinates of TA(S).
There is a natural way to assign a positive real number for every (including
boundary arcs) edge, e ? ? using our decoration. Let le be the signed length of e
measured between the horocycles on the endpoints of e, where the sign is positive
if the lifts of the horocycles do not intersect in H2 after developing and negative
otherwise. We assign the ?? length?
ae = exp(le/2) (2.18)
to each arc in ?.
Proposition 2.5.3 (Penner [31]). Every collection of N = 6g ? g + 3b + 3p +
2n positive real numbers appears as the lambda lengths of a decorated hyperbolic
45
structure on S. In other words, TA(S) parameterizes decorated hyperbolic structures
on S
Proposition 2.5.4 (Penner [31]). The natural map TA(S) ? T (S) forgetting the
decoration is a fibration, with fibers homeomorphic to (R>0)n+p parameterized by
the lengths of the decorating horocycles
There is a natural relationship between the lambda lengths of a decorated
hyperbolic structure and cross ratios. We have a map ? : TA(S) ? TX (S) defined
on coordinates by fixing ? an ideal triangulation of S and defining
? ?12?34? (xe) = (2.19)
?23?14
when xe = r(z1, z2, z3, z4) and ?ij is the lambda length of the arc connecting the
marked points corresponding to zi and zj. This map is the inspiration of the map
? : A ? X of a cluster ensemble.
2.5.3 Representations from points in Teichmu?ller spaces
Points in TA(S) and TX (S) naturally give rise to representations ?1(S) ?
PSL(2,R) with desired properties. Part of the proofs of the propositions in the
previous two sections is an explicit calculation of this representation. We will review
these ideas here.
Given S a marked hyperbolic surface with ideal triangulation ? we can con-
struct a ?fatgraph? on S as follows. Let F? be the dual graph to ?, i.e, F has one
vertex for each triangle in ? and edges between vertices whenever two triangles
46
share an edge, and edges to the boundary whenever a triangle contains a boundary
component . Then we generate the fatgraph, F from F? by replacing each vertex
with a triangle of three ?short edges? connecting the three ?long edges? edges at its
vertices, see figure 2.9 for an example.
This fatgraph can be used to construct a representation ? : ?1(S)? PSL(2,R)
for a point in TX (S) in the following way. We will first assign matrices to the edges
of F . To each? long edge of F ?which crosses an interior arc e ? ?, we assign the??? ? ?0 ? xe??
matrix Xe = ???? ???? where xe is the X coordinate associated with the?
( x )?1e 0
arc e. The three short edges w?ill be a?ssigned th?e matrices? ??????
1 1?????
? ??? ?
R = ? ? , L = ??
0 1 ?
? ?? ???? (2.20)
?1 0 ?1 ?1
Let ? ? ?1(S). We may homotope ? to our fat so that ? may be obtained by
following along segments of F . We then obtain ?(?) by first picking a segment to
begin and then multiplying the matrices for each consecutive segment obtained from
following ?, using R whenever we use a short edge to turn right and L whenever we
turn left.
Example 2.5.1. Continuing example 2.4.1, the fatgraph for ? is shown in figure
47
Figure 2.9: The fatgraph for a triangulation of a punctured digon.
2.9. We can see that the element ? ???1(S) can be repre?sented by the matrix??? ? ?x1x2 0 ??
X1RX2R = ???? ???? . (2.21)
?1 + ?1 ? 1
x1 x2 x1x2
This matrix has trace equal to 2 exactly when x1x2 = 1
We may use this to compute the traces of monodromy operators associated
with a hyperbolic structure on a surface in terms of X coordinates, and we may use
the map ? : TA(S)? TX (S) to write these formulas in terms of the A coordinates.
2.6 Cluster Ensembles associated with Surfaces
In many cases, cluster ensemble invariants have a geometric interpretation
coming from the relationship between cluster ensembles and the Teichmu?ller theory
48
of surfaces. We refer to [13] and [32, 33] for background on the ingredients of a
cluster ensemble associated to a hyperbolic surface. We recall some of these ideas
here.
2.6.1 Quivers from triangulations
Let S be a marked surface. Given ? a triangulation of S, we associate a
quiver, Q?, as follows: For each edge e ? ? we add a node Ne and for each triangle
t ? ? we add a clockwise oriented cycle of arrows between the nodes associated with
the edges of t. In the situation where we have arrows between two nodes in opposite
directions, we cancel them, as shown in example 2.6.2. The nodes associated to
boundary edges are frozen. There are ?3?(S) + 2n total nodes and n frozen nodes.
There is one caveat to this construction when S has punctures. In this case
it may be possible to have a ?self folded? triangle in an ideal triangulation of S see
figure 2.10a. In this case, the construction mentioned above does not produce the
correct quiver. However, we can always find a triangulation of S with no self folded
triangles, and use this to construct a quiver associated with the triangulation.
Mutation of nodes in Q? corresponds to a ?flip? or ?Whitehead Move? in
? at the corresponding arc. A flip is the operation of removing the specified arc,
and replacing it with the unique new arc which forms a new triangulation. Again,
there is a caveat to this when S has punctures. The interior arc of a self folded
triangle cannot be flipped, but the corresponding node in the quiver can be. This
is addressed in [32] by the addition of ?tagged? arcs. Essentially, we replace the
49
(a) Arcs in a punctured digon.
(b) The tagged arc flip graph.
Figure 2.10: Untagged vs tagged arcs in a punctured digon.
outside arc of a self folded triangulation with a tagged arc as shown in figure 2.10.
There is then a rule for flipping tagged arcs which agrees with the mutation rule for
quivers. With this addition, we may always flip any arc and this always agrees with
mutation of corresponding quivers. We do not need the details of this in general.
We can generate a graph associated to S where the nodes correspond to tri-
angulations (possibly with tagged arcs) and the edges correspond to flips of trian-
gulations. We call this graph the ?flip graph? of S.
Let i be the seed obtained by using Q? as its underlying quiver.
Proposition 2.6.1 ( [32]). The flip graph of S agrees with the cluster modular
groupoid of i in the following way: If the triangulation of S, ??, is obtained from ? by
two different paths of flips, then the two seeds obtained from i by the corresponding
50
paths of mutations are isomorphic.
The cluster ensemble generated by using Q? as a seed is closely related to the
Teichmu?ller theory of S. We will usually refer to this seed simply as ? when it is
understood that we mean the seed associated with the triangulation ?.
The positive real valued points of the positive varieties A? and X? are exactly
the Teichmu?ller spaces TA(S) and TX (S) defined in the previous section. We write
(AS,XS) for the cluster ensemble generated from any triangulation of S.
Importantly, one checks that the associated A and X exchange relations cor-
respond exactly to the change in the lambda lengths and cross ratios after flipping.
Thus, the cluster ensemble encodes the spaces TA(S) and TX (S) along with specified
coordinate systems for each ideal triangulation.
Example 2.6.1. Let S be a disk with n marked points on the boundary. The
cluster ensemble encodes some well known algebraic objects. The cluster algebra
associated with AS is the affine cone coordinate ring of the Grassmannian Gr(2, n),
see [11]. The space XS is essentially the moduli space M0,n of genus zero curves with
n distinguished points see example 1.3 of [12].
For any n, there are finitely many triangulations, and thereby finitely many
clusters. We can also see that there is always a triangulation for which the associated
quiver is an orientation of an An Dynkin diagram. Thus this cluster ensemble is of
type An.
Example 2.6.2. Following example 2.4.1, we can construct the quiver for this
surface and triangulation. Figure 2.11 shows the construction of the quiver. Notice
51
Figure 2.11: Generating the quiver for the surface S and ?. The two middle arrows
are removed.
that we cancel the arrows between nodes 1 and 2. The quiver, Q?, we obtain is
1
[3] [4] . (2.22)
2
We can see that flips at arcs 1 or 2 will generate self folded triangles. We can see
what the quivers associated to these triangulations should be by mutating the corre-
sponding nodes of Q?. We cannot flip arc 1 and then flip arc 2 without introducing
tagged arcs. This is necessary since we can mutate node 1 and then node 2 of Q?
without issue.
The cluster ensemble associated with S has four clusters and the flip graph of
S has four nodes. This is shown in figure 2.11.
52
2.6.2 Action of the mapping class group
We can define an action of the mapping class group, MCG(S), on the triangu-
lations of S and hence identify the mapping class group as a subgroup of the cluster
modular group, ?S of our cluster ensemble (AS,XS). We give an explicit construc-
tion of this subgroup here. We refer to [34] section 2 for computations involving the
mapping class group of selected surfaces.
Given f ? MCG(S) we can define ?f ? ?S as follows: f gives a new triangula-
tion of S and hence by [32] there is a path of flips, Pf , taking ? to f(?). f defines
a map between the edges of ? and f(?) and it preserves the adjacency relations
between the triangles of ?. This means that ? and f(?) have the same associated
quivers. P defines a map between the nodes of Qf(?) and P (Q?) since these quivers
come from the same triangulation. Let ?f,P be the isomorphism of quivers Q? to
P (Q?) defined by the composition
f P
?f,P : Q? ?? Qf(?) ?? P (Q?). (2.23)
Thus to f we associate ?f = {Pf , ?f,P}.
It is not immediately clear that this does not depend on the choice of path,
P . Let {P, ?} and {R, ?} be two possible representatives of ?f . Then we have
{P, ?}{R, ?}?1 = {P, ?}{??1(R?1), ??1} = {P???1(R?1), ???1}. (2.24)
We need to show that this element is a trivial cluster transformation. ???1 is
the quiver isomorphism from R(Q?) to P (Q?) coming from the fact that these
both correspond to the same triangulation of S. The composite mutation path,
53
P???1(R?1), consists of following P and then following R?1 back to our initial
cluster. This introduces a permutation on the cluster variables determined by the
map ???1 : P (Q?) ? R(Q?). Together these permutations act trivially on the
cluster variables, and ?f is well defined in the cluster modular group.
For all but finitely many quivers associated with surfaces, the cluster modular
group is essentially equal to the mapping class group, see [35] proposition 8.5. For
the remaining surfaces, one may check case by case that MCG(S) is always a finite
index normal subgroup of ?.
2.6.3 Skein algebras of surfaces
The skein algebra associated with a marked surface will play a key role in
understanding the relationships between invariant functions on a cluster ensemble
associated with a surface. We refer to [26] for full details on the relationship between
skein algebras and cluster algebras associated with surfaces. We note that we will
always be considering the unquantized version of the skein algebra.
Let S be a marked surface.
Definition 2.6.1. A Skein on S is any collection of arcs on S or closed curves on
S. We do not require that these arcs or closed curves do not intersect each other or
themselves.
Definition 2.6.2. The Skein Algebra, Sk(S) is the algebra with generators given
by the skeins on S subject to the Skein relations, see figure 2.12
The cluster algebra associated with S, A(S), is naturally contained in the
54
(a) The skein relation for a crossing.
,
(b) A contractible closed curve is equal to
?2.
(c) A contractible arc is equal to zero.
Figure 2.12: The skein relations for arcs and curves on a marked surface.
55
Figure 2.13: Computing the element ?a1a2 in Sk(S).
skein algebra. The skein relation is simply another description of the A exchange
relation.
Let T be the collection of trace functions for closed curves on S. These func-
tions also naturally form a subalgebra of the skein algebra associated with skeins
which do not include marked points. The following proposition is a natural descrip-
tion of the skein algebra of a surface, see [6].
Proposition 2.6.2. The skein algebra of S is generated by the elements of A(S)
and T .
This proposition essentially means that we may use skein relations to write
any skein as products and sums of skeins with no crossings.
Example 2.6.3. We can compute the element of the skein algebra corresponding
to the arc ? on the surface S from example 2.4.1. We can see that
?a1a2 = a1a2 + a1a2 (2.25)
as shown in figure 2.13. Thus ? = 2 in the skein algebra. This is expected, since
this element must be the same as the trace of the monodromy operator associated
56
with ? which is 2, since this arc is homotopic to a puncture and we are computing
this from a point in TA(S).
57
Chapter 3: Examples of Mutation Invariant Functions
We now compile many interesting examples of mutation invariants.
3.1 First Examples
Example 3.1.1. First we cover example 1.2.1 in a bit more detail.
Let Q be a quiver of type A?1 shown in figure 3.1. Lets first understand the
cluster ensemble associated with this quiver. The seed groupoid looks essentially as
follows, with pairs of A and X coordinates shown on the nodes of Q to represent
the seeds i, ?1(i) = i
? and ?2(i) = i
??:
(a??, x??1 1) (a1, x1) (a
?
1, x
?
1)
?1 ?2 ?1 ?2
. (3.1)
(a??, x?? ? ?2 2) (a2, x2) (a2, x2)
Each of these seeds has the same underlying quiver so the seed groupoid only has one
object and has maps indexed by Z. Thus, the cluster modular group is isomorphic
to Z and has the same presentation at every seed. A generator of this group is given
by
? = {1, (12)} ? ?Q ' Z. (3.2)
The positive varieties Ai and Xi are both homeomorphic to (Gm)2. The mu-
58
1 2
Figure 3.1: Quiver of type A?1.
tations induce birational maps according to the mutation rules. For example, we
have
1 + a2
??(a? , a?1 1 2) = (
2 , a2) (3.3)
a1
??1(x
?
1, x
? ?1 2
2) = (x1 , x2(1 + x1) ). (3.4)
The map ? : Ai ? Xi is given on coordinates by
??(x , x ) = (a2, a?21 2 2 1 ). (3.5)
The action of ?Q on F(Ai) and F(Xi) is given by
1 + a2
?(a1, a2) = (a ,
2
2 ) (3.6)
a1
?(x1, x2) = (x2(1 + x
2 ?1
1) , x1 ). (3.7)
It is not too difficult to write an invariant for ?Q on AQ. The function
1 + a21 + a
2
F (a 21, a2) = (3.8)
a1a2
is invariant. We also have an invariant for this group on XQ :
(x (x + 1) + 1)22 1
G(x1, x2) = . (3.9)
x1x2
It is easy to check that ??(G) = F 2
Let us interpret these invariants in terms of hyperbolic geometry. The A?1
affine ensemble is associated to an annulus with one marked point on each boundary
59
component (see figure 3.2). The quiver for this triangulation is shown in figure 3.3.
There are two frozen nodes and coefficients to account for the boundary arcs on S,
call the variables on the A space associated with them c, d. The cluster modular
group corresponds exactly to the mapping class group of S and the generator ? =
{1, (12)} corresponds to a Dehn twist about ?. If we take the trace of the monodromy
operator, ?(?), associated with the closed curve ?, we find that on the X space
x2(x1 + 1) + 1
Tr(?(?)) = ? . (3.10)
x1x2
This is exactly the square root of G(x1, x2) from before. Furthermore, we have
? a2 + a2
??( G(x , x )) = 1 2
+ cd
1 2 = F (a1, a2, c, d) (3.11)
a1a2
which is the previous invariant we found, taking into account the new coefficients.
We can also investigate the limiting behavior of the cluster variables via the
hyperbolic geometry of S and the skein algebra of S.
Let us explicitly compute the element of Sk(S) corresponding to the simple
closed curve ? on S. Call this element Fsk. We find using a skein relation that
a2 + cd a2 + a21 1 2 + cda1Fsk = a2 + ?2(a2) = a2 + = . (3.12)
a2 a2
Thus Fsk = F is simply another incarnation of the same invariant found earlier.
The invariance of F can be seen since the closed curve ? does not change when we
do a Dehn twist.
{ n?1 1+a
2
Let a 2n} = ? (a1) = {a1, a2, , . . . } be the sequence of cluster variablesa1
obtained after repeated applications of ?. Then, applying ?n?1 to our previous skein
60
Figure 3.2: The surface S, along with a choice of triangulation. ? is the generator of
the fundamental group and the mapping class ? corresponds to a Dehn twist about
?.
relation and using the invariance of F , we find the linear recurrence
an+1 = anF ? an?1. (3.13)
We can explicitly write a solution for this linear recurrence in terms of the initial
[3]
1 2
[4]
Figure 3.3: The quiver associated with a marked annulus.
61
two cluster variables a1, a2 and the coefficients c, d. Let
?
F + F 2 ? 4
? = = exp(arccosh(F/2)) (3.14)
2
be such that ?+ ??1 = F . ? is the eigenvalue of a matrix representing ?(?). Let
w = a2 ? a ?11? w? = ?a2 + a1?. (3.15)
Then we have that
1
a = n ? ?nn
?? ??
(w? + w ? ). (3.16)
1
The A coordinates correspond to lambda lengths associated to the two interior
arcs on S. After many applications of ?, their lambda lengths grow like multiples
of ?. One may check that there is exactly 1 closed geodesic on S with the same
homotopy class as ? and that its hyperbolic length is equal to 2 arccosh(F/2). Thus,
the multiplier ? is essentially the lambda length of the arc ?. This should make
intuitive sense, since each time we twist, we are adding approximately the length
of this geodesic to each arc, so in terms of lambda-lengths, we multiply by the
exponential of half of the length.
Example 3.1.2. The mutation class of a quiver of type Dn has an element that
looks like a directed n-cycle, as shown in figure 3.4. Let Q be this quiver. The
cluster modular group for Q is Z/nZ ? Z/2Z generated by r and t in each of the
factors1. The function
1 1 1
F (a1, a2, . . . , an) = + + ? ? ?+ (3.17)
a1a2 a2a3 ana1
1This is not quite true when n = 4, but the given function is still an invariant.
62
3 4
2 5
1 n
Figure 3.4: Qn?cycle. This quiver is simply an n cycle of nodes and arrows.
is an element of F(A )<r>Q and satisfies t(F ) = F?1. This is a generalization of
example 2.3.3.
Example 3.1.3. Let (AQ,XQ) be the cluster ensemble with trivial coefficients as-
sociated with the Markov quiver, Q, of figure 3.6a. The function
a2 + a2 + a2
F (a1, a2, a ) =
1 2 3
3 (3.18)
a1a2a3
is an invariant function for all of ?Q = PSL(2,Z) on AQ. The function
G(x1, x2, x3) = x1x2x3 (3.19)
is an invariant for ?Q on XQ.
The function F encapsulates the Diophantine properties of the Markov num-
bers. The Markov numbers are generated by evaluating theA coordinates at (1, 1, 1).
We write these as triples of integers (x, y, z). Since F (1, 1, 1) = 3, we have that the
Markov numbers all satisfy x2 + y2 + z2 ? 3xyz = 0.
If we freeze any one of the nodes in Q, then remaining mutable portion is a
quiver of type A?1. We can use the invariant in the exact same way as example 3.1.1
to study the limiting behavior of mutations on this modified quiver. In this way we
recover similar analysis of [36].
63
Figure 3.5: The surface S1,0,1,0 with a choice of triangulation and associated quiver.
There is a simple geometric interpretation of the invariant of the Markov
quiver. If S = S1,1 is a torus with one puncture then the cluster ensemble asso-
(1,1)
ciated to S is of type A1 see figure 3.5. If we have a hyperbolic metric on S
with the puncture at infinity and a horocycle around the puncture, then the theory
of lambda lengths implies that the function F is simply the formula for half the
length of the horocycle in terms of the A coordinates. Since the length of the horo-
cycle is independent on the triangulation and there is only one topological type of
triangulation, F must be invariant under exchanges.
3.2 Invariants for Doubly Extended Quivers
Quivers associated with doubly extended Dynkin diagrams all have similar
(1,1)
properties to the Markov quiver, which is of type A1 . We will show several
examples of invariants which are analogous to the Markov invariant.
64
1
2 2
1[4] 1
3 2 2
1,4 4,1 4,1 1,4
(a) The Markov quiver.
Also associated with a 3 2 2 3[4] 2 2[4]
punctured torus and of (2,1) (2,4)(b) BC1 . (c) BC1 .
doubly extended type
(1,1)
A1 .
Figure 3.6: Quivers associated with doubly extended Dynkin diagrams with 3 nodes.
(2,1)
Example 3.2.1. Let Q be a quiver of type BC1 Then the function
a41 + (a2 + a3)
2
F (a1, a2, a3) = (3.20)
a21a2a3
is an element of F(AQ)?Q .
(2,4)
Example 3.2.2. Let Q be a quiver of type BC1 The function
a21 + 2a
2 2
1(a2 + a3) + a
4 + a4
F (a1, a2, a
2 3
3) = (3.21)
a 2 21a2a3
is an element of F(A )?QQ . This function and its Diophantine properties were studied
by Lampe in [20]. It was shown that all solutions for the equation F (x, y, z) = 7
could be obtained from the initial solution F (1, 1, 1) = 7 by cluster mutations. It
would be interesting to study this type of Diophantine problem for other invariant
functions.
65
(3,3)
Example 3.2.3. A quiver of type G2 has two quiver isomorphism classes. Let
Q be the isomorphism class shown in figure 3.7. we can see that the element ? =
{1, (12)} is in ?Q. It is not hard to compute that
1? N(?)? ?Q ? D12 ? 1 (3.22)
where N(?) is the normalizer of the element ? and D12 is the dihedral group with
12 elements. The functions
(a3 + a3)(a + a ) + a a (2a2 + a a + 2a2) + a2 21 2 3 4 1 2 a
F1(a , a , a , a ) =
1 2 1 2 3 4
1 2 3 4 (3.23)
a21a
2
2a3a4
and
(a 21 + a2) + a3a4
F2(a1, a2, a3, a4) = (3.24)
a a a21 2 3
are elements of F(A )N(?)Q .
(1,1)
Example 3.2.4. A quiver of typeD4 has 4 mutation classes, call themQ1, Q2, Q3, Q4,
see figure 3.8. Let Q4 be our seed. The cluster modular group can be written as an
extension
1? Z ? Z? ?Q4 ? Aut(F4)+ ? 1 (3.25)
where Aut(F )+4 is orientation preserving group of automorphisms of the F4 root
1[3]
1,3
4 2 3[3]
3,1
2[3]
(3,3)
Figure 3.7: Quiver of type G2 .
66
system of order 1152. This group is generated by the orientation preserving elements
of the Weyl group W (F4) along with the duality autormorphism.
We will see below why Z ?Z is a normal subgroup. W (F4) is generated in the
quotient by the elements of Aut(Q4) and the paths {1231, (23)} and {14, (2356)}.
An element of F(A )Z?ZQ4 is
(a1a4 + a
2
2a5 + a3a6)
F (a1, a2, a3, a4, a5, a6) = (3.26)
a1a2a3a4a5a6
There is also an X invariant
G(x1, . . . , x6) = x1x2x3x4x5x6. (3.27)
These are not the only functions in its exchange class, that is to say that
Aut(F +4) acts non trivially on this function. After applying the element {1231, (23)},
we obtain the functions
a1a4 + a2a5 + a3a6
F456 = G456 = x4x5x6 (3.28)
a4a5a6
There are 24 different functions in the exchange class of F . They are
F, F?1
a1
, F456, F
?1
456, (3.29)a4
along with each of their images under the automorphism group of Q4.
There is a relationship with these functions and the invariant of the Markov
quiver. If we fold Q4 by associating nodes 1 and 4, 2 and 5, 3 and 6, we obtain
(1,1)
a quiver of type A1 . Under this folding, each of these functions becomes an
(1,1)
invariant function for a cluster algebra of type A1 . In other words, if we set
a1 = a4, a2 = a5, a3 = a6 then our functions become either 1, F
? or F ?2, where F ? is
the invariant of example 3.1.3.
67
There is also a relationship between these invariants and the invariants of the
(2,1) (2,1)
BC1 ensemble of example 3.2.1. We can fold Q1 to obtain a quiver of type BC1
by associating nodes 2, 3, 5, and 6. We can find invariants for AQ1 by pulling our
set of invariants on AQ4 back along a mutation path between these two quivers. In
doing so, we obtain the functions
(a1 + a
2
4) + a2a3a5a6 ?1 a2F1425 = , F
a a a a 1425
, F23 = (3.30)
1 4 2 5 a3
and each of their images under the action of Aut(Q1), as elements of F(A )Z?ZQ1 .
(2,1)
Each of these clearly folds to an invariant of the BC1 algebra by setting a2 =
a3 = a5 = a6.
Example 3.2.5. The Weyl group, W (D4) = Z32 o S4, is a normal subgroup of the
(1,1)
modular group of the D4 cluster ensemble. For a simple presentation for this, take
Q1 as our initial quiver and look at the subgroup generated by the Aut(Q1) = S4
and the path {214, (142)} ? {314, (143)}?1. This generates W (D4) as a subgroup of
Z42 oS4 generated by the elements of S4 and the element ((1,?1, 0, 0), id). We have
an exact sequence:
1? W (D4)? ?? PSL(2,Z)? 1 (3.31)
Consider Q4 as our seed. The function
(a1a
3
4 + a2a5 + a3a6)
F (a1, a2, a3, a4, a5, a6) = (3.32)
a1a2a3a4a5a6
is an element of F(A )W (D4)Q4 .
68
1 1
6 2 6 2
2
5 3 5 3
4 4
(a) Q1 (b) Q2
1 1
6 2 6 2
5 3 5 3
4 4
(c) Q3 (d) Q4
(1,1)
Figure 3.8: Quivers of type D4 .
3.3 Examples from Somos Sequences
Example 3.3.1. The Somos 4, 5 and 6 sequences can be associated to cluster
algebras in a simple way2. If we look at the variables on the A space generated
by starting with quivers Qs4 , Qs5 or Qs6 and following the mutation paths ?4 =
{1, (1234)} , ?5 = {1, (12345)} or ?6 = {1, (123456)}, we obtain the respective
Somos sequences by evaluating all of the initial cluster variables at 1. Explicitly,
2This Somos 6 sequence is not the most famous instance, see [37] for a discussion of the example
shown here.
69
1 2 3 3 4
3 2 4 2 5
4 3 1 5 1 6
(a) Qs4. (b) Qs5. (c) Qs6.
Figure 3.9: Quivers for the Somos 4, 5 and 6 sequences.
these sequences can be obtained by the recurrence
Xn+kXn = Xn+1Xn+k?1 +Xn+bk/2cXn+dk/2e (3.33)
with {X1, X2, . . . , Xk} all equal to 1 for k = 4, 5, 6 respectively. This gives the
sequences
S4 = {1, 1, 1, 1, 2, 3, 7, 23, 59, 314, . . . } (3.34)
S5 = {1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, . . . } (3.35)
S6 = {1, 1, 1, 1, 1, 1, 2, 3, 4, 8, 17, 50, 107 . . . }. (3.36)
There are invariant functions on each of these cluster ensembles for these
mutation paths. The functions
a2a2 + a1a
3 3 2 2
F (a , a , a , a ) = 1 4 3
+ a4a2 + a2a3
4 1 2 3 4 (3.37)
a1a2a3a4
a2a2a + a a2a2 + a a2a2 + a2 2 35 1 1 a a5 + a2a a4
F5(a1, a2, a , a , a ) =
1 4 2 5 3 4 2 3 3
3 4 5 (3.38)
a1a2a3a4a5
70
and
F6(a1, a2, a3, a4, a5, a6) =
a2a a a2 + a2a a3 + a3a a2 + a a2a a2 + a2a a2a + a a a3a + a a3a a + a3a31 2 5 6 1 4 5 2 3 6 1 3 4 5 2 3 4 6 1 3 4 5 2 3 4 6 3 4
a1a2a3a4a5a6
(3.39)
are elements of F(A <?4> <?5> <?6>Qs4) , F(AQs5) and F(AQs6) respectively.
Explicitly, this means that each of these functions has the same value when
applied to a consecutive collection of numbers from its respective sequence. For
example, we see that
?
F6(1, 1, 1, 1, 1, 1) = 8 = F6(1, 2, 3, 4, 8, 17) (3.40)
4624 + 2048 + 6936 + 2304 + 3264 + 1536 + 3672 + 1728
= = 8 (3.41)
3264
The functions F4 and F5 have appeared in [21] in relation to the number
theoretic properties of the Somos sequences e.g. one may compute the j?invariant
of an elliptic curve associated with the sequences using the values of these functions.
While the cluster algebra interpretation of the Somos sequences has been studied,
the mutation invariant formulation of these functions is new.
These functions for general periodic quivers are studied in [38].
71
Chapter 4: First Properties and Construction of Invariants
In this chapter we prove some theorems about the abstract structure of mu-
tation invariants, and show some basic constructions of them. We also do some
computations which will be help full in the proof of our main theorem.
4.1 Basic properties of mutation invariant functions
First, we consider properties and constructions of invariants that can be used
in general.
4.1.1 Trivial Invariants
Since it respects mutations, the map ? : A ? X gives a map
?? : F(X )?? ? {?} ? F(A)?? ? {?} (4.1)
by pullback. This is very useful for studying the invariants on the A space when
there are nontrivial coefficients. However, since ? is not surjective or injective in
general, we cannot say much about the image of the set of X invariants under ??.
Definition 4.1.1. A rational function in the coefficients of A is called a trivial A
invariant.
72
These functions are clearly unchanged by the action of cluster modular group
elements.
We can understand what functions pull-back to trivial A invariants via ?? as
follows: any vecto?r, v = (v1, . . . , vn) of the kernel of the exchange matrix gives such a
function since ??( xvii ) is a function just of frozen variables, see section 2.3 of [12].
These functions are called ?Casimir? functions.
Definition 4.1.2. A X invariant that is also a Casimir function will be called a
Casimir X invariant.
Example 4.1.1. The type An cluster ensembles have an interesting Casimir X
invariant when n is odd. The cluster modular group of the type An cluster ensemble
is Z/(n + 3)Z, generated by ?. Take Q to be the quiver in figure 4.1 and let
G(x1, . . . , x
Z/mZ
2n+1) = x1x3 . . . x2n+1. Then G is a trivial Casimir in F(XQ) , where
m = n+3 . G satisfies ?(G) = G?1.
2
1 2 ... 2n+ 1
Figure 4.1: Quiver of type A2n+1.
4.1.2 Invariants of Subensembles
The invariants of the A space and X space of an ensemble respect inclusions
of ?subensembles? in different ways.
Definition 4.1.3. Let R? ? Q? be a subquiver of the mutable portion of Q. Let
R be the quiver obtained from Q by freezing all the nodes in N(Q) ? N(R?). We
73
consider (AR,XR) to be a subensemble of (AQ,XQ).
Let ?R = {PR, ?R} be an element of the cluster modular group of R that
extends trivially to an element of the modular group of Q, i.e. there exists ?Q =
{PR, ?Q} where ?Q|R = ?R and ?Q does not permute any of the frozen nodes in R.
Remark 4.1.1. In all our applications, we will need to construct a unique extension
of cluster modular group elements in ?R.
We can now state the following pair of propositions:
Proposition 4.1.1. F(A )<?Q>Q is exactly F(A )<?R>R . Furthermore, evaluating
the variables associated to the nodes in N(Q) ? N(R?) at 1 gives a map of sets
F(A )<?Q>Q ? {?} ? F(A <?R>R?) ? {?}.
Proposition 4.1.2. We have a natural inclusion F(X )<?R>R ? F(X <?Q>Q) . Fur-
thermore, evaluating the variables associated to nodes in N(Q?)?N(R?) at 0 gives
a surjection of sets F(X )<?Q>Q ? {?} ? F(X )<?R>R ? {?}.
Proof. The first proposition follows from the fact that the mutation rule on the
A space only changes the variable being mutated. The first part of the second
proposition follows since the X coordinates are unchanged when frozen variables are
added. The only thing left to prove is that evaluating the nodes in N(Q?)?N(R?)
at 0 gives a map from F(X )<?Q> to F(X )<?R>Q R . The surjection is obvious because
of the inclusion of rings.
Let r = #N(R?) and renumber the nodes of Q so that the first r nodes are
the nodes in N(R?). Let Gi(x1, . . . , xn) = xi and let j = ?
?1(i). Then for any i > r,
74
we have
?Q(Gi) = xjhj(x1, . . . , xr) (4.2)
, where hj is some rational function that does not depend on any xi, i > r. This fol-
lows since we never mutate at node i in ?Q, and the X mutation rule only multiplies
coordinates by functions of those which are mutated.
Let K = F[xr+1, . . . , xn] and L = F(x1, . . . , xr) = F(XR). Then, ?Q acts on
any monomial term in K by multiplication by an element of L. Now we write any
function, g = p ? F(X
q Q
) = F(x1, . . . , xn), as a ratio of two polynomials in K with
coefficients in L. Let p0, q0 ? L be the constant terms of p and q. Now we may see
that if g is invariant then we have
p?Q(q) = q?Q(p) (4.3)
which implies by comparing constant terms that
p0?Q(q0) = q0?Q(p0) (4.4)
and so we have that p0 ? L is invariant.
q0
Corollary 4.1.1. The dimension of the field extension F(XQ)<?Q>/F(X )<?R>R ?
n? r.
Proof. This follows from the proof of the previous theorem. Since ?Q acts on each
monomial in K independently by multiplication by elements in L, any basis of
F(X )??Q?/F(X )<?R>Q R can be written as a basis of monomials in K with coefficients
in L. Now we may see that any particular monomial in K can only be associated
75
with one independent invariant, since the ratio of two such invariants is in F(X )??Q?R .
The corollary now follows from realizing that there are at most n? r algebraically
independent monomials in K.
Remark 4.1.2. The evaluation maps should be considered to be related to pro-
jective maps of varieties which have the corresponding invariant rings as function
fields. However, it is not clear that these varieties actually exist.
Combining these two propositions together, we may state the following useful
theorem.
Theorem 4.1.1. Let Q be a quiver and let R be a quiver obtained by deleting some
subset D ? N(Q) of the nodes of Q. Let A(D) and X (D) be the collections of A
and X coordinates associated with the nodes in D. Suppose that there is an element
? ? ?R such that ? does not mutate at any node in R adjacent to a node in D. Then
there exists ?Q ? ?Q which restricts to ?R and we have
1. F(A )??Q?Q = F(A )??R?R (A(D))
2. F(X )??Q?Q = F(X )??R?R (X (D))
or to write simply
F(A ,X ??Q? ??R?Q Q) = F(AR,XR) (D). (4.5)
Proof. Our element ?Q can be constructed from ?R by extending the quiver isomor-
phism of ?R to all of Q.
76
Since ?R does not mutate at any node adjacent to the nodes in D, we may
construct a new quiver R? where we freeze every node of R which is adjacent to any
node in D and naturally think of ?R as an element in ?R? . Then adding the nodes
of D to R? as frozen nodes does not change the mutation structure of the ensemble
associated with F? since the new nodes are only connected to frozen nodes. Adding
these nodes simply adds coefficients which appear in every cluster, but do not effect
the exchange relations in any way.
Now, since ?R returns R to an isomorphic quiver, it also does the same for R
?.
Finally, we may find our desired isomorphism of Q by attaching the nodes in D to
R? before and after applying ?R and choosing the map which gives an isomorphism
between these two quivers.
Denote by Q? the quiver obtained from Q by freezing all the nodes in D along
with nodes adjacent to those in D. We have ?(Q?) = ?(R?). Now we have
F(A )??Q? = F(A ?)??Q? = F(A ?)??R?(A(D)) = F(A )??R?Q Q R R (A(D)) (4.6)
where the first and third equality follow from proposition 4.1.1 the second equality
follows from our realization that the A coordinates for D are trivially invariant and
that AR? and AR? have the same mutation structure.
For the X space, we have by proposition 4.1.2
F(X )??R?R ? F(X )??Q?Q (4.7)
is a field extension of degree less than #D by corollary 4.1.1. We can also can see
that
F(X )??R?R (X (D)) ? F(X )??Q?Q . (4.8)
77
4 1 m? 1
3 2 m
Figure 4.2: Quiver with double edge and oriented cycles.
This must be an equality since we may write a basis of R(X )??Q? over F(X )??R?Q R in
terms of monomials in X (D), but every such monomial is contained in F(X )??R?R (X (D)).
Example 4.1.2. Let R be the quiver shown in figure 4.2, and let Q be any quiver
which is obtained by adding nodes to R which are only connected to nodes 3, . . . ,m.
Let D be the set of these new nodes. We have ? = {1, (12)} ? ?R,?Q. Then theorem
4.1.1 implies that
F(A ,X )???Q Q = F(AR,X )???R (D). (4.9)
4.2 Invariants Of Ensembles Associated With Surfaces
Our main source of invariants of cluster ensembles associated with surfaces will
be from trace functions associated to closed curves on S. Our formulas in section
2.5.3 show how we may write the monodromy operator associated with traversing a
closed curve, ?, in terms of the A and X coordinates of a triangulation of S. This
allows us to assign a trace function of the A and X coordinates for each closed curve
? on S. The formula for a trace function only depends on the order that ? touches
the arcs of the triangulation. Therefore, this formula must remain the same if we
78
Figure 4.3: The arcs near a boundary with one marked point.
apply a mapping class to S which fixes ?, and it thereby gives rise to an invariant
function.
4.2.1 Calculating trace functions
It will be important to know what the exact formula for the trace of a mon-
odromy operator for a closed loop on S is in terms of the A and X coordinates in
a special case.
Let S be a marked surface and assume that S has a boundary component with
exactly one marked point. Let ? be a closed curve on S homotopic to this boundary
component and let {e1, e2, . . . , ek} be the arcs in an ideal triangulation of S which
touch the marked point on this boundary, see figure 4.3.
We can write an explicit formula for the trace of the monodromy operator
associated with ? in terms of the X coordinates. Recall the matrices associated to
arcs in ? and the matrices R and L from section 2.5.3. Call the matrices associated
to {e1, e2, . . . , ek} {X1, . . . , Xk}. Choosing the orientation of ? shown in figure 4.3,
79
we compute that
?(?) = X1RX2R . . .Xk. (4.10)
With a little bit of work, we find
(?i=k ?j=i1 )
Tr (?(?)) = ? xj + 1 . (4.11)
x1x2 . . . xk i=1 j=1
This is a rational function in the X coordinates exactly when the product under the
square root is a square.
4.2.2 Other types of surface invariants
We can produce A invariants by clever use of the following fact about lambda
lengths. Let S be a hyperbolic surface decorated with horocycles, t ? ? be a
triangle, and let h3 be the segment of the horocycle bounded between edges 1 and
?3
2, see figure 4.4. Then the hyperbolic length h3 is , see lemma 4.4 of [31].
?1?2
We can use this formula to write a function that evaluates to the length of a
horocycle about a marked point on our surface in terms of the A coordinates of a
given triangulation. This function is preserved under the action of mapping classes
since the total length of the horocycle and the topology of the triangulation are
preserved. It is important to notice that this expression will not be invariant under
any mapping class that permutes this marked point with another.
We can construct an invariant for the entire mapping class group by summing
all of the lengths of horocycle segments on S. Let et1, et2, et3 be the three edges of
80
Figure 4.4: An ideal hyperbolic triangle with horocycle segments around its three
vertices.
a triangle t ? ?. This invariant is given by the formula
? a2e + a2e + a2t1 t2 et3 . (4.12)
a
? et1
ae
t ? t2
aet3
We can easily produce Casimir X invariants associated to each puncture on
S by taking the product of the X coordinates associated to edges touching the
puncture. These are the functions pX mentioned in 2.17.
Example 4.2.1. We can interpret the invariants of example 3.2.4 using a surface.
The cluster ensemble associated to S = S0,0,4,0, a four punctured sphere, is of type
(1,1)
D4 . If we take a triangulation of S that topologically looks like a tetrahedron,
then the quiver associated with this triangulation is Q4 from figure 3.8 and the
length of a particular horocycle around a puncture is given by a function like F456.
The function G456 is the X invariant associated a puncture on S. We have that
PMCG(S) = Z ? Z (4.13)
and hence we have that these functions are invariant under the subgroup claimed.
81
Remark 4.2.1. We note that specifying a values for horocycle lengths and fixin
pX = 1 for every puncture are the relations between TA(S), TX (S) and T (S). We
may think of these functions as being the ?gluing equations? for a hyperbolic struc-
ture on S.
One can interpret the invariance of these expressions as a way of stating that
the action of the mapping class group does not change the topological structure of
the surface and hence does not change the conditions that must be satisfied for there
to be a hyperbolic structure on it.
4.3 Sequences of A coordinates from Dehn twists
As we saw in example 3.1.1, the invariant function (the trace function) of the
A?1 affine ensemble was essential for understanding the sequence of cluster variables
obtained along the mutation path we were considering. We will need to understand
the sequences of A coordinates found by doing Dehn twists on general surfaces.
In most cases, we can arrange that the closed arc which we are twisting lives in a
cylinder on S and thereby simply apply the analysis of 3.1.1. The only new case we
will need to consider is when our surface is a pair of pants with one marked point
on one of its boundaries. This surface is not a marked surface, since there are no
marked points on two of its boundaries, but we may still consider its skein algebra.
Let S be this surface, shown in figure 4.5. Two arcs from the marked point to
itself is shown, call these arcs a0 and b0. Let A,B,K be the elements of the skein
algebra of S associated with closed curves homotopic to the three boundaries as
82
Figure 4.5: A pair of pants with one marked point before and after performing a
Dehn twist about the bottom boundary
shown in the figure. Furthermore let f be the arc from the marked point to itself
which is homotopic to K. Let {an}, {bn} be the sequence of arcs obtained after
performing n clockwise Dehn twists about K. Using several Skein relations, we may
first compute that
a0K = b0 + b1 + fA (4.14)
see figure 4.6. Multiplying through by K and using similar skein relations, we find
a0K
2 = b0K + b1K + fAK = a?1 + a0 + fB + a0 + a1 + fB + fAK. (4.15)
Thus, we find that {an} again satisfies a linear recurrence:
a = a (K2n+1 n ? 2)? an?1 ? f(AK + 2B). (4.16)
We may solve this recurrence explicitly. Let K = ?+ ??1 and let
? ?2 f(AK + 2B)w = a1 a ? + (??20 ? 1) (4.17)
K2 ? 4
? ? 2 ? f(AK + 2B)w = a1 + a0? (?2 ? 1). (4.18)
K2 ? 4
83
Figure 4.6: The Skein relations obtained from computing a0K. This is a zoomed in
view near the points where the crossings occur
Then we have
1 2n ? ?2n f(AK + 2B)an = ? (w? + w ? ) + . (4.19)?2 ? ? 2 K2 ? 4
Furthermore, using more skein relations we may compute that
K2? (A
2 +B2 +K2 + ABK ? 4)
ww = (4.20)
K2 ? 4
w ? w? = 2a1 ? a 20(K ? 2) + f(AK + 2B) = a1 ? a?1. (4.21)
Thus w and ?w? are roots of the following polynomial equation
2 2 2 2
2 ? ? ? K (A +B +K + ABK ? 4)w (a1 a?1)w = 0. (4.22)
K2 ? 4
This analysis can be applied whenever we are in a situation where we have an
A coordinate in a surface cluster algebra which lives in a pair of pants on S. This
will allow us to compute sequences of A coordinates for Dehn twists which do not
84
live in cylinders on S. We simply need to evaluate the trace functions for K,A,B in
terms of the other A coordinates, then we may use our explicit formula to compute
the sequence.
85
Chapter 5: Surface Invariants
Now we will state and prove our main theorems. We concentrate first on
ensembles associated with surfaces, but we show that the same proof ideas work for
mutation finite cluster ensembles as well.
5.1 Invariants of Surface Cluster Ensembles
In this section, we will classify the invariants in the situation where our clus-
ter ensemble is associated with any surface and we are considering invariants with
respect to the action of a single Dehn Twist.
Let S = Sg,b,p,n be a marked surface. We will construct invariants rings for
the cluster ensembles associated with ideal triangulations of S by using particular
initial triangulations obtained by ?excising? our given Dehn twist.
Definition 5.1.1. We say a simple closed curve, ? on S is genus 0 on S if cutting
along ? leaves S connected, or if we can find a marked point or puncture on both
connected components of S after cutting.
When ? is genus 0, we may pick an ideal triangulation of S which cuts excises
a cylinder containing ? in the following way: pick any two, not necessarily distinct
marked points or punctures on S, p1, p2 along with curves c1 from p1 to one side of
86
Figure 5.1: The construction of arcs which excise a given closed curve.
? and c2 to the other side of ?. We can always find these curves since ? is excisable.
We then begin a triangulation of S by first using the arc e1 obtained by following
c1, traversing ? and then following c1 back to p1 and a similarly constructed arc e2.
This procedure is shown in figure 5.1.
When ? is not genus 0, we may still find one marked point and curve coming
to ? from one side. In this case we can construct e1 and we cut out a surface of
genus g from S with one boundary. We say that ? is genus g, and we may again
complete
Definition 5.1.2. A triangulation of S which contains the arcs ei arcs is called an
excising triangulation for ? The arcs of an excising triangulation which intersect ?
are called the interior arcs and the arcs who?s associated quiver nodes are connected
to the nodes of the interior arcs are called boundary arcs.
See figure 5.2 for a examples of various genus curves on a surface.
Every simple closed curve on S and choice of excising triangulation determines
87
Figure 5.2: The blue closed curve is genus 0, and the red closed curve is genus 1.
two important subensembles. We denote by (A?,X?) the subensemble obtained by
freezing all of the nodes other than the interior arcs. When delta is genus 0, this
subensemble is of type A?1, and we say that it is of genus g type otherwise.
We denote the subensemble obtained by freezing all of the nodes other than
the interior arcs and boundary arcs by (Ab?,X b? ). We have
(A?,X?) ? (Ab,X b? ? ) ? (A,X ) (5.1)
.
Definition 5.1.3. We call the coefficients of A? for any choice of excising triangu-
lation the invariant A coordinates for ?
Let m be the total number of interior and boundary arcs. Suppose that ?
is genus 0 on S. Then there are 2 interior arcs of our triangulation and between
2 and 4 boundary arcs. Number nodes of the quiver associated with our excising
88
triangulation with the first being the interior arc on the source of the double edge,
then the second interior arc, then the boundary arcs, then the rest.
Definition 5.1.4. The functions x3(x2(x1+1)+1), . . . , xm(x2(x1+1)+1), xm+1 . . . , xN
are called invariant X coordinates for our excising triangulation when ? has genus
0 on S
Suppose now that ? is genus g > 0. There are now 1 or 2 boundary arcs. let
xI be the product of the X coordinates associated with the interior arcs. Recall
that the total number of interior arcs is 6g? 2, and number the nodes of the quiver
with the boundary arcs first, then interior, then the rest.
Definition 5.1.5. The functions xIx1, . . . , xIxm?(6g?2), xm+1, . . . xN are called the
invariant X coordinates for our excising triangulation when ? has genus g on S.
Definition 5.1.6. In any case, the A coordinate functions for arcs of our excis-
ing triangulation which do not intersect ? (non interior arcs) are the invariant A
coordinates for our excising triangulation.
Definition 5.1.7. The closed curves that have representatives that are fully con-
tained in the excised surface are called the excised closed curves.
In figure 5.2, the excised curve for the blue curve are simply curves which wrap
around the excised cylinder a number of times. The excised curves for excising the
red curve are curves which are contained in the genus 1 surface.
Let ? be the cluster modular group element corresponding to a Dehn twist
about ?. Our main theorem can be stated as follows:
89
B
R
Y P
[f ]
Figure 5.3: The triangulation and associated quiver when g = 1 quiver
Theorem 5.1.1. 1. The ring F(X )???S is generated by traces of monodromy oper-
ators of excised closed curves on S and invariant X coordinates for an excising
triangulation of ?.
2. The ring F(AS)??? is generated by traces of monodromy operators of excised
closed curves on S and invariant A coordinates of an excising triangulation of
?.
We note that in the X case, not every trace function is itself a rational function
because of the square roots that we have seen 4.2.1. However, the generators we
need are simply those trace functions which are rational.
Before tackling the main theorem, we will treat the case where S is either
a cylinder or surface of genus g with one boundary component. This surface for
g = 0, g = 1, g > 1 is shown in figure 5.3. We also fix a triangulation ? and
associated quiver Q of each of these surfaces, shown in figures 5.3,5.5 and 5.4
Lemma 5.1.1. Let S = S0,2,0,2 be an annulus with one marked point on each bound-
ary or let S = Sg,1,0,1 be a surface of genus g with one boundary component and a
90
Figure 5.4: The triangulation for when g > 1. Each of the g handles is associated
with 4 arcs of the triangulation as figure 5.3, and each of non-handle pairs of pants
is associated with 2 arcs if the triangulation.
9
4 10
5 8 11
3 6 7
1 2 11
[f ] 12
Figure 5.5: The quiver Q when g > 1
91
single marked point on the boundary. Let ? be a simple closed curve on S homo-
topic to the boundary component. Let ? ? ?S be the cluster modular group element
corresponding to a Dehn twist about ?.
1. The ring R(X )???S is generated by traces of of monodromy operators about
simple non-excisable closed curves and by squares of traces of monodromy op-
erators about simple excisable closed curves on S.
2. Then the ring R(A )???S is generated by traces of monodromy operators about
simple closed curves on S and frozen variables.
Proof. We will first treat the A case with g > 0. Let m = 6g ? 2 and let
h(x1, . . . , xm+1) ? R(AS)??? be an invariant function. Let {a1, . . . , a6g?2, f} be our
collection of A coordinates for ? triangulation of S show in figure. The coordinate
f is the frozen variable.
Then let {ain} be the sequence of coordinates obtained from acting on our
initial set by ?n. The invariance of h implies that
h(a1, . . . , am, f) = h(a1n , . . . , amn , f). (5.2)
. Let h? be defined by
1 y2 y3 ym
h?( , , , . . . , , ym+1) = h(y1, . . . , ym+1). (5.3)
y1 y1 y1 y1
Then from the invariance of h we have
h(a1, . . . , am, f) = lim h(a1n , . . . , amn , f) = h?(0, ?2, . . . , ?m, f) (5.4)
n??
where ?i = lim
yi
n?? . We can compute ?i explicitly using the analysis of sectiony1
4.3.
92
Figure 5.6: Two arcs in ? along with the pants that they live in.
Each of the arcs in ? can be seen to live in a pair of pants in the following
way. We may generate each arc by following a path from the marked point to a
closed curve, Ai, then traveling around this closed curve and finally going back to
the marked point. This arc lives in a pair of pants with boundaries Ai, Bi, K where
K = ?S and Bi is the closed curve which is homotopic to the generator of H1(S)
equal to [A] + [K]. This situation is shown in figure 5.6.
In the continuation, we will use Ai, Bi, K to refer interchangeably to the el-
ements of the skein algebra of S or the traces of monodromy operators associated
with their respective closed curves. Let K = ? + ??1 as before. By our analysis of
sequences of A coordinates we have
1 2n f(AiK + 2Bi)ain = (wi? + w
???2n) + (5.5)
?2 ? ??2 i K2 ? 4
93
where
? ?2 f(AiK + 2Bi)w ?2i = ai1 ai0? + (? ? 1) (5.6)K2 ? 4
? ? 2 ? f(AiK + 2Bi)wi = a 2i1 + ai0? (? ? 1). (5.7)K2 ? 4
Thus, we find that
wi
?i = (5.8)
w1
Thus, any invariant function only depends on {?2, . . . , ?m, f}. Thus our field
of invariants contains all of the rational functions in the lambda?s which are also
rational functions in the A coordinates. Clearly, wi is an element in the field ex-
tension R(AS)(?) over R(AS). Let T ? R(AS) be the field generated by traces of
closed curves on S and f . We will show that each ?i is an element in T (?). The
following tower of fields shows the situation:
R(AS)(?)
? 2
? (5.9)
?i ? T (?) h ? R(AS)
2 ?
T
Let ? : R(AS)(?) ? R(AS)(?) be the field automorphism which sends ? to
??1. Then we have
(K2(A2 +B2 +K2? ? ? ?1 i i + AiBiK ? 4))?(wi) = wi = wi (5.10)K2 ? 4
and so
w? ( 2 2 2 )
?(? ) = i = ??1
Ai +Bi +K + AiBiK ? 4
i ? i . (5.11)w A21 1 +B
2
1 +K
2 + A1B1K ? 4
94
(a) ai is separate from a (b) a1 i is contained in a1
Figure 5.7: The two possibilities for the arrangement of arcs a1 and ai
Thus ?i?(?i) ? T . We wish to now prove that ?i + ?(?i) ? T . This calculation is
more involved. There are two possibilities for how the arcs a1 and ai are situated
on S. ai is either contained in a1 or is separate, as figure 5.7 shows
We treat the separate case here. The non separate case follows since if ai is
contained in a1 then we can find an arc which is separate from both, b, and write
ai ai b
= . (5.12)
a1 b a1
For the sake simplifying notation lets write an = a1n ,cn = ain , w1 = wa, wi =
wc and A1 = A,B1 = B,Ai = C,Bi = D, as figure 5.8 shows.
We need to compute W := waw
?
c + w
?
awc . From our formulas we first find
W = ?2a1c1 ? 2a0c0 + (K2 ? 2)(a1c0 + a0c1)? (a1 + a0)f(CK + 2D)
2
? f (AK + 2B)(CK + 2D)(c1 + c0)f(AK + 2B) + .
K2 ? 4
Next we will calculate K2(a1c0 + a0c1) using skein relations. There are essen-
tially two generic possibilities for how the arcs a0and c0 are situated on S. The arc
95
Figure 5.8: The collections of arc and closed curves for the calculation of W in the
separate case
c0 is either contained in a0, like the four colored arc of figure 5.4, or they are sep
We will treat the first case shown in figure 5.8
For the first case, we compute
K(a0c1) = b1c1 + a0d0 + f(c1A+ e1 + a0C) (5.13)
K(a1c0) = a1d0 + b1c0 + f(a1C + e0 + c0A+ a0C + b1D) (5.14)
+ f 2(AD + CB + E +KF +KAC). (5.15)
So we have
K(a1c0 + a0c1) = (c0 + c1)fA+ (a0 + a1)fC (5.16)
+ b1(c1 + c0 + fD) + d0(a0 + a1) + a0fC + (e0 + e1)f (5.17)
+ f 2(AD + CB + E +KF +KAC). (5.18)
96
Using skein relations for Kd0 and Kb1 we find
K(a1c0 + a0c1) = (c0 + c1)fA+ (a0 + a1)fC (5.19)
+ 2b1d0K ? d0fB + a0fC + (e0 + e1)f (5.20)
+ f 2(AD + CB + E +KF +KAC). (5.21)
And using a different skein relation for d0fB we find
d0fB = a0fC + (e0 + e1)f + f
2E. (5.22)
So now multiplying through by K and using the above, we have
K2(a1c0 + a0c1) = (c0 + c1)fKA+ (a0 + a1)fCK (5.23)
+ 2b1d
2 2
0K + f (AD + CB +KF +KAC). (5.24)
Lastly we have
b d K21 0 = (a0 + a1 + fB)(c0 + c1 + fD) (5.25)
= a0c0 + a1c1 + a0c
2
1 + a1c0 + (c0 + c1)fB + (a0 + a1)fD + f BD. (5.26)
So finally we have
K2(a1c0 + a0c1) = (c0 + c1)(fKA+ 2B) + (a0 + a1)(fCK + 2D) (5.27)
+ 2a0c0 + 2a1c1 + 2a0c1 + 2a1c0 (5.28)
+ f 2(AD + CB +KF +KAC + 2BD). (5.29)
At long last we have
2f 2(AK + 2B)(CK + 2D)
W = f 2(AD + CB +KF +KAC + 2BD) + . (5.30)
K2 ? 4
97
Thus ?i satisfies a quadratic polynomial equation with coefficients in T , and is
not fixed by ?. Moreover, any other automorphism which moves ? must also move
?i. Thus T (?i) = T (?) for all i and so h ? T (?). Then since h ? R(AS) is fixed by
?, we have h ? T .
The A case when g = 0 follows by essentially the same argument, only simpler.
Now we will treat the X case. We can see that in each case the map ?? :
R(X ) ? R(A) is injective since there are are no punctures and there is only one
marked point on each boundary. Furthermore, one may check that this map is still
injective even if we remove the variables associated with the frozen nodes.
Thus we may compute the invariant ring by determining which functions pull
back to A invariants after evaluating the frozen variables at 1. Clearly, we may
obtain all such functions by pulling back trace functions in the X coordinates, since
the only other elements of the A invariant ring are generated by the frozen variables.
Now, we will determine which closed curves have traces which are rational
functions in the X coordinates. Let ? be a closed curve on S which is excisable.
Since there is only one marked point on S, these curves are exactly the closed curves
which leave S connected after cutting. We may pick a triangulation of S which only
touches ? once, since we may find two paths from the marked point to ether side
of ? and pick the arc which concatenates these paths. Based on the analysis of
section 4.2.1, the trace of the monodromy operator associated with ? must have a
square root in the denominator when computed with a triangulation which has an
arc which only touches it once.
Thus the only closed skeins which have trace functions that are rational func-
98
tions in the X coordinates are those for which have even intersection number with
every arc. This completes the theorem on the X space.
Now we are ready to prove the general case.
Proof. First, we pick an excising triangulation of ?. We will first compute the ring
of invariants for (Ab b?,X? ) associated with this triangulation. Let (A?,X ?) be the
cluster ensemble obtained by using the same quiver associated with (Ab b?,X? ) but
with all of the frozen node removed. Call by D the set of these nodes. We can find
a mutation sequence representing ? which does not mutate at any of the boundary
arcs, since we could cut along the boundary arcs to make a surface with them as
actual boundaries and consider ? in this surface.
Thus by theorem 4.1.1 we have
R(A ,X )??? = R(Ab,X bS S ? ? )???(D). (5.31)
We only need to compute R(Ab,X b)???? ?
Lemma 5.1.1 exactly computes R(A ????,X?) . The theorem on the A space now
follows from proposition 4.1.1 applied to A? ? Ab?.
The invariant X functions for ? defined in 5.1.5 can be checked to be invariant
under the action of ?. Since these invariant X coordinates are degree 1 in the
boundary X coordinates, theorem 4.1.2 implies that these functions together with
the elements of R(X )???? must generate all of R(X b? )???. This completes the theorem
on the X space
99
The following corollary follows from direct inspection.
Corollary 5.1.1. The invariant ring for a Dehn twist about ? on AS is exactly
the the subalgebra of the skein algebra of S consisting of elements corresponding to
skeins which do not intersect ?
5.2 Invariants on Affine, Doubly Extended and Exotic Ensembles
We will give a classification of the invariants for analogues of Dehn twists
on affine type and doubly extended type cluster ensembles with trivial coefficients.
These classifications will all follow from the following corollary of theorem 5.1.1
Following example 4.1.2, let R be the quiver shown in figure 4.2, and let Q be
any quiver which is obtained by adding nodes to R which are only connected to nodes
3, . . . ,m. Let D be the set of these new nodes. We have ? = {1, (12)} ? ?R,?Q.
Let
(x2(x1 + 1) + 1)
2
G(x1, x2) = ( ) (5.32)x1x2? a2 + a2 + a3 ? ? ? am
F (a1, a2, a3, . . . , am) = ?
? G = 1 2 (5.33)
a1a2
Then the proof of theorem 5.1.1 and theorem 4.1.1 implies the following corol-
lary:
Corollary 5.2.1.
F(A )???R = F(F, a3, . . . , am) (5.34)
F(X )???R = F (G, x3(x2(x1 + 1) + 1), . . . , xm(x2(x1 + 1) + 1)) (5.35)
F(A ,X )??? = F(A ,X )???Q Q R R (D). (5.36)
100
(1,1) (1,1)
The A?p,q, D?n, A1 and D4 affine and doubly extended type cluster ensem-
bles are associated to an annulus with p and q marked points on each boundary
and a twice punctured disk with n? 2 marked points on the boundary respectively,
see [32] examples 6.9 and 6.10. The E type affine and doubly extended cluster en-
sembles are not associated to surfaces, but there is an analogue coming from the
notion of a ?cluster Dehn Twist? of Ishibashi in [39].
5.2.1 Tp,q,r Quivers
Each of the affine ADE mutation classes contain the quiver Tp,q,r, where
1+ 1+ 1 < 1, see figure 5.9 , [40]. For A?p,q, we have (p, q, r) = (p, q, 1) , for D?n we havep q r
(p, q, r) = (n? 2, 2, 2), and for E?n we have (p, q, r) = (n? 3, 3, 2). We will use this
quiver as our initial seed. We associate variables (a1, a2, b2, . . . , bp, c2, . . . , cq, d2, . . . , dr)
for the variables on theA space and variables (x1, x2, y2, . . . , yp, z2, . . . , zq, w2, . . . , wr)
for the variables on the X space in an obvious way.
(1,1) (1,1) (1,1)
The E6 , E7 , E8 cluster ensembles are associated with Tp,q,r quivers with
(p, q, r) equal to (3, 3, 3), (4, 4, 2), (6, 3, 2) respectively. The A and D type doubly
extended ensembles are not quite explicitly Tp,q,r quivers, but have quivers with
analogous properties, see figures 3.6a and 3.8a
The cluster modular groups of the cluster ensembles associated with Tp,q,r
quivers and the X6, X7 (see figure 2.3) quivers have the element ? = {1, (12)} ? ?Q.
In the case that this ensemble is associated with a surface, this element corresponds
to a Dehn twist. In the non surface cases, we refer to this element as a cluster Dehn
101
1 P2 ... Pp?1 Pp
Rr Rr?1 ... R2
2 Q2 ... Qq?1 Qq
Figure 5.9: General form of a Tp,q,r quiver.
twist.
Let Q be a non-surface type mutation finite quiver with, as in table 2.2.
Definition 5.2.1. An element ? ? ?Q is called a cluster Dehn twist if there is some
seed i for which ? ? ?i is represented by a single mutation at a node on a double
edge. An element ? ? ?Q is called a partial cluster Dehn twist if some power of it is
a cluster Dehn twist.
Remark 5.2.1. Ishibashi does not make a distinction between partial and non-
partial cluster Dehn twists.
Remark 5.2.2. A finite index subgroup of the cluster modular group ?Q for any
mutation finite quiver is always generated by partial cluster Dehn twists as shown
in [35] for surfaces, [40] for doubly-extended types and [41] for the affine and exotic
types. In most cases, we actually obtain the full cluster modular group this way.
When we have a cluster Dehn twist, ? = {1, (12)}, represented by mutation
at a node on a double edge, we may define A and X trace functions for ?.
2
Definition 5.2.2. The function G(x , x ) = (x2(x1+1)+1)1 2 is the cluster trace functionx1x2
?
for ? on the X space and ?? G is the cluster trace function for ? on the A space.
102
Again, there is a notion of invariant A and X coordinates for ?
Definition 5.2.3. The collection of A coordinates associated to nodes other than
node 1 and 2 are the invariant A coordinates for ?. The functions {xk(x2(x1 + 1) +
1), xl} when the node for xk is connected to nodes 1 and 2, and the node for xl is
not connected to nodes 1 or 2 are the invariant X coordinates for ?.
We will classify the invariants for these ensembles for the subgroup generated
by ? = {1, (12)}
Theorem 5.2.1. The invariant functions for any cluster Dehn twist are generated
by the cluster trace function for ? and the invariant A and X coordinate functions.
This follows from corollary 5.2.1.
5.2.2 Invariants for partial Dehn twists
Suppose that (A,X ) is a finite mutation type cluster ensemble and ? ? ? is
a partial cluster Dehn twist satisfying ?n = ? for some cluster Dehn twist ? and n
minimal.
The cyclic group Cn = Z/nZ acts on these ? invariant A and X functions in
an obvious way: since ?n = ? we can see that the action of ? on these coordinates
descends to an action of Cn = ???/???. This gives us the following proposition
Proposition 5.2.1. R(A,X )??? = (R(A,X ))???)Cn .
This can be used to compute the invariant rings for any partial Dehn twist by
studying the action of Cn on the invariant ring associated with ?. These invariant
rings will simply consist of symmetric functions in the elements of (R(A,X ))???).
103
Chapter 6: Further Problems and Applications
We will discuss some various applications, properties and conjectures about
mutation invariant functions.
6.1 Affine A coordinates
We can use the theorems at the end of chapter 5 to answer a question posed
in [27] about the A coordinates appearing on the tails of a Tp,q,r quiver. Let Q be
a Tp,q,r quiver which is of affine type (so 1/p + 1/q + 1/r ? 1 > 0). Then by the
calculation of the cluster modular group of affine cluster algebras done in [40], we
have that the cluster modular group is generated by the commuting partial cluster
Dehn twists ?p, ?q, ?r satisfying ?
i
i = ? = {1, (12)} along with the elements of AutQ.
Thus, there is only one distinct cluster Dehn twist for any affine cluster ensemble.
We call the A coordinates which appear on nodes other than 1 and 2 for any
seed with quiver isomorphic to Q ?affine A coordinates?. Let F be the cluster trace
function for ?, {?} be the set of initial affine A coordinates and let R = F[??1, F ]
be the ring generated by F and Laurent polynomials in {?}
Theorem 6.1.1. The affine A coordinates are elements of R
Proof. We know from the Laurent phenomenon that all of the A coordinates are
104
Laurent polynomials in the initial A coordinates. Furthermore, we know that all of
the affine A coordinates are invariant functions under the action of ?. Thus, these
affine coordinates are elements of F(?, F ). Let A be the ring of Laurent polynomials
in the initial cluster variables.
Now we will show that F(?, F ) ? A = R. Let x = a1 . First, we notice that
a2
b2c2d2
F = x+ x?1 + (6.1)
a1a2
and
F n = (x)n + (x)?n + . . . . (6.2)
Now we show that the only elements of F(F ) ? A are polynomials in F . Suppose
that
P (F ) h
= ? A (6.3)
Q(F ) b
is a Laurent polynomial in A with Q and P relatively prime. Then if Q(F ) is
not a monomial in {?}, then there must be a cancellation between the numerator
and denominator. This cannot happen if Q and P are relatively prime, since F is
algebraically independent over F. Thus we have that Q(F ) must be a monomial in
A. This can not happen unless Q is constant since F = x + x?1 modulo b2 and we
see that every non constant polynomial in F must have xn +x?n as its leading term
in x. Thus F(?, F ) ? A = R.
Now we see that the elements of F(F ) ?A and F(?)?A generate F(?, F )?A
since the elements in ? are algebraically independent with a1 and a2.
105
This theorem naturally has an analogue for surface cluster algebras. Let S be a
marked surface and let ? ? ?S be the cluster modular group element corresponding
to a Dehn twist about a simple closed curve ? and let F be the trace function for
this curve. Let {?} be the set of A coordinates in an excising triangulation for ?
that do not intersect ? and let R = F[??1, F ]. Then we have
Theorem 6.1.2. The A coordinates for arcs on S that do not intersect ? are ele-
ments of R
The proof of this theorem follows from the same argument as the previous
theorem.
6.2 Invariants for Cluster Dehn Twists on General Mutation Finite
Types via folding
We have only completed our classification for simply laced finite mutation type
ensembles. We will not treat the general case here, but it should be easy to obtain
from what we have already completed. We may conjecture that the folding map of
invariants is always surjective.
Conjecture 6.2.1. Let R be a quiver obtained by folding a quiver Q. Then the
map:
F(A ,X )??? ?fQ Q ?
o?ld F(AR,X )???R (6.4)
is surjective.
106
We believe that this should follow from our classification in the case where Q
is a surface or mutation finite quiver, and ? is a cluster Dehn twist.
6.3 Further Geometric structures related to invariants
We suspect that the analysis of section 5 extends to cluster ensembles associ-
ated to the higher Teichmu?ller spaces of Fock and Goncharov in [5]. The mapping
class group of the surface will again be a canonical subgroup of the cluster modular
group of these ensembles. As before, it is possible to compute traces of monodromy
operators about loops on the surface in terms of the X coordinates, and these will
give X invariants for mapping classes that preserve the given loop.
There is also evidence that invariant functions may aid in the analysis of the
gluing equations for more general geometric structures on manifolds, generalizing
remark 4.2.1. The work of Nagao, Terashima, and Yamazaki in [42] gives indications
of this via their notion of a ?parameter periodicity equation?.
6.4 Correspondence between A and X Invariants
We will give some evidence for a correspondence between the invariants of the
A and X spaces via ?denominator vectors?.
Definition 6.4.1. The denominator vector of a Laurent polynomial is the vector
of exponents appearing in the denominator of the polynomial written as a single
fraction. If there are monomial factors in the numerator, we include these as negative
valued exponents in the denominator.
107
Definition 6.4.2. We call a pair of bases or partial bases for A and X invariants
that correspond via denominator vectors a correspondence basis.
Through all of the examples of section 3, we can explicitly see examples of
correspondences between theA and X invariants. In the cases where no X invariants
are given, there are trivial X invariants corresponding via denominator vectors to
each of the A invariants.
Example 6.4.1. Explicitly, this correspondence for the invariants of example 3.2.4
are as follows:
(a1a4 + a2a5 + a
2
3a6) ????? (x1x2x3x4x5x )?16
a1a2a3a4a5a6
(a1a4 + a2a5 + a3a6) ????? (x4x5x6)?1
a4a5a6
a1 ????? x1
a4 x4
We can show this correspondence when our cluster ensemble is associated with
a surface, S, and we are considering invariants for a Dehn twist about ? on S. This
correspondence is very clear when ? has genus zero on S. Let ? be an excising
triangulation of ? and number the arcs as definition 5.1.5.
In this case, the correspondence basis is given by the following table:
6.5 Laurent Property of Invariants
Many of the examples of A invariants are Laurent polynomials in the A coor-
dinates. This is not be true in every case, as the functions of example 3.2.4 include
inverses of Laurent polynomials too. However we may conjecture the following:
108
Type of invariant A X
?
Trace function of ? ??( G(x , x )) G(x , x ) = x2(x1+1)+11 2 1 2 x1x2
Boundary arcs a3, . . . , am x3(x2(x1 + 1) + 1), . . . , xm(x2(x1 + 1) + 1)
Nonboundary arcs am+1, . . . , aN xm+1, . . . , xN
Conjecture 6.5.1. There is a basis of F(X )??Q with corresponding basis of F(A )??Q
such that ? acts on the basis of F(AQ)?? by positive Laurent polynomials.
The examples of section 3, other than example 3.2.5, give ample evidence for
this conjecture. This conjecture also implies, following theorem 6.1, That all of the
A coordinates appearing on the tails of the Tp,q,r quiver are Laurent polynomials in
the the initial invariants. This is easy to verify case by case, and is probably not
too difficult to prove in general.
It seems that in some cases that there is a stronger version of this phenomenon.
We may occasionally find a correspondence basis such that the action of the cluster
modular group sends the basis to Laurent monomials in the basis. In other words,
we can pick a basis that gives a representation
?/?? ? GL(2,Z) (6.5)
Surprisingly, we have that the action of ? on the X invariants is also by Laurent
monomials and the induced representation seems to be identical in this case.
109
Example 6.5.1. Continuing example 3.2.3, we can take (F1, F2) as a possibly par-
tial basis of invariants. Then one may check that the paths ? = {34, (12)} and
r = {414, (132)} generate D12 and we have that ?((F1, F2)) = (F1, F1/F2) and
r((F1, F2)) = (F1/F2, F1). Thus we have the representation
? : D12 ? GL(2,Z) (6.6)
given by ?? ?? ?? ????1 1 ?? ?? 1 1??
?(?) = ???? ???? ?(r) = ???? ???? (6.7)
0 ?1 ?1 0
6.6 Coefficients and Canonical Bases
Throughout our study of mutation invariants, we have mostly ignored coeffi-
cients on the A space. It is natural to wish to study invariants for more general
coefficients, but there are some immediate problems one encounters. For example,
take the Markov quiver, example 3.1.3, and add ?principal coeffiecents? i.e. one
frozen node for each unfrozen connected with a single arrow. Mutations return us
to a quiver with isomorphic mutable portion, but the frozen nodes are connected
differently. We therefore, cannot expect to construct a mutation invariant function
which is identical after each mutation.
There should still be some more general notion of invariant which addresses
this problem. If we are in a situation where there are non-trivial X invariants, one
may pull them back to the A space and obtain functions which are invariant in some
110
[f1] [f2] [f1] [f2]
1 2 1 2
Figure 6.1: A?1 quiver with principal coefficients before and after applying ?.
sense. We expect that these functions will be elements of the ?Canonical basis? of
the cluster algebra associated with the A space.
Example 6.6.1. For example, if we tak?e an A?1 cluster ensemble with principal
coefficients, we can pull our X invariant, G(x1, x2), of example 3.1.1 back to the
A space. We find
? a
2f1f2 + f2 + a
2
? (G(x1, x2)) = F (a1, a2, f1, f2) =
2 ? 1 (6.8)
a1a2 f1f2
Mutations will change how each mutable node is connected with the frozen nodes,
so this function will only be invariant up to multiplying its arguments by products of
the coefficients. The element ? = {1, (12)} is still an element of the cluster modular
group. We find
a2 22f2 +?1 + a1f
? ?
1
?(F ) = = F (a1 f1f2, a2 f2/f1) (6.9)
a1a2 f1f2
Thus F is invariant up to multiplying its arguments by functions of the frozen
variables. This should be related to the notion of a ?cluster quasi-homorphism?
of [43].
The function F appears as theta function in the sense of [18]. The computation
of the theta function of the limiting ray in the cluster scattering diagram associated
111
with the A?1 cluster algebra in [44], shows
?(a1, a2, x1, x
?1
2) = a1a2 (x2(x1 + 1) + 1) (6.10)
which after applying ?? to the X coordinates we have
a2f1f
2
2 + f2 + a
?(a1, a2, f1, f2) =
2 1 (6.11)
a1a2
which is essentially the function F .
Remark 6.6.1. The expression of the theta function in equation 6.10 is essentially
the form of the invariant X coordinates of 5.1.5, imagining that there is a third
mutable node connected to nodes 1 and 2 in an oriented cycle.
The fact that a theta function associated with a limiting ray of the scattering
diagram is in some way an invariant function should make intuitive sense. Whenever
the scattering diagram is mutation invariant under the mutation path which takes us
towards a limiting ray, the theta function associated with this ray will be invariant
too. Our classification theorem for surface cluster algebras shows us that the theta
functions associated with Dehn twists should be related to the trace function for
the curve we are twisting. This gives a natural relation between theta functions and
trace functions.
It is conjectured in [18] that the ?theta basis? of a surface cluster algebra
should be the same as the basis of trace functions constructed in [5]. Our classi-
fication of mutation invariants should act as a stepping stone between these two
constructions. Interestingly, the construction theta basis makes no reference to any
surface underlying the combinatorial structure of the algebra. Similarly, our notion
112
of mutation invariant does not require any data which is not intrinsic to the cluster
algebra, but the classification of invariants is only visible when considering surfaces.
There are several other instances of the same invariant function. In [8], we
can see the appearance of the cluster trace function of the affine cluster algebras
appearing as the ?cluster character? of a module with dimension vector given by
the longest root in the affine root system in the associated cluster category. This
cluster character can be seen to be mutation invariant by theorem 3.1 of [45]. Again,
our classification of mutation invariants helps explain why we see the same function
appear.
We can consider the following conjecture as a way of organizing these ideas:
Conjecture 6.6.1. Let Q be a mutation finite quiver. There is a canonical basis of
invariant functions for any cluster Dehn twist. The union of these bases over every
cluster Dehn twist is the canonical basis of the cluster algebra C(Q).
113
Appendix A: Dynkin Diagrams
A.1 Finite Dynkin Diagrams
(a) An
(b) Dn
(c) E6 (d) E7
(e) E8
Figure A.1: Simply Laced Finite Dynkin Diagrams
114
(a) Bn
(b) Cn
(c) F4 (d) G2
Figure A.2: Folded Finite Dynkin Diagrams
A.2 Affine Dynkin Diagrams
(a) A?n (b) D?n
(d) E?7
(c) E?6
(e) E?7
Figure A.3: Simply Laced Affine Dynkin Diagrams
115
(a) B?n
(b) C?n (c) F?4
(d) G?n
Figure A.4: Folded Affine Dynkin Diagrams
A.3 Doubly-Extended Dynkin Diagrams
(1,1)
(a) A (1,1)1 (b) D4
(1,1) (1,1)
(c) E6 (d) E7
(1,1)
(e) E8
Figure A.5: Simply Laced Doubly Extended Dynkin Diagrams
116
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