Five Lectures on Cluster Theory

Ray Maresca

###### Abstract

In this paper, we will present the author’s interpretation and embellishment
of five lectures on cluster theory given by Kiyoshi Igusa during the Spring
semester of 2022 at Brandeis University. They are meant to be used as an
introduction to cluster theory from a representation-theoretic point of view.

##  1 Introduction

###  1.1 Some History

Since its introduction in the early 2000’s by Fomin and Zelevinsky, cluster
theory has been an active area of research. Some of the first results in
cluster theory, such Fomin and Zelevinsky’s classification of finite type
cluster algebras in [17], bore striking resemblances to theorems in
representation theory such as Gabriel’s theorem in [20] and [21]. In
particular, every seed or cluster in a cluster algebra has some number, say
$n$, cluster variables. This allows us to represent the variables of a cluster
algebra in a graph in which there is an edge between any two variables that
occur in a cluster. In this graph, which we will see in Section 4, we see that
if we have $n-1$ cluster variables connected by edges, there are precisely 2
ways to complete the the graph.

On the other hand, many of these phenomena were also being seen in
representation theory. For instance in [40], Skowroński showed that every
basic tilting module has $n$ indecomposable summands. Moreover, Happel and
Unger showed in [24] that every basic partial tilting module having $n-1$
indecomposable direct summands can always be completed into a tilting module
and that there are at most two ways that this can be done. Therefore, one may
think that the ‘right’ way to connect cluster theory and representation theory
is through tilting theory, though this fails in several ways, one being that
there are more cluster variables then there are indecomposable rigid modules
and more clusters than tilting modules.

To attain a categorification of cluster theory in terms of representation
theory, we thus need to extend the module category in some way. This was done
by Buan, Marsh, Reineke, Reiten, and Todorov in [8] where they constructed a
larger category called the cluster category in which the original module
category can be embedded. In [8], it was assumed that the quiver $Q$
associated to the initial seed was acyclic, so the path algebra is hereditary,
which is an assumption we will also make throughout this paper. This need not
be the case and in [2], Amoit removes the condition of $\Bbbk Q$ being
hereditary and constructs the cluster category for non-hereditary algebras of
global dimension 2 and quivers with potential.

There however is still one thing to notice. In the cluster category of a
hereditary algebra, we have the following isomorphism from Auslander-Reiten
duality: Ext${}^{1}(M,N)\cong$ Hom$(N,\tau M)$. One can show that this
isomorphism actually also holds in module categories of non-hereditary cluster
algebras. Therefore, there is a correspondence between the tilting objects in
the cluster category and modules in the module category of a cluster-tilted
algebra. The issue with this is that these modules may not be partial tilting
objects due to having infinite projective dimension. By dropping the
requirement on projective dimension and loosening rigidity to $\tau$-rigidity,
Adachi, Iyama, and Reiten introduced $\tau$-tilting theory in [1].

###  1.2 Framework of These Notes

Although $\tau$-tilting theory is one of the most active areas of current
research, in this paper, we will focus on classical tilting theory and cluster
theory. For a survey on $\tau$-tilting theory, we suggest [43] by Treffinger,
where many of the references and much of the background information in these
notes were found. In this article, we will illuminate some connections between
cluster theory and representation theory while working through the process of
categorifying cluster theory when the initial seed corresponds to a quiver
with no loops or two cycles. We will do this by working through the author’s
interpretation and embellishment of 5 lectures given by Kiyoshi Igusa during
the spring semester of 2022 at Brandeis University which contain several
motivating examples and provide some intuition behind results. One thing to
note is that all proofs in the first six sections of this paper are meant to
provide the main idea and intuition behind the proof and should be taken as
nothing more than sketches. We will assume that the reader has some background
in the foundations of representation theory and suggest [3], [12], [38], and
[39] as references for this material.

We will begin these notes with Section 2 in which we give some examples that
provide intuition behind what a cluster algebra is and how clusters and
cluster variables are connected to representation theory. We then provide
definitions of cluster algebras and mutations in terms of a quiver $Q$. In
particular, we will see a connection between cluster variables (characters)
and the Auslander-Reiten (AR) quiver of the corresponding initial quiver.
Afterward in Section 3 we will explicitly provide the correspondence by
showing how to associate a cluster character to a $\Bbbk Q$-module. We will
moreover see how the coefficients of this cluster character are related to the
module itself.

In Section 4 we will introduce two questions that we will spend the rest of
the notes trying to answer; namely, which sets of modules are sent to clusters
and which algebraic objects correspond to the initial cluster variables? These
two questions motivated the categorification of cluster theory in terms of
representation theory. It is here we will introduce the notion of tilting
modules and how they fall short of describing cluster theory in its entirety.
We will need to extend the idea of a tilting module by introducing support
tilting modules, shifted projectives, and silting pairs. We will do this
without introducing the bounded derived category or explicitly creating the
cluster category. At this point, we will provide the bijection between
clusters and silting pairs. We will finish this section by constructing the
wall and chamber structure (stability picture) and show how this structure
connects to cluster theory.

After this, in Section 5, we will describe in fact why the stability picture
introduced in the previous section is accurate by showing that rigidity is a
Zariski open condition. To do this, we will introduce the category of 2-term
silting complexes. We finish this section by introducing the notion of stable
barcodes which won’t actually be used for the remainder of the notes. In the
final section, Section 6, we introduce the notion of maximal green sequences
using exchange matrices and ice quivers. The definition of these sequences
rely on notions like sign coherence of $g$-vectors and $c$-vectors, which we
will also explain in this section. Throughout the notes, we will attempt to
provide as much referencing as possible to both history and proofs of the
results.

Before we begin, we would like to remark that not all connections between
cluster and representation theory are made in these notes. For instance, there
is a beautiful connection between functorially finite torsion classes in
mod-$\Bbbk Q$ and clusters, namely that they are in bijection. One way to see
this is through the fact that the dual graph of the stability picture is
precisely the Hasse quiver of functorially finite torsion classes in
mod-$\Bbbk Q$. For more on the lattice of torsion classes, we suggest Thomas’s
exposition [41]. For more details on the bijection, Treffinger’s survey [43]
is a great place to start.

##  2 Cluster Theory

###  2.1 Examples

Before presenting the formal definition of a cluster algebra, we provide some
intuition, motivation, and examples. Intuitively, ‘clusters’ are sets of $n$
objects which can be mutated. Each of these $n$ objects are a cluster variable
or cluster character. Throughout these lectures, we will provide two methods
of thinking about clusters, the former is the original definition and the
later is a categorification of it.

  1. (a)

Clusters are transcendence bases for $\mathbb{Q}(x_{1},x_{2},\dots,x_{n})$
given by a quiver $Q$.

  2. (b)

Clusters are objects in a category of the form $T=T_{1}\oplus
T_{2}\oplus\dots\oplus T_{n}$.

One relationship between the above two methods is through something called the
‘cluster character’ denoted by
$\chi(T_{i})\in\mathbb{Q}(x_{1},x_{2},\dots,x_{n})$. We begin our studies of
cluster algebras using method (a) through an example.

Until otherwise stated, let $Q$ be the quiver $2\rightarrow 1\leftarrow 3$.
Note that this quiver consists of only descending arrows, that is, there is an
arrow from $i$ to $j$ if and only if $j<i$. Below are three depictions of the
Auslander-Reiten (AR) quiver for this quiver. Note that the maps between the
projectives are ascending with respect to their vertices; that is, there is an
arrow from $P_{i}$ to $P_{j}$ in the AR quiver if and only if $i<j$. This is
one reason to always take the arrows to be descending when the quiver does not
have oriented cycles. Moreover, note that the quiver formed by the three
projectives in the AR quiver is the opposite quiver of $Q$. On the top left we
have the standard projective/injective at vertex $i$ notation. On the top
right we have another standard notation indicating the tops and socles of the
modules on the left. Finally, below these two is a depiction of the AR quiver
using dimension vectors. In each quiver the dotted lines indicate the AR
translate $\tau$:

$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{3}}$$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}}$ | $\textstyle{{2\atop 1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{2\,3\atop 1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{3\atop 1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2}$
---|---

$\textstyle{(1,1,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,0,1)}$$\textstyle{(1,0,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,1,0)}$
---

The form that uses the dimension vectors is computationally useful and sheds
some light onto the relationship between AR theory and cluster theory. Recall
that given a short exact sequence of modules $0\rightarrow A\rightarrow
B\rightarrow C\rightarrow 0$, the dimension vectors satisfy the equality
dim$A\,+$ dim$C=$ dim$B$. Since each mesh in the above AR quiver forms an
almost split sequence, which is short exact, we can construct the AR quiver
using this relationship and in some sense extend it as follows.

|
---
$\textstyle{(1,1,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(-1,0,-1)}$$\textstyle{(1,0,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(-1,0,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,1,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(-1,-1,0)}$

These newly added vectors are not dimension vectors in the usual sense since
they contain negative entries. From a representation theoretic point of view,
we will see in Section 4 that these negative ‘dimension vectors’ correspond to
‘shifted projectives’ in the categorification method of studying cluster
algebras. Now to the dimension vector $(i_{1},i_{2},\dots,i_{n})$, we
associate the symbol $x_{1}^{i_{1}}x_{2}^{i_{2}}\dots x_{n}^{i_{n}}$. Then
from the aforementioned additive relationship given by the dimension vectors,
we have that given a short exact sequence of modules $0\rightarrow
A\rightarrow B\rightarrow C\rightarrow 0$, the symbols satisfy the equality
$x^{\text{dim}A}x^{\text{dim}C}=x^{\text{dim}B}$. In this notation, by
$x^{\text{dim}A}$ we mean $\prod_{i=1}^{n}x_{i}^{\text{dim}A_{i}}$ where
$\text{dim}A_{i}$ is the $i$th entry in the dimension vector of $A$. Then the
above AR quiver can be rewritten in terms of the symbols as follows.

|
---
$\textstyle{x_{1}x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\over
x_{1}x_{3}}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}x_{2}x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{1\over
x_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\over
x_{1}x_{2}}$

###  2.2 Cluster Variables

These symbols are the cluster variables or cluster characters for the cluster
algebra whose initial quiver is $Q$. Though we have not yet defined the
cluster variables, they satisfy a nice property. As was shown by Caldero and
Chapoton in [9], given an almost split sequence of modules $0\rightarrow
A\rightarrow B\rightarrow C\rightarrow 0$, the corresponding cluster
characters satisfy the relationship

$\chi(A)\chi(C)=\chi(B)+1$

where $\chi(B)=\chi(\oplus_{i}B_{i})=\sum_{i}\chi(B_{i})$. Note that if the
sequence is neither split nor almost split, we would need to add more than one
on the right hand side and if the sequence is split, we would not need to add
anything, providing some intuition on why the sequences are almost split but
not split. We will see in Section 3 where this plus 1 is coming from. Using
this formula for cluster characters, we can reconstruct the AR quiver for our
type $\mathbb{A}$ quiver $Q$, along with those ‘shifted projectives’, for the
corresponding path algebra by beginning with the opposite quiver as follows.

|
---
$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

To fill the left most oval, we need a cluster variable $y$ such that
$x_{1}y=x_{2}x_{3}+1$, so $y={x_{2}x_{3}+1\over x_{1}}$. We get:

|
---
$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{x_{2}x_{3}+1\over
x_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Continuing in this way will provide us the entire AR quiver and more for $Q$
in terms of the cluster variables, though it will get quite messy quickly. To
reduce unnecessary computations, we examine a simpler example. Let $Q$ now be
the quiver $1\leftarrow 2$. Then the AR quiver is

$\textstyle{(1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,1)}$
---

By using the relationship between cluster characters and beginning with the
opposite quiver $x_{1}\rightarrow x_{2}$, we get the following.

$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{x_{2}+1+x_{1}\over
x_{1}x_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{x_{1}+1\over
x_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{1+x_{1}\over
x_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{2}}$
---

Notice that if we restrict our attention to only the cluster variables with
nontrivial denominators, we are looking at the AR quiver of $Q$. These cluster
characters correspond to $\Bbbk Q$-modules and the intimate relationship
between the dimension vectors and the cluster variables will be revealed in
Section 3. The cluster variables with trivial denominator correspond to the
shifted projectives which will be explained in Section 4.

###  2.3 Cluster Algebras and Mutation

Now that we have a sense of what we want cluster variables to be, we are ready
to provide a formal definition. The definitions of a cluster algebra and
mutation was first written down by Fomin and Zelevinski in [16]. The original
definition used the notion of an exchange matrix instead of a quiver; however
as we will soon see, these two notions are one in the same.

######  Definition 2.1.

A cluster algebra is a subalgebra
$A\subset\mathbb{Q}(x_{1},x_{2},\dots,x_{n})$ generated by the cluster
variables given by mutating a seed $(Q,x_{*})$ where

  * •

$Q$ is a quiver with no loops or two cycles.

  * •

$x_{*}=(x_{1},x_{2},\dots,x_{n})$ is a transcendence basis for
$\mathbb{Q}(x_{1},x_{2},\dots,x_{n})=\\{{f(x)\over
g(x)}:f,g\in\mathbb{Q}(x_{1},x_{2},\dots,x_{n})\\}$.

In order for this definition to be complete, we must define mutation which
consists of two parts, mutation of the quiver and the seed. We begin by
defining mutation of a quiver $Q$ with a running example. For the following
definition, let $Q^{\prime}$ be the quiver

$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

######  Definition 2.2.

We define the mutation of $Q$ at vertex $k$, denoted by $\mu_{k}Q$, as
follows.

  1. 1.

We first compose any length two paths through $k$ by introducing a new arrow.
In the running example, let $k=2$.

$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha\beta}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

  2. 2.

Reverse all arrows at $k$:

$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{*}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\beta}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{*}}$
---

  3. 3.

Eliminate all two cycles:

$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{*}}$$\textstyle{1}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{*}}$
---

After all three steps, the resulting quiver is $\mu_{k}Q$, or in this example
$\mu_{2}Q^{\prime}$.

Notice that the procedure of mutation does not preserve representation type of
$Q$. To provide another example of mutation:

######  Example 2.1.

Let $Q$ be the quiver $2\rightarrow 1\leftarrow 3$. Then $\mu_{1}Q$ is the
quiver $2\leftarrow 1\rightarrow 3$.

Now is a good time to introduce the notion of exchange matrices, which will
become important in the study of maximal green sequences which will be defined
in Section 6.

######  Definition 2.3.

Let $Q$ be a quiver. The exchange matrix of $Q$ is the matrix $B=[b_{ij}]$
where $b_{ij}=$ the number of arrows $i\rightarrow j$ \- the number of arrows
$j\rightarrow i$.

######  Example 2.2.

For the running example in Definition 2.2, the exchange matrix is

$\left[\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.27776pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern 21.05551pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1}$}}}}}}}{\hbox{\kern 44.83327pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{-1}$}}}}}}}{\hbox{\kern-8.27776pt\raise-22.44443pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{-1}$}}}}}}}{\hbox{\kern
21.05551pt\raise-22.44443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern
47.61102pt\raise-22.44443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{2}$}}}}}}}{\hbox{\kern-5.5pt\raise-44.88885pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1}$}}}}}}}{\hbox{\kern
18.27776pt\raise-44.88885pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{-2}$}}}}}}}{\hbox{\kern
47.61102pt\raise-44.88885pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces}\right]$

Notice that the exchange matrix in Example 2.2 is skew symmetric, that is, all
entries are integers and $b_{ij}=-b_{ji}$. This holds more generally; in fact,
any skew symmetric matrix gives a quiver with no loops or two cycles and any
quiver with no loops or two-cycles has a skew symmetric exchange matrix, so
really in our definition of cluster algebra, we are only considering ‘skew
symmetric’ cluster algebras. We are now ready to define mutation of the seed
$x_{*}$.

######  Definition 2.4.

We define the mutation of $x_{*}$ at vertex $k$, denoted by $\mu_{k}(x_{*})$,
as $\mu_{k}(x_{*})=(x_{1},x_{2},\dots,x^{\prime}_{k},x_{k+1},\dots,x_{n})$
where

$x^{\prime}_{k}={\displaystyle\prod_{i\rightarrow
k}x_{i}^{b_{ik}}+\displaystyle\prod_{k\rightarrow j}x_{j}^{b_{kj}}\over
x_{k}}.$

The notation $\displaystyle\prod_{i\rightarrow k}$ means we take the product
over all vertices $i$ such that there is an arrow from $i$ to $k$. We adopt
the standard convention that the empty product equals 1.

######  Example 2.3.

Let $Q$ be the quiver $2\rightarrow 1\leftarrow 3$ and consider the seed
$(x_{1},x_{2},x_{3})$. Then mutation at vertex 1 gives
$\mu_{1}(Q,(x_{1},x_{2},x_{3}))=(\mu_{1}Q,({x_{2}x_{3}+1\over
x_{1}},x_{2},x_{3}))$ with $\mu_{1}Q$ the quiver from Example 2.1. Notice that
the plus one in the mutated seed comes from the fact that vertex 1 in $Q$ is a
sink, so there are no arrows out of it, and the empty product is always taken
to be 1. Continuing by mutating at vertex 2 then 3 gives the following:

$\textstyle{(2\leftarrow 1\rightarrow 3,({x_{2}x_{3}+1\over
x_{1}},x_{2},x_{3}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\textstyle{(2\rightarrow
1\rightarrow 3,({x_{2}x_{3}+1\over x_{1}},{x_{2}x_{3}+1+x_{1}\over
x_{1}x_{2}},x_{3}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{3}}$$\textstyle{(2\rightarrow
1\leftarrow 3,({x_{2}x_{3}+1\over x_{1}},{x_{2}x_{3}+1+x_{1}\over
x_{1}x_{2}},{x_{2}x_{3}+1+x_{1}\over x_{1}x_{3}}))}$
---

We can visualize the mutations of these cluster variables as follows. Recall
the construction of the AR quiver for $Q:2\rightarrow 1\leftarrow 3$ from
Section 2.2. Continuing this procedure would give the picture in Figure 1.
Within each loop we have the opposite quiver of the mutated quiver along with
the mutated seed after performing the indicated mutation. Moreover, each loop
encloses a cluster; that is, a collection of $n$ cluster variables attained by
mutations.

$x_{1}$$x_{2}$$x_{3}$$\frac{x_{2}x_{3}+1}{x_{1}}$$\frac{x_{2}x_{3}+1+x_{1}}{x_{1}x_{2}}$$\frac{x_{2}x_{3}+1+x_{1}}{x_{1}x_{3}}$$\mu_{1}$Initial
seed$\mu_{2}\circ\mu_{1}$$\mu_{3}\circ\mu_{2}\circ\mu_{1}$ Figure 1: A
visualization of mutation in the AR quiver

In all of the examples we have done so far, the cluster variables were all
Laurent polynomials; that is, they are of the form ${f(x)\over x^{\alpha}}$
where $f(x)$ is a polynomial in $\mathbb{N}(x_{1},x_{2},\dots,x_{n})$ and
$\alpha\in\mathbb{Z}^{n}$. It turns out that whether or not this phenomenon
holds in general has been an open question called the positivity conjecture
first stated by Fomin and Zelevinsky in 2002 in [16]. Around 2015 in [35], Lee
and Schiffler proved the positivity conjecture for all cluster algebras for
which the exchange matrix is skew symmetric, called skew symmetric cluster
algebras, which is the convention we take in these notes for all our cluster
algebras. The general case was proven in 2017 by Gross, Hacking, Keel, and
Kontsevich in [22]. We have the following theorem.

######  Theorem 2.1.

.

  * •

Cluster variables are Laurant polynomials.

  * •

The numerator of the Laurant polynomial always has nonnegative integral
coefficients.

The fact that the coefficients of the numerator are nonnegative integers may
lead us to believe that they count something. This is indeed the case as we
will see in Section 3.

##  3 Cluster Character

In this section, we begin by showing how to attain the cluster character
associated to a $\Bbbk Q$-module where $Q$ is a connected quiver with $n$
vertices and no oriented cycles; that is, $\Bbbk Q$ is a hereditary algebra
which is an assumption we will take throughout the remainder of the notes
unless otherwise specified. We then explore the connections between the
cluster character and notions like quiver grassmannians and the Fomin-
Zelevinsky formula for mutation of cluster variables. To do this, we need the
notion of $g$-vectors, which arose from a purely cluster-algebraic perspective
by Fomin and Zelevinsky in [18]. Afterward, we realized that $g$-vectors can
be studied in a representation-theoretic way through projective presentations.
The first such realization was done by Dehy and Keller in [13]. Let
$\Bbbk=\overline{\Bbbk}$ be an algebraically closed field.

######  Definition 3.1.

.

  * •

The $g$-vector of a projective $\Bbbk Q$-module $P=\oplus a_{i}P_{i}$ is the
vector $g(P):=\vec{a}=(a_{1},a_{2},\dots,a_{n})$.

  * •

Let $M$ be a $\Bbbk Q$ module. Then $M$ admits a minimal projective
presentation of the form $0\rightarrow P^{\prime}_{M}\rightarrow
P_{M}\rightarrow M\rightarrow 0$ where both $P^{\prime}_{M}$ and $P_{M}$ are
projective. The $g$-vector of $M$ is $g(M):=g(P_{M})-g(P^{\prime}_{M})$.

Now we provide some intuition behind how to think of the cluster character
with an example.

######  Example 3.1.

Let $Q=2\rightarrow 1\leftarrow 3$. Referring back to the AR quiver given in
Section 2.1 and Figure 1, we see that the cluster character of $S_{1}=P_{1}$
should be $\chi(S_{1})={x_{2}x_{3}+1\over x_{1}}={x_{2}x_{3}\over
x_{1}}+{1\over x_{1}}$. The module $S_{1}$ has two submodules, namely $0$ and
$S_{1}$, and these should correspond to the left and right terms in the above
sum respectively. The module $P_{3}={3\atop 1}$ has three submodules:
$0,S_{1},P_{3}$. Again comparing the AR quiver and Figure 1, we see that
$\chi(P_{3})={x_{2}x_{3}+1+x_{1}\over x_{1}x_{3}}={x_{2}\over x_{1}}+{1\over
x_{1}x_{3}}+{1\over x_{3}}$ and from left to right, these three terms should
correspond to $0,S_{1},$ and $P_{3}$ respectively.

To see how this correspondence works in general, we make the following
definition which can be found in [9].

######  Definition 3.2.

The cluster character of the $\Bbbk Q$-module $M$ is
$\chi(M):=\displaystyle\sum_{V\subset M}\chi(M,V)$, with

$\chi(M,V):=x^{-g(V)}x^{-g(D(M/V))}$

where $D:\Bbbk Q$-mod$\rightarrow$ mod-$\Bbbk Q^{op}$ denotes the duality
functor and $D(M/V)$ denotes the dual of the quotient module $M/V$.

Let’s now verify the results from the previous example.

######  Example 3.2.

We begin by computing the necessary $g$-vectors. Since $S_{1}=P_{1}$ is
projective, by definition we have $g(S_{1})=(1,0,0)$. Similarly,
$g(P_{3})=(0,0,1)$. To compute $\chi(P_{3})$ we also need
$g(D(P_{3}/S_{1}))=g(D(S_{3}))$ and $g(D(P_{3}/0))=g(D(P_{3}))$. The former is
easier than the later since $D(S_{3})$ is the $\Bbbk Q^{op}$ representation
$0\leftarrow 0\rightarrow\Bbbk$, hence it is a projective $\Bbbk
Q^{op}$-module. Therefore $g(D(S_{3}))=(0,0,1)$. To compute $g(D(P_{3}))$,
note that $D(P_{3}))$ is the $\Bbbk Q^{op}$ representation
$0\leftarrow\Bbbk\rightarrow\Bbbk={1\atop 3}$. We have a $\Bbbk Q^{op}$
minimal projective presentation given by $0\rightarrow 2\rightarrow{1\atop
2\,3}\rightarrow{1\atop 3}\rightarrow 0$. Thus
$g(D(P_{3}))=(1,0,0)-(0,1,0)=(1,-1,0)$. Finally, we need $g(D(S_{1}))$ to
compute $\chi(S_{1})$. Since $D(S_{1})$ is the $\Bbbk Q^{op}$ representation
$0\leftarrow\Bbbk\rightarrow 0=I_{1}$, we have a $\Bbbk Q^{op}$ minimal
projective presentation given by $0\rightarrow 2\oplus 3\rightarrow{1\atop
2\,3}\rightarrow 1\rightarrow 0$. Therefore
$g(D(S_{1}))=(1,0,0)-(0,1,0)-(0,0,1)=(1,-1,-1)$. We are now ready to compute
the cluster characters:

$\displaystyle\chi(P_{3})$
$\displaystyle=\chi(P_{3},0)+\chi(P_{3},S_{1})+\chi(P_{3},P_{3})$
$\displaystyle=x^{0}x^{-g(D(P_{3}))}+x^{-g(S_{1})}x^{-g(D(S_{3}))}+x^{-g(P_{3})}x^{-g(D(0))}$
$\displaystyle=x^{0}x^{(-1,1,0)}+x^{(-1,0,0)}x^{(0,0,-1)}+x^{(-1,0,0)}x^{0}$
$\displaystyle={x_{2}\over x_{1}}+{1\over x_{1}x_{3}}+{1\over x_{1}}$

$\displaystyle\chi(S_{1})$ $\displaystyle=\chi(S_{1},0)+\chi(S_{1},S_{1})$
$\displaystyle=x^{0}x^{-g(D(S_{1}))}+x^{-g(S_{1})}x^{0}$
$\displaystyle=x^{0}x^{(-1,1,1)}+x^{(-1,0,0)}x^{0}$
$\displaystyle={x_{2}x_{3}\over x_{1}}+{1\over x_{1}}$

Notice that $\chi(S_{1})$ in Example 3.2 is the same as
$\mu_{1}((x_{1},x_{2},x_{3}))$ from Example 2.3. This is indeed not a
coincidence. Suppose we have a have a general quiver without oriented cycles.
Then at vertex $k$, $Q$ and $Q^{op}$ appear locally as depicted on the left
and right of the following diagram.

$\textstyle{i_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots}$$\textstyle{i_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{1}}$$\textstyle{j_{2}}$$\textstyle{\dots}$$\textstyle{j_{r}}$ | $\textstyle{i_{1}}$$\textstyle{i_{2}}$$\textstyle{\dots}$$\textstyle{i_{l}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots}$$\textstyle{j_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---|---

We attain a minimal projective $\Bbbk Q$ resolution of $S_{k}$ by
$P_{j_{1}}\oplus P_{j_{2}}\oplus\dots\oplus P_{j_{r}}\hookrightarrow
P_{k}\twoheadrightarrow S_{k}$ and a minimal projective $\Bbbk Q^{op}$
resolution of $D(S_{k})$ given by $P_{i_{1}}\oplus P_{i_{2}}\oplus\dots\oplus
P_{i_{l}}\hookrightarrow P_{k}\twoheadrightarrow S_{k}$. Therefore
$x^{-g(D(S_{k}))}={\prod x_{i}\over x_{k}}$ and $x^{-g(S_{k})}={\prod
x_{j}\over x_{k}}$. Finally, we attain a formula for $\chi(S_{k})$ given by

$\chi(S_{k})=\chi(S_{k},0)+\chi(S_{k},S_{k})={\displaystyle\prod_{i\rightarrow
k}x_{i}\over x_{k}}+{\displaystyle\prod_{k\rightarrow j}x_{j}\over
x_{k}}={\displaystyle\prod_{i\rightarrow
k}x_{i}+\displaystyle\prod_{k\rightarrow j}x_{j}\over x_{k}}.$

But this is precisely the formula for mutation at vertex $k$ given by Fomin
and Zelevinsky in Definition 2.4! We conclude that mutation at vertex $k$ of
the seed $(x_{1},x_{2},\dots,x_{n})$ is precisely the cluster character of the
simple at vertex $k$. As we have seen from the previous examples, the cluster
characters seem to count the number of submodules. This is the case for
modules that have finitely many submodules. The following theorem, which
follows from Theorem 3.2, provides this result.

######  Theorem 3.1.

By setting all $x_{i}=1$, we get that $\chi(M)|_{(1,1,\dots,1)}$ is precisely
the number of submodules of $M$ so long as $M$ has finitely many submodules.

Using this theorem, we can can reconstruct the AR quiver for $Q$ when all
$\Bbbk Q$-modules have finitely many submodules. Suppose the module
$B=B_{1}\oplus B_{2}$. By evaluating the cluster character at 1 and using the
relationship $\chi(A)\chi(C)=\chi(B)+1$ for almost split sequences
$A\hookrightarrow B\twoheadrightarrow C$, we have that the information of
being an almost split sequence is encoded in the fact that the determinant of
the following matrix is one

$\left[\begin{array}[]{cc}\chi(A)&\chi(B_{1})\\\ \chi(B_{2})&\chi(C)\\\
\end{array}\right].$

Moreover note that $\chi(M)=0$ if and only if $M=0$.

######  Example 3.3.

Let $Q$ be the quiver $1\leftarrow 2\leftarrow 3$. We begin with the array

$1$$1$$1$$1$$1$$1$$1$$1$

The first rhombus indicates the first matrix we must complete, namely,
$\left[\begin{array}[]{cc}1&1\\\ 1&\\\ \end{array}\right].$ Solving for the
missing entry to ensure this matrix has determinant one gives the first
missing entry.

$1$$1$$1$$1$$1$$1$$1$$1$$2$

Continuing to find the missing entry in the matrix given by the rhombi will
give the following picture.

$1$$1$$1$$1$$1$$1$$1$$1$$2$$3$$4$$2$$3$$2$$1$$1$$1$$1$

Inside the triangle is a depiction of the AR quiver for $Q$. Notice that each
number provides the number of submodules of the corresponding module in the AR
quiver:

$\textstyle{{3\atop{2\atop
1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{2\atop
1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{3\atop
2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3}$
---

Note that in the statement of Theorem 3.1, we restrict ourselves to the case
in which $M$ has only finitely many submodules. This naturally raises the
question, what does the cluster character of $M$ tell us when $M$ has
infinitely many submodules? We proceed with an example.

######  Example 3.4.

Let $Q$ be the quiver $1\leftarrow 2$. Then $\chi(S_{2})|_{1}={x_{1}+1\over
x_{2}}\big{|}_{1}=2$. Consider the module $M=S_{2}\oplus S_{2}$. Then $M$ has
three types of submodules, namely $0,S_{2},$ and $M$. There is only one
submodule of the form $0$ and $M$; however, there are infinitely many of the
form $S_{2}$. Given $(a,b)\in\Bbbk^{2}-\\{(0,0)\\}$ we have an embedding
$S_{2}\hookrightarrow S_{2}\oplus S_{2}$ given by $x\mapsto(ax,bx)$. Note that
scaling this embedding gives the same image. Therefore, we can realize the
image of this embedding as the equivalence class of
$(a,b)\in\Bbbk^{2}-\\{(0,0)\\}$ up to scaling; that is, we can realize each
submodule of $M$ of the form $S_{2}$ as an element of the projective line
$\Bbbk P^{1}$. There are two methods to find the cluster character.

  1. 1.

We take $\Bbbk=\mathbb{C}$. Therefore the set of submodules of $M$ isomorphic
to $S_{2}$ equals $\mathbb{C}P^{1}\cong S^{2}$. We then set the cluster
character $\chi(M,S_{2})$ equal to the Euler characteristic of
$\mathbb{C}P^{1}$. Recall the definition of Euler characteristic is
$\chi(\mathbb{C}P^{1})=\displaystyle\sum_{k}(-1)^{k}\text{dim}H_{k}(\mathbb{C}P^{1})=1-0+1$
where $H_{k}$ denotes the $k$th homology. There is a shortcut to compute this;
namely, the Euler characteristic of a surface of genus $g$ is given by
$\chi(\Sigma_{g})=2-2g$. In this case, our surface is of genus 0, so we attain
the result. One thing to observe is that this number can be negative; however
by Theorem 2.1, the coefficients in the Laurant polynomial are always
positive. This provides us a restriction on the genus of this surface whenever
the modules correspond to cluster characters!

  2. 2.

Let $\Bbbk=\mathbb{F}_{q}$ be the finite field with $q$ elements. The number
of elements in $\Bbbk P^{1}$ is ${q^{2}-1\over q-1}=q+1$. One way to see this
is through the fact that the points are given by $[1,a]$ for $a\in\Bbbk$ and
the point at infinity $[0,1]$. By taking the field with one element, we get
that $q=1$ and that the number of elements in $\Bbbk P^{1}$ is 2. The reason
why we get the same number as the Euler characteristic of $\mathbb{C}P^{1}$
follows from a deep theorem in number theory that is out of the scope of these
notes.

As it turns out, the collection of submodules of fixed dimension of a given
module has been thoroughly studied.

######  Definition 3.3.

Let $Q$ be a quiver and $M$ a $\Bbbk Q$-module. The space of all submodules of
$M$ with dimension vector $\bm{e}$, denoted by Gr$(M,\bm{e})$, is called a
quiver grassmannian.

This is indeed a space, in fact, it is a projective variety. A special case is
when $Q$ is just a point with no arrows. Then for the representation
$M=\Bbbk^{n}$, the quiver grassmannian Gr$(M,k)$ gives the ordinary
grassmannian of $k$-planes in $n$-space. For more on quiver grassmannians, see
Cerulli Irelli’s lectures on quiver grassmannians [10]. We have already seen
another example of a quiver grassmannian:

######  Example 3.5.

For the $M$ and $Q$ from the previous example, Gr$(M,(0,1))\cong\Bbbk P^{1}$.
Moreover, we’ve computed its Euler characteristic:
$\chi(\text{Gr}(M,(0,1)))=2$.

We can use the notion of quiver grassmannians to define the cluster character
of any module $M$ as was done by Caldero and Chapoton in [9] as follows.

######  Theorem 3.2 (Caldero-Chapoton).

.
The cluster character of $M$ is given by

$\chi(M)=\displaystyle\sum_{\bm{e}}\chi(\text{Gr}(M,\bm{e}))x^{-g(\bm{e})}x^{-g(D(M/\bm{e}))}$

where the sum is taken over all vectors $\bm{e}$ such that there is a
submodule of $M$ of dimension vector $\bm{e}$.

At this point, this definition may not seem well defined because it stipulates
that $g$ vectors depend only on the dimension vectors of modules and not the
actual module itself. This is indeed the case since we can also define the $g$
vector of a $\Bbbk Q$\- module $V$ by $g(V):=C^{-1}_{\Bbbk Q}\cdot\text{dim}V$
where $C_{\Bbbk Q}$ is the Cartan matrix of $Q$, which is defined as the
matrix whose $i$th column is the dimension vector of the projective
representation at vertex $i$. This gives the following lemma.

######  Lemma 3.3.

The $g$ vector $g(V)$ depends only on the dimension vector
$\bm{e}=\text{dim}V$.

Notice that in Example 3.3, we used the fact that $\chi(B_{1}\oplus
B_{2})=\chi(B_{1})\chi(B_{2})$. We now have the necessary tools to provide a
representation theoretic proof of this fact.

######  Theorem 3.4.

For two $\Bbbk Q$ modules $A$ and $B$, the cluster character satisfies the
following equation $\chi(A\oplus B)=\chi(A)\chi(B)$.

###### Sketch of Proof.

It suffices to prove this when we set the $x_{i}=1$. We begin by providing a
correspondence between submodules of $A\oplus B$ and pairs of submodules of
the form $(X,Y)$ where $X\leq A$ and $Y\leq B$. Define $\psi((X,Y))=X\oplus
Y$. Then this is an injection into the collection of submodules of $A\oplus
B$. Now, we have a split exact sequence $A\hookrightarrow A\oplus
B\overset{p_{B}}{\twoheadrightarrow}B$ and given any submodule $V\leq A\oplus
B$, we have another exact sequence $V\cap A\hookrightarrow V\twoheadrightarrow
P_{B}(V)$. We define a map $\phi(V)=(V\cap A,p_{B}(V))$. Note that
$\phi\circ\psi=1$, so $\phi$ is surjective.

Now $\phi$ is not necessarily injective; however, we will show that it is
injective on the level of Euler characteristics. Consider the pair of
submodules $(X,Y)$. Then $\phi^{-1}((X,Y))=\\{V\subset A\oplus B:V\cap
A=X\text{ and }p_{B}(V)=Y\\}$. By the Noether isomorphism theorem, this set is
in bijection with $\\{W\subset A/X\oplus B:W\cap A/X=0\text{ and
}p_{b}(W)=Y\\}$ via the correspondence $V\mapsto V/X$. We can uniquely realize
all submodules $W$ in this latter set as the graph of a function from $Y$ to
$A/X$:

$W$$\frac{A}{X}$$B$$Y$$p_{B}$

In particular, with a little more effort, we have that
$\chi(\phi^{-1}(X,Y))=\chi(\text{Hom}(Y,A/X))$. When we consider
$\text{Hom}(Y,A/X)$ over a field with $q$ elements,
$\chi(\text{Hom}(Y,A/X))=q^{l}$ for some $l$. By taking $q=1$, we have that
$\chi(\text{Hom}(Y,A/X))=1$ and therefore, $\chi(\text{Gr}(A\oplus
B,\bm{e}))=\displaystyle\sum_{\bm{e_{1}}+\bm{e_{2}}=\bm{e}}\chi(\text{Gr}(A,\bm{e_{1}})\times\text{Gr}(B,\bm{e}_{2}))=\displaystyle\sum_{\bm{e_{1}}+\bm{e_{2}}=\bm{e}}\chi(\text{Gr}(A,\bm{e_{1}})\chi(\text{Gr}(B,\bm{e}_{2}))$.

Finally, since $g$-vectors are linear; that is, $g(V)=g(X)+g(Y)$, we have that
$g(D(A\oplus B)/X)=g(D(A/X))+g(D(B/Y))$. By summing over all possible
dimension vectors $\bm{e}$, we have the result. ∎

Recall from Section 2.2 that when we have an almost spit sequence
$A\hookrightarrow B\twoheadrightarrow C$, the cluster characters satisfy the
equation $\chi(A)\chi(C)=\chi(B)+1$ where
$\chi(B)=\chi(\oplus_{i}B_{i})=\sum_{i}\chi(B_{i})$. We now have the tools to
prove this.

######  Theorem 3.5.

For an almost split sequence of $\Bbbk Q$-modules
$A\overset{q}{\hookrightarrow}B\overset{p}{\twoheadrightarrow}C$, we have

$\chi(A\oplus C)=\chi(A)\chi(C)=\chi(B)+1.$

###### Sketch of Proof.

We proceed as in the proof of the previous theorem. Consider the collection of
tuples $(X,Y)$ where $X\leq A$ and $Y\leq B$ are submodules, and let $V\leq B$
be a submodule. We define a map $\phi:\\{\text{submodules of
$B$}\\}\rightarrow\\{\text{pairs $(X,Y)$}\\}$ by $\phi(V)=(V\cap A,p(V))$.
Consider a tuple $(X,Y)$ such that $Y\neq 0$. Then since $p$ is irreducible
and the inclusion of $Y$ into $C$ is not a retraction, we have the following
commutative diagram.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q\circ
i}$$\scriptstyle{i}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{i}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{C}$
---

Define $V=q(X)+f(Y)$, so that $\phi(V)=(X,Y)$ and $(X,Y)$ is in the image of
$\phi$. Now suppose that $X\neq 0$. Then since the quotient map is not a
section, we have the following pushout diagram.

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$id$\textstyle{A/X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/X\oplus
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C}$
---

Define $V=\pi^{-1}((0,Y))+X$, thus $\phi(V)=(X,Y)$ and we conclude that
$(X,Y)$ is in the image of $\phi$. Therefore the map $\phi$ is onto
$\\{\text{pairs $(X,Y)$}\\}-\\{(0,C)\\}$ since $C$ can’t be lifted due to the
irreducibilty of $p$. With some more work, we conclude that
$\chi(B)=\chi(A)\chi(C)-1$ where the minus one comes from the fact that the
submodule $(0,C)$ can’t be lifted. ∎

We have techniques to compute cluster characters of modules that lie in an
almost split sequence; however, we do not have any techniques to compute
cluster characters of modules that do not lie at the end of an almost split
sequence. Although we know how to use the definition to compute cluster
characters of any $\Bbbk Q$-module, we will now provide a theorem which will
provide a technique to compute the cluster characters of projectives.

######  Theorem 3.6.

Let $P_{i}$ denote the projective $\Bbbk Q$-module at vertex $i$. Then we have

$\chi(P_{i})=\chi(\text{rad}P_{i})x^{-g(DS_{i})}+x_{i}^{-1}$

where $S_{i}$ is the simple top of $P_{i}$.

We will not prove this, but instead provide an example. As the reader may have
guessed, the proof is given by Caldero and Chapoton in [9].

######  Example 3.6.

Let $Q$ be the quiver $1\leftarrow 2\leftarrow 3$ so that $Q^{op}$ is
$1\rightarrow 2\rightarrow 3$. We compute $\chi(P_{3})$ where
$P_{3}={3\atop{2\atop 1}}$, $\text{rad}P_{3}={2\atop 1}=P_{2}$, and
$\text{rad}P_{2}=1=P_{1}=S_{1}$. Thus

$\chi(P_{3})=\chi(P_{2})x^{-g(DS_{3})}+x_{3}^{-1}=(\chi(S_{1})x^{-g(DS_{2})}+x_{2}^{-1})x^{-g(DS_{3})}+x_{3}^{-1}.$

We compute

$\chi(S_{1})=x^{-g(0)}x^{-g(D(S_{1}))}+x^{-g(S_{1})}x^{-g(0)}=x^{(-1,1,0)}+x^{(-1,0,0)}={x_{2}+1\over
x_{1}}.$

Now we need $\chi(P_{2})=\chi(S_{1})x^{-g(DS_{2})}+x_{2}^{-1}$. We have

$\chi(P_{2})={x_{2}+1\over
x_{1}}x^{(0,-1,1)}+x_{2}^{-1}={x_{2}x_{3}+x_{3}+x_{1}\over x_{1}x_{2}}.$

Finally, we compute

$\chi(P_{3})=\chi(P_{2})x^{-g(DS_{3})}+x_{3}^{-1}={x_{2}x_{3}+x_{3}+x_{1}\over
x_{1}x_{2}}x^{(0,0,-1)}+x_{3}^{-1}={x_{2}x_{3}+x_{3}+x_{1}+x_{1}x_{2}\over
x_{1}x_{2}}.$

In the case in which the AR quiver of our algebra is connected, we can get the
cluster character of injectives simply by completing the AR quiver. However,
in the case in which the AR quiver is disconnected, we can’t get the cluster
character of injectives by beginning at projectives and completing meshes. For
instance in the tame case, we would never leave the preprojective component.
By duality, we have a dual theorem to Theorem 3.5 that provides us with a
technique to compute the cluster character of injectives.

######  Theorem 3.7.

Let $I_{i}$ denote the injective $\Bbbk Q$-module at vertex $i$. Then we have
$\chi(I_{i})=\chi(I_{i}/S_{i})x^{-g(S_{i})}+x_{i}^{-1}$.

######  Example 3.7.

Let’s take $Q$ to be the Kronecker $1\leftleftarrows 2$ and compute
$\chi(I_{1})=\chi({2\,2\atop 1})$. By the previous theorem, we compute
$\chi(S_{2}\oplus S_{2})x^{-g(S_{1})}+x_{1}^{-1}$. Note by Theorem 3.4, we
have that $\chi(S_{2}\oplus S_{2})=\chi(S_{2})\chi(S_{2})$. We begin by
computing $\chi(S_{2})={x_{1}^{2}+1\over x_{2}}$. Since $g(S_{1})=(1,0)$, we
have that

$\chi(I_{1})=\bigg{(}{x_{1}^{2}+1\over x_{2}}\bigg{)}\bigg{(}{x_{1}^{2}+1\over
x_{2}}\bigg{)}x^{(-1,0)}+{1\over x_{1}}={x_{1}^{4}+2x_{1}^{2}+x_{2}^{2}+1\over
x_{2}^{2}x_{1}}.$

Notice that this is a case analogous to Example 3.4 in which there are
infinitely many submodules of $I_{1}$ of the form $S_{2}\oplus S_{2}$. So in
this case, the coefficient of $x_{1}^{2}$ in the cluster character provides
the Euler characteristic of Gr$(I_{1},(1,1))$.

##  4 Clusters with Modules

Although not yet explicitly stated, thus far we have seen that the cluster
character $\chi$ sends indecomposable rigid modules to cluster variables,
where by rigid we mean Ext${}^{1}(M,M)=0$. The two questions we wish to answer
in this section are

  * •

Which sets of modules are sent to clusters?

  * •

Given a rigid indecomposable module $M$, we get
$\chi(M)={\text{something}\over x^{\text{dim}M}}$. Which algebraic objects
correspond cluster variables of the form $x_{i}$ and what object corresponds
to the initial cluster?

To do this, we need to introduce and establish some more algebraic machinery.
For more on classical tilting theory, see [3].

######  Definition 4.1.

Let $\Lambda=\Bbbk Q$ where $Q$ is a quiver with $n$ vertices and no oriented
cycles. Then a tilting module is a rigid module $T$ with $n$ indecomposable
non-isomorphic summands $T=T_{1}\oplus T_{2}\oplus\dots\oplus T_{n}$.

The condition that $T$ is rigid implies that
Ext${}^{1}(T,T)=\displaystyle\oplus_{i,j}$Ext${}^{1}(T_{i},T_{j})=0$. This
happens if and only if each $T_{i}$ is rigid and they don’t extend each other,
that is, Ext${}^{1}(T_{i},T_{j})=0$ for all $i$ and $j$.

######  Example 4.1.

Some examples of tilting modules are the following.

  * •

The sum of the projective modules $\Lambda=P_{1}\oplus P_{2}\oplus\dots\oplus
P_{n}$.

  * •

The sum of the injective modules $\nu\Lambda=I_{1}\oplus
I_{2}\oplus\dots\oplus I_{n}$ where $\nu$ is the Nakayama functor.

  * •

Let $Q$ be the quiver $1\leftarrow 2$. Then AR quiver is as follows.

$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}}$
---

Then there are 2 tilting $\Bbbk Q$-modules given by $S_{1}\oplus P_{2}$ and
$P_{2}\oplus S_{2}$. Note that $S_{1}\oplus S_{2}$ is not tilting since there
is an extension of $S_{2}$ by $S_{1}$.

  * •

Let $Q$ be the quiver $1\leftarrow 2\leftarrow 3$. Then the AR quiver is as
follows.

$\textstyle{P_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}=S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{3}=S_{3}}$
---

To find all 5 tilting $\Bbbk Q$-modules, we must look for sets of three
modules in the AR quiver that do not form a mesh, or almost split sequence. To
do this, we can draw the compatibility graph, which is the graph in which
there is an edge between any two indecomposable modules that form a rigid
pair. This idea first originated as generalized associahedra (Stasheff
polytopes) by Chapoton, Fomin, and Zelevinsky in [11]. Later in [36], Marsh,
Reineke, and Zelevinsky constructed these generalized associahedra using the
representation theory of quivers and something called the category of
decorated representations. Below is the compatibility graph for this example.

$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

The 5 tilting $\Bbbk Q$-modules are given by the triangles in the
compatibility graph. We can re-write this using dimension vectors as follows:

$\textstyle{(0,1,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,1,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,0,0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

Notice that any dimension vector that is a linear combination of any two other
dimension vectors lies on the line connecting the two dimension vectors.

###  4.1 Cluster Variables

A cluster algebra whose initial quiver is of type $A_{3}$ has 9 cluster
variables, 6 from the cluster character of the 6 indecomposable rigid modules
and 3 from the initial cluster $(x_{1},x_{2},x_{3})$. We can put all these
cluster variables in an extended compatibility graph where we draw an edge
between any two clusters variables that occur in a cluster.

$\textstyle{x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$---------------------------$\textstyle{\chi(S_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(P_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(I_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(P_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(S_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(S_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

Including the trivial outer triangle that corresponds to the initial cluster,
there are 14 triangles in the compatibility graph. We can visualize cluster
mutation as ‘wall crossing’ in the compatibility graph and moreover, from the
fact that the compatibility graph is a manifold, we see that we can get
between any two clusters, or triangles, by a sequence of mutations. Consider
the sequence of mutations of the initial cluster
$(x_{1},x_{2},x_{3})\overset{\mu_{1}}{\rightarrow}(\chi(x_{1}),x_{2},x_{3})\overset{\mu_{3}}{\rightarrow}(\chi(x_{1}),x_{2},\chi(x_{3}))$.
Then this is seen in the compatibility graph as the following sequence of
‘wall crossings’ where the corresponding triangle has edges given by the
cluster variables in the mutated cluster.

$\textstyle{x_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$---------------------------$\textstyle{\chi(S_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(P_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(I_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(P_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(S_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi(S_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{3}}$$\textstyle{\cdot}$$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$
---

###  4.2 Support Tilting Modules and Shifted Projectives

We will now try to understand why there are 14 clusters, but only 5 tilting
modules. To do this, we need to weaken the notion of a tilting module to a
support tilting module which was introduced by Ingalls and Thomas in [32].

######  Definition 4.2.

A $\Bbbk Q$ module $T=T_{1}\oplus T_{2}\oplus\dots\oplus T_{k}$ where $k\leq
n$ is called support tilting if the following hold.

  1. 1.

$T$ is rigid.

  2. 2.

The $T_{i}$ are non-isomorphic.

  3. 3.

The support of $T$ has $k$ elements.

It is condition 3 that motivates the name support tilting because this
condition stipulates that the module is tilting on its support. Recall that
the support of a module is the set of vertices of the quiver at which the
corresponding representation has a non-zero vector space.

######  Example 4.2.

Take $Q$ to be $1\leftarrow 2\leftarrow 3$.

  * •

$P_{1}\oplus P_{2}$ is support tilting.

  * •

Any $S_{i}$ is support tilting.

  * •

$P_{2}\oplus P_{3}$ is not support tilting. Note that $|$supp$(P_{2}\oplus
P_{3})|=|\\{1,2,3\\}|=3\neq 2$.

We will now introduce the algebraic objects that correspond to the initial
cluster variables. A shifted projective $P_{i}[1]$ is an object with
projective presentation $P_{i}\rightarrow 0\rightarrow P_{i}[1]$. These are
more naturally realized in the bounded derived category of mod-$\Lambda$ where
the shift $[1]$ denotes the shift functor in $D^{b}(\text{mod-}\Lambda)$ and
the presentation is a distinguished triangle in the category. After applying
the Nakayama functor to the aforementioned projective presentation, we get
$0\rightarrow I_{i}=\tau P_{i}[1]\rightarrow I_{i}\rightarrow 0$, so we
conclude that $\tau(P_{i}[1])=I_{i}$. For $Q:1\leftarrow 2\leftarrow 3$, the
AR quiver with the shifted projectives is the following:

$\textstyle{P_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{3}[1]}$
---

Really what we are looking at here is the ‘cluster category’ intoduced by
Buan, Marsh, Reineke, Reiten, and Todorov in [8].

###  4.3 Support Tilting Pairs and Extending the Cluster Character

The notion of support tilting (silting) pairs was first introduced in [5] by
Broomhead, Pauksztello and Ploog, though as we will see, the notion of silting
objects is older.

######  Definition 4.3.

A silting pair is a tuple of $\Bbbk Q$-modules $(T,P)$ where $T=T_{1}\oplus
T_{2}\oplus\dots\oplus T_{k}$ is a support tilting module and
$P=P_{j_{1}}\oplus P_{j_{2}}\oplus\dots\oplus P_{j_{n-k}}$ is a projective
module with $n-k$ components whose simple tops $S_{j_{1}},\dots,S_{j_{n-k}}$
at the vertices of $Q$ are not in the support of $T$.

Note the condition on the tops of the projective summands is equivalent to
Hom$(P,T)=0$.

######  Example 4.3.

Taking $Q$ to be the linear $\mathbb{A}_{3}$ quiver from above, we have the
following silting pairs.

  * •

$(P_{1}\oplus P_{3},P_{3})$ is a silting pair.

  * •

$(S_{1},P_{2}\oplus P_{3})$ is a silting pair.

We are now ready to completely explain the compatibility diagram in terms of
algebraic objects. The first part of the bijection in the next theorem was
first proven by Buan, Marsh, Reineke, Reiten, and Todorov in [8]. They
moreover showed that the tilting objects in the cluster category, which are
formed by taking direct sums of indecomposables and shifted projectives, are
in bijection with clusters. Adachi, Iyama, and Reiten in [1] showed that
silting pairs $(T_{1}\oplus T_{2}\oplus\dots\oplus T_{k},P_{j_{1}}\oplus
P_{j_{2}}\oplus\dots\oplus P_{j_{n-k}})$ and the aforementioned tilting
objects are in bijection via the map $(T,P)\mapsto T_{1}\oplus
T_{2}\oplus\dots\oplus T_{k}\oplus P_{j_{1}}[1]\oplus
P_{j_{2}}[1]\oplus\dots\oplus P_{j_{n-k}}[1]$. This is one way to get the
latter part of the below bijection.

######  Theorem 4.1.

There is a bijection between the collections $\\{$rigid indecomposable $\Bbbk
Q$-modules $M$ and shifted projectives$\\}$ and $\\{$cluster variables in the
corresponding cluster algebra$\\}$. This bijection induces a bijection
$\varphi:\\{\text{silting pairs}\\}\rightarrow\\{\text{clusters}\\}$ by
$\varphi((T,P))=(\chi(T_{1}),\chi(T_{2}),\dots,\chi(T_{k}),x_{j_{1}},\dots,x_{j_{n-k}})$.

######  Remark 4.1.

Recall in Example 3.4, we mentioned that the Euler characteristic of the
quiver grassmannian of any module can be computed using the shortcut $2-2g$
where $g$ is the genus of the quiver grassmannian. By the positivity
conjecture, Theorem 2.1, this number is always nonnegative whenever $M$
corresponds to a cluster variable. The previous theorem allows us to conclude
that the genus of the quiver grassmannian of a rigid module $M$ is at most 1.

We can now define the extended compatibility graph in terms of representation
theory, which was done by Buan, Marsh, Reineke, Reiten, and Todorov in [8], as
follows. We connect $P_{i}[1]$ and $M$ with an edge if Hom$(P_{i},M)=0$, we
connect $M$ and $N$ with an edge if Ext$(M,N)=$ Ext$(N,M)=0$, and we connect
$P_{i}[1]$ and $P_{j}[1]$ with an edge so long as $i\neq j$. The extended
compatibility graph for the linear $\mathbb{A}_{3}$ quiver is as follows.

$\textstyle{P_{3}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$---------------------$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

Notice that in the above extended compatibility graph, a nontrivial edge is
only shared by at most two triangles. For instance if we consider the silting
pair $(S_{1}\oplus S_{3},P_{2})$, we are analyzing the triangle with vertices
given by $S_{1},S_{3},$ and $P_{2}[1]$. We see that if we remove $P_{2}[1]$,
there is only one other triangle that has two vertices given by $S_{1}$ and
$S_{3}$, namely the triangle whose third vertex is $P_{3}$. This is an example
of the following not algebraically obvious theorem proven by Buan, Marsh,
Reineke, Reiten, and Todorov in [8].

######  Theorem 4.2.

Given a silting pair $(T,P)$ and one object in the cluster $T_{i}$ or
$P_{i}[1]$, there is exactly one way to replace that object with another
object such that the new collection of objects is a silting pair.

###  4.4 Stability Pictures

The stability picture, more commonly known as the wall and chamber structure,
is another method of studying quiver algebras. We will now define the
stability picture and see how it is related to the extended cluster diagram.
We first need the notion of stability conditions, which was first studied by
King in [34]. In [4], Bridgeland used these stability conditions to construct
a scattering diagram whose support is the so-called wall and chamber structure
which we define below.

######  Definition 4.4.

For $\theta\in\mathbb{R}^{n}$, a non-zero module $M\in\text{mod}$-$\Bbbk Q$ is
called $\theta$-stable if it is orthogonal to $\theta$, that is, $\theta\cdot
M=\theta\cdot\text{dim}M=0$, and $\theta\cdot L<0$ for every proper submodule
$L$ of $M$. Moreover, a module $M$ orthogonal to $\theta$ is called
$\theta$-semistable if $\theta\cdot L\leq 0$ for every submodule $L$ of $M$.

We wish to study all the values of $\theta$ such that a fixed module $M$ is
$\theta$-stable.

######  Definition 4.5.

The stability space of a $\Bbbk Q$-module $M$ is
$\mathcal{D}(M)=\\{\theta\in\mathbb{R}^{n}:M\,\text{is}\,\theta\text{-semistable}\\}$.

When $\mathcal{D}(M)$ has codimension 1, we call it a wall. It is not the case
that the stability space of every indecomposable module always gives a wall.
Precisely which modules have walls as their stability space has been worked
out by Treffinger in [42].

######  Definition 4.6.

Let

$\mathcal{R}=\mathbb{R}^{n}-\overline{\bigcup_{M\in\text{mod}\Bbbk
Q}\mathcal{D}(M)}$

denote the maximal open set of $\theta$ having no $\theta$-semistable non-zero
modules. Then a connected component $\mathcal{C}$ of $\mathcal{R}$ is called a
chamber.

######  Example 4.4.

Let $Q=1\leftarrow 2$. Then we have three indecomposable modules
$S_{1},P_{2},$ and $S_{2}$ with respective dimension vectors $(1,0),(1,1),$
and $(0,1)$. Given $\theta\in\mathbb{R}^{2}$, we have
$(a,b)\cdot(1,0)=0\Longleftrightarrow a=0$, so
$\mathcal{D}(S_{1})=\\{(a,b):a=0\\}$. Similarly,
$\mathcal{D}(S_{2})=\\{(a,b):b=0\\}$. Finally,
$(a,b)\cdot(1,1)=0\Longleftrightarrow a=-b$. But note that since $S_{1}\leq
P_{1}$, we also require that $(a,b)\cdot(1,0)\leq 0\Longleftrightarrow a\leq
0$. Therefore $\mathcal{D}(P_{2})=\\{(a,b):a=-b\,\text{and}\,a\leq 0\\}$.
Below is a depiction of the wall and chamber structure, which we also call the
stability picture. In this stability picture we also have the $g$-vectors of
each indecomposable module and shifted projective. Note that the $g$-vector of
the shifted projective $P_{i}[1]$ is $(0,\dots,0,-1,0,\dots,0)$ where the $-1$
is in the $i$th postition, since any shifted projective has a projective
presentation $P_{i}\rightarrow 0\rightarrow P_{i}[1]$.

$\mathcal{D}(S_{1})$$\mathcal{D}(S_{1})$$\mathcal{D}(S_{2})$$\mathcal{D}(S_{2})$$\mathcal{D}(P_{2})$$\displaystyle
g(P_{1})$$\displaystyle g(P_{2})$$\displaystyle g(P_{2}[1])$$\displaystyle
g(P_{1}[1])$$\displaystyle g(I_{2})$

Notice in the previous example that the $g$-vectors of all indecomposable
modules and shifted projectives lie on a wall in the stability picture. This
is indeed not a coincidence and was proven in more generality by Brüstle,
Smith, and Treffinger in [7].

######  Theorem 4.3.

The $g$-vectors of indecomposable rigid modules and shifted projectives lie on
walls in the stability picture.

######  Example 4.5.

Take $Q=1\leftarrow 2\leftarrow 3$. Then a stereographic projection of the
stability picture is depicted below. The $i$th coordinate is negative inside
the wall of the $i$th simple, it is positive outside the wall, and it is zero
on the wall. The vertices depicted are the $g$-vectors of the corresponding
indecomposable modules and their shifted projectives. Notice that there are 14
chambers including the trivial outer chamber, each bounded by three
$g$-vectors of three indecomposable modules or shifted projectives. This is no
coincidence since the objects corresponding to the vertices of each chamber
form a silting pair and hence a cluster. Notice that the stability picture is
actually homeomorphic to the extended compatibility graph that we have already
computed by straightening the walls that connect any two $g$-vectors.

$\mathcal{D}(S_{1})$$\mathcal{D}(S_{2})$$\mathcal{D}(S_{3})$$\mathcal{D}(P_{2})$$\mathcal{D}(P_{3})$$\mathcal{D}(I_{2})$$g(P_{3})$$g(I_{2})$$g(P_{3}[1])$$g(S_{3})$$g(P_{2})$$g(P_{1})$$g(P_{2}[1])$$g(P_{1}[1])$$g(S_{2})$

In general, there is a bijection between chambers in the stability picture and
silting pairs proven by Brüstle, Smith, and Treffinger in [7].

######  Theorem 4.4.

The chambers in the stability picture are in bijection with silting pairs, and
hence clusters, where each chamber is sent to the direct sum of the
modules/shifted projectives whose $g$-vectors enclose said chamber.

It follows from this theorem that two $g$-vectors that lie on a wall in the
stability picture with no $g$-vector between them form a rigid pair. Thus the
extended compatibility graph and the stability picture are homeomorphic for
any finite dimensional hereditary algebra. Notice that the wall and chamber
structure in the previous example triangulates the 2-sphere. This was proven
true by Demonet, Iyama, and Jasso in [14] for all algebras that have finitely
many silting pairs.

######  Theorem 4.5.

The compatibility graph and stability picture of a hereditary algebra with
finitely many silting pairs form a triangulation of an $n-1$ sphere.

##  5 Rigidity

One may question whether the stability picture in Example 4.5 is accurate;
that is, whether there is nothing inside $\mathcal{D}(S_{1})$ and outside both
$\mathcal{D}(S_{3})$ and $\mathcal{D}(S_{2})$ for instance. The answer to this
question is yes, the diagram is accurate and the reason for this is because
‘rigidity is an open condition’. We will spend this section proving this fact,
but first, we begin with an example.

######  Example 5.1.

Take $\Bbbk=\mathbb{C}$ and $Q:1\rightarrow 2$. Then the collection of all
representations of $Q$ with dimension $(3,2)$ is

$\text{Rep}_{\mathbb{C}Q}(3,2):=\\{\mathbb{C}^{3}\overset{f}{\rightarrow}\mathbb{C}^{2}\\}\cong\mathbb{C}^{6}.$

The morphism $f$ is given by a $3\times 2$ matrix with entries in
$\mathbb{C}$, say $\begin{bmatrix}a&c&e\\\ b&d&f\end{bmatrix}$. The condition
of $f$ being onto is equivalent to this matrix having full rank of 2, which is
further equivalent to the non-vanishing of the minors: $ad-bc\neq 0$, $cf-
de\neq 0$, and $af-be\neq 0$. Since these are polynomial inequalities,
$\\{f:f\,\text{is onto}\\}$ is a Zariski open subset of
$\text{Rep}_{\mathbb{C}Q}(3,2)\cong\mathbb{C}^{6}$. With the usual topology on
$\mathbb{C}^{6}$, we see that $\\{f:f\text{ is onto}\\}$ is open, dense, and
has full measure.

If $f$ is onto, then the corresponding representation decomposes:

$\mathbb{C}^{3}\twoheadrightarrow\mathbb{C}^{2}\cong\mathbb{C}^{2}\overset{\cong}{\rightarrow}\mathbb{C}^{2}\oplus\mathbb{C}\rightarrow
0=2P_{1}\oplus S_{1}.$

Therefore all the $f$ in this open, dense, full measure set give the same
rigid module, namely $2P_{1}\oplus S_{1}$.

It is this idea that surjectivity of a morphism is a Zariski open condition,
illuminated by the previous example, that will play a key role in showing that
the stability picture in Example 4.5 is accurate. To continue, we must further
develop the notion of 2-term silting complexes.

###  5.1 Category of 2-term Silting Complexes

The notion of a silting object in a category was first used studied by Keller
and Vossieck in [33]. The notion of two-term silting complexes was first
studied several years later by Hoshino, Kato, and Miyachi in [25]. Some time
later, Adachi, Iyama, and Reiten in [1] noticed that the zero cohomologies of
such complexes connect with cluster theory and more generally, $\tau$-tilting
theory. For more on the topic, we suggest Hügel’s survey [26].

The category of 2-term silting complexes is a subcategory of the bounded
derived category of $\Bbbk Q$. Its objects, called 2-term silting complexes,
are segments of chain complexes of the form
$C_{1}\overset{P}{\rightarrow}C_{0}$ where $C_{0}$ and $C_{1}$ are both
projective $\Bbbk Q$-modules. The morphisms are morphisms of chain complexes
up to chain homotopy. More concretely, they are pairs of module morphisms
$f_{*}=(f_{1},f_{0})$ up to equivalence such that the following diagram
commutes:

$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{f_{1}}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{D_{0}}$
---

Two morphisms $f_{*}$ and $g_{*}$ are equivalent, or chain homotopic denoted
by $f_{*}\simeq g_{*}$, if there exists a module morphism $h:C_{0}\rightarrow
D_{1}$ such that $qh=g_{0}-f_{0}$ and $hp=g_{1}-f_{1}$. The following diagram
may be useful.

$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{g_{1}}$$\scriptstyle{f_{1}}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{g_{0}}$$\scriptstyle{f_{0}}$$\textstyle{D_{0}}$
---

The morphism $h$ is a chain homotopy. The indecomposable objects of this
category are completely described:

######  Theorem 5.1.

The indecomposable objects in the category of 2-term silting complexes are

  1. 1.

Projective presentations of indecomposable $\Bbbk Q$-modules $M$:
$C_{1}\overset{p}{\hookrightarrow}C_{0}$ where the cokernel of $p$ is $M$.

  2. 2.

Objects of the form $P\rightarrow 0$ where $P$ is an indecomposable
projective, that is, the projective presentations of the shifted projectives.

###### Sketch of proof.

Let $C_{1}\overset{p}{\rightarrow}C_{0}$ be a 2-term silting complex. Then we
have a commutative diagram:

$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{C_{0}}$$\textstyle{\text{im}(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

Since $\Bbbk Q$ is hereditary, we have that im$(p)$ is a projective module, so
we have a projective presentation of $M=\text{coker}(p)$ given by
$0\rightarrow\text{im}(p)\rightarrow C_{0}\rightarrow M\rightarrow 0$.
Moreover, the exact sequence $0\rightarrow\text{ker}(p)\rightarrow
C_{1}\rightarrow\text{im}(p)\rightarrow 0$ splits by the projectivity of
im$(p)$. Therefore, we can decompose the original 2-term silting complex into
the direct sum of projective presentations of a shifted projective and $M$
respectively:

$C_{1}\overset{p}{\rightarrow}C_{0}\cong\text{ker}(p)\rightarrow
0\oplus\text{im}(p)\rightarrow C_{0}.$

∎

We will now recall some facts from homological algebra and category theory.

######  Lemma 5.2.

The projective presentation of a $\Bbbk Q$-module $M$ is unique up to chain
homotopy.

######  Theorem 5.3.

Let $C_{1}\rightarrow C_{0}$ and $D_{1}\rightarrow D_{0}$ be two projective
presentations of $M$ and $N$ respectively. Then Ext${}^{1}(M,N)$ is the set of
homotopy classes of maps $h:C_{1}\rightarrow D_{0}$.

What this theorem is saying is that Ext$(M,N)$ is equivalent to the set of all
$h$ up to homotopy of the form

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{g}$$\scriptstyle{h}$$\textstyle{D_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{0}$
---

In this diagram, we have shifted the projective presentation of $N$ by one.
Really what we are looking at here is Hom$(M,N[1])$ in the derived category,
which is well known to be Ext$(M,N)$; however, from this diagram we can
conclude that Ext$(M,N)$ is the cokernel of the map Hom${}_{\Bbbk
Q}(C_{0},D_{0})\oplus$ Hom${}_{\Bbbk
Q}(C_{1},D_{1})\overset{(p,q)}{\rightarrow}\text{Hom}_{\Bbbk Q}(C_{1},D_{0})$
that sends $(f,g)\mapsto f\circ p+q\circ g$. If Ext$(M,N)=0$, then this map is
surjective, a Zariski open condition on the maps $p$ and $q$. We have the
following corollary whose proof is written down by Igusa, Orr, Todorov, and
Weyman in [29].

######  Corollary 5.4.

Suppose Ext$(M,N)=0$. Then there exist open neighborhoods $U$ and $V$ of $p$
and $q$ respectively in Hom${}_{\Bbbk Q}(C_{1},C_{0})$ and Hom${}_{\Bbbk
Q}(D_{1},D_{0})$ such that all $f\in U$ and $g\in V$ are monomorphisms whose
cokernels satisfy Ext$(\text{coker}f,\text{coker}g)=0$.

To continue, we require the following lemma originally due to Happel and
Ringel in [23]. A proof of which can also be found in [12].

######  Lemma 5.5 (Happel-Ringel).

Let $A,B$ be two indecomposable $\Bbbk Q$-modules such that Ext$(B,A)=0$. Then
any nonzero morphism $f:A\rightarrow B$ is either mono or epi.

###### Proof.

Let $f:A\rightarrow B$ be a morphism and let $C=\text{im}f$ and
$X=\text{ker}f$. Then we have a short exact sequence $0\rightarrow
X\rightarrow A\overset{g}{\rightarrow}C\rightarrow 0$. This induces a long
exact sequence that contains the chain complex
Ext$(B/C,X)\rightarrow\text{Ext}(B/C,A)\overset{g_{\\#}}{\rightarrow}\text{Ext}(B/C,C)\rightarrow
0$ where the last term vanishes because $\Bbbk Q$ is hereditary. We conclude
that the map $g_{\\#}$ is a surjection. In particular, this implies that there
exists a module $D$ such that we have the following diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
---

This yields a Mayer-Vietoris sequence in which every third map is an
isomorphism, hence we attain a short exact sequence $0\rightarrow A\rightarrow
C\oplus D\rightarrow B\rightarrow 0\in\text{Ext}(B,A)$. Since
$\text{Ext}(B,A)=0$ by assumption, we have that $C\oplus D\cong A\oplus B$. We
conclude that $C=A$ in which case $f$ is mono, or $C=B$, in which case $f$ is
epi. ∎

An example of how we can use this lemma is the following.

######  Example 5.2.

Let $Q=1\leftarrow 2\leftarrow 3$. Then there is a map from $P_{2}$ to
$I_{2}$, namely $(0,1,0)$. This map is neither mono nor epi, so by the Happel-
Ringel lemma we conclude that there is a nontrivial extension of $I_{2}$ by
$P_{1}$, namely $P_{2}\hookrightarrow S_{2}\oplus P_{3}\twoheadrightarrow
I_{2}$, which is an almost split sequence.

Moreover, if we consider a rigid module $M$, then any map from $M$ to itself
is either mono or epi by the Happel-Ringel lemma. Then we have the following
corollary which can also be found in [12].

######  Corollary 5.6.

If $M$ is rigid and indecomposable, then End$(M)=\Bbbk$; that is, $M$ is a
brick.

###  5.2 Rigidity is Open

The next and main result of this section provides some intuition behind why
the term rigid is chosen. Intuitively it comes down to the fact that if you
shake $M$ a little bit, $M$ won’t change. The proof is written by Igusa, Orr,
Todorov, and Weyman in [29].

######  Theorem 5.7.

Let $0\rightarrow P^{\prime}\rightarrow P\rightarrow M\rightarrow 0$ be a
projective presentation of a rigid $\Bbbk Q$ module $M$. Then there is a
Zariski open neighborhood $U\subset\text{Hom}(P^{\prime},P)$ such that all
$f\in U$, are mono and $M_{f}:=\text{coker}f\cong M$. In other words, If $M$
is rigid, almost all modules $M^{\prime}$ with the same dimension vector as
$M$ are isomorphic to $M$.

###### Sketch of Proof.

Suppose $M$ is indecomposable, we will prove the general case in Section 6. By
Corollary 5.4, we know there is an open $U\subset\text{Hom}(P^{\prime},P)$
such that all $f,g\in U$ are mono and Ext$(M_{f},M_{g})=0$. It remains to show
that $M_{f}$ and $M_{g}$ are isomorphic. We have two short exact sequences

$0\rightarrow P^{\prime}\rightarrow P\rightarrow M_{g}\rightarrow 0$
$0\rightarrow P^{\prime}\rightarrow P\rightarrow M_{f}\rightarrow 0$

These short exact sequences allow us to conclude that the dimension vectors of
$M_{f}$ and $M_{g}$ are the same. Moreover, they induce long exact sequences

$0\rightarrow\text{Hom}(M_{g},M_{f})\rightarrow\text{Hom}(P,M_{f})\overset{g^{\\#}}{\rightarrow}\text{Hom}(P^{\prime},M_{f})\rightarrow
0$
$0\rightarrow\text{Hom}(M_{f},M_{f})\rightarrow\text{Hom}(P,M_{f})\overset{f^{\\#}}{\rightarrow}\text{Hom}(P^{\prime},M_{f})\rightarrow
0$

where the last term is zero since there are no extensions. Since $M_{f}$ is
rigid and indecomposable, it is a brick by Corollary 5.6, so
$\text{Hom}(M_{f},M_{f})=\Bbbk$. This forces Hom$(M_{g},M_{f})\neq 0$, so
there is a nontrivial morphism $h:M_{g}\rightarrow M_{f}$ that is either mono
or epi by the Happel-Ringel lemma. But since these two modules have the same
dimension vector, it must be an isomorphism. ∎

We will now list some consequences of this theorem, all of which have also
been proven using different methods by Brüstle, and Treffinger in [7] and by
Adachi, Iyama, and Reiten in [1].

######  Corollary 5.8.

The components $T_{i}$ of a rigid module have linearly independent $g$-vectors
and dimension vectors.

###### Sketch of Proof.

Since dim$M=C_{\Bbbk Q}g(M)$ where $C_{\Bbbk Q}$ is the Cartan matrix, it
suffices to prove this for the $g$-vectors. For a contradiction, suppose that
the components of a rigid module have linearly dependent dimension vectors.
Then $\sum n_{i}g(T_{i})=0$. We then collect the positive and negative terms
as in the following example. Suppose for example we have
$2g(T_{1})+g(T_{2})=3g(T_{3})+g(T_{4})=(4,-2,-1,5)$. Then this implies we have
two maps $\textstyle{2P_{2}\oplus
P_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{4P_{1}\oplus
5P_{4}}$ such that coker$f=2T_{1}\oplus T_{2}$ and coker$g=3T_{3}\oplus
T_{4}$. By the previous theorem, we have Zariski open
$U,V\subset\text{Hom}(2P_{2}\oplus P_{3},4P_{1}\oplus 5P_{4})$ such that all
$h\in U$ have the same cokernel as $f$ and all $j\in V$ have the same cokernel
as $g$. Since $U$ and $V$ are Zariski open, their intersection is not empty.
But then the previous theorem implies that $3T_{3}\oplus T_{4}\cong
2T_{1}\oplus T_{2}$, a contradiction. ∎

The following three facts are immediate consequences from the previous
corollary.

######  Corollary 5.9.

.

  1. 1.

Rigid modules can have at most $n$ nonisomorphic indecomposable summands.

  2. 2.

The $g$-vectors of the components of a silting pair span an $n-1$ simplex.

  3. 3.

Simplices can’t overlap.

In particular, the fact that these simplices don’t overlap allows us to
conclude that the stability pictures we have drawn in Section 4.4 are indeed
accurate. Moreover, it implies that we cannot have simplices of the following
form in our extended compatibility graph.

$T_{1}$$T_{2}$$T_{3}$$T_{4}$$T_{5}$

If we did have such simplices, then the mutation of $T_{1}$ would not be well
defined since we would not know whether to send it to $T_{4}$ or $T_{5}$.

###  5.3 Stable Barcode

In this subsection, we will present a connection between rigidity and
persistent homology through stable barcodes. For more on the topic we suggest
Vejdemo-Johansson’s survey on the topic [44]. Throughout this subsection, let
$Q$ be the linearly oriented quiver of type $\mathbb{A}_{n}$ given by
$1\leftarrow 2\leftarrow\dots\leftarrow n$. Recall that the indecomposable
modules are uniquely determined by their dimension vector. This leads to the
notion of interval modules where $M_{ab}$ denotes the indecomposable module
with support $[a,b]$. We will construct the stable barcode associated to the
rigid modules. To do this, we need a lemma that classifies extensions of
interval modules.

######  Lemma 5.10.

Ext$(M_{cd},M_{ab})\neq 0$ if and only if one of the following hold:

  1. 1.

$a<c\leq b<d$.

  2. 2.

$b+1=c$.

In case 1 we have a short exact sequence given by $M_{ab}\hookrightarrow
M_{ad}\oplus M_{cb}\twoheadrightarrow M_{cd}$. In case 2 we have an exact
sequence $M_{ab}\hookrightarrow M_{ad}\twoheadrightarrow M_{cd}$. This lemma
allows us to prove the following theorem.

######  Theorem 5.11.

The unique rigid module with dimension vector $v$ is given by placing $v_{i}$
spots above the point $(i,0)$ on the $x$-axis and joining adjacent spots
horizontally. This is called the stable barcode associated to $v$.

######  Example 5.3.

Take $v=(3,4,2)$. Then the corresponding rigid module is $M_{22}\oplus
M_{12}\oplus 2M_{13}$ and the stable barcode is as follows.

$M_{12}$$M_{22}$$2M_{13}$$v_{i}:\ \ \ \ \ \ \ 3\ \ \ \ \ \ \ \ \ \ 4\ \ \ \ \
\ \ \ \ \ 2$$i\ \ :\ \ \ \ \ \ \ 1\ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ \ \ \ 3$

##  6 Maximal Green Sequences

In this section we will begin by removing the indecomposability assumption in
the proof of Theorem 5.7. We begin with a lemma that was first proven by
Schofield in [39], then proved in the language used here by Igusa and
Schiffler in [30].

######  Lemma 6.1.

Let $T_{1},T_{2},\dots,T_{k}$ be indecomposable ext-orthogonal $\Bbbk
Q$-modules; that is, Ext$(T_{i},T_{j})=0$ for all $i$ and $j$. Then the
$T_{i}$ can be ordered such that Hom$(T_{j},T_{i})=0$ for all $i<j$. Such a
sequence of modules is an example of an exceptional sequence, which is a
sequence of modules $(M_{1},M_{2},\dots,M_{k})$ such that
Hom$(M_{j},M_{i})=0=$ Ext$(M_{j},M_{i})$ for all $i<j$.

###### Sketch of Proof.

It suffices to show that there are no oriented cycles of hom:

$\textstyle{T_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

If there were such a cycle, by the Happel-Ringel lemma, each map is a
monomorphism or an epimorphism. This forces an epimorphism followed by a
monomorphism whose composition is nonzero and neither mono nor epi, a
contradiction of the Happel-Ringel lemma. ∎

######  Corollary 6.2.

The set $\\{T_{1},T_{2},\dots,T_{k}\\}$ form an exceptional collection, that
is, there exists at least one ordering $(T_{i_{1}},T_{i_{2}},\dots,T_{i_{k}})$
that is an exceptional sequence.

We are now ready to prove Theorem 5.7.

###### Proof Sketch of Theorem 5.7.

Let $M$ be a rigid $\Bbbk Q$-module and
$P_{1}\overset{p}{\rightarrow}P_{0}\rightarrow M$ be a projective presentation
of $M$. By Corollary 5.4, there exists an open
$U\subset\text{Hom}(P_{1},P_{0})$ such that any $f,g\in U$ is such that
Ext$(M_{f},M_{g})=0$ where $M_{f}$ is the cokernel of $f$ and similarly for
$g$. Therefore, $M_{f}\oplus M_{g}$ is rigid. Let $E_{1},E_{2},\dots E_{k}$ be
the list of nonisomorphic indecomposable summands of $M_{f}\oplus M_{g}$. By
Lemma 6.1, we may without loss of generality assume that Hom$(E_{j},E_{i})=0$
for $j>i$. By applying Hom$(-,E_{i})$ to the short exact sequence
$0\rightarrow P_{1}\overset{f}{\rightarrow}P_{0}\rightarrow M_{f}\rightarrow
0$ and the analogous one for $g$, we get the long exact sequences

$0\rightarrow\text{Hom}(M_{f},E_{i})\rightarrow\text{Hom}(P_{0},E_{i})\rightarrow\text{Hom}(P_{1},E_{i})\rightarrow
0$
$0\rightarrow\text{Hom}(M_{g},E_{i})\rightarrow\text{Hom}(P_{0},E_{i})\rightarrow\text{Hom}(P_{1},E_{i})\rightarrow
0$

where the last term is zero since Ext$(M_{f},E_{i})=0$ and analogously for
$g$. By analyzing dimensions we conclude that dim(Hom$(M_{f},E_{i}))=$
dim(Hom$(M_{g},E_{i}))=\Bbbk^{a_{i}}$. Suppose that $M_{f}=b_{1}E_{1}\oplus
b_{2}E_{2}\oplus\dots\oplus b_{k}E_{k}$. Then we have that
$a_{1}=b_{1},a_{2}=b_{2}+c_{12}b_{1},$ and so on where
$c_{ij}=\text{dim(Hom(}E_{i},E_{j}))$. Therefore the $a_{i}$ determine the
$b_{i}$ and we conclude $M_{f}\cong M_{g}$. ∎

###  6.1 Sign Coherence of $g$-vectors

The sign coherence of $g$-vectors was a conjecture until proven in [22] by
Gross, Hacking, Keel, and Kontsevich for skew-symmetrizable cluster algebras.
The now theorem is as follows.

######  Theorem 6.3.

For any tilting module $T=T_{1}\oplus T_{2}\oplus\dots\oplus T_{n}$ and any
$k$, the $k$th coordinate of $g(T_{i})$ have the same sign.

######  Example 6.1.

Consider $Q:1\leftarrow 2\leftarrow 3$ and $T=I_{2}\oplus S_{2}\oplus P_{3}$.
Then by placing the corresponding $g$-vectors in the first, second, and third
column respectively of a matrix, we get the following.

$\begin{bmatrix}-1&-1&0\\\ 0&1&0\\\ 1&0&1\end{bmatrix}$

Note in each row, the signs of all the entries are the same (or zero). The
first row is all negative and the last two are all positive.

The sign coherence of $g$-vectors is equivalent to the statement that
$D(S_{k})=\\{x\in\mathbb{R}^{n}:x_{k}=0\\}$ does not cut the simplex with
vertices $g(T_{i})$. That is, we can’t have the following in the stability
picture:

$\textstyle{g(T_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D(S_{k})}$$\textstyle{g(T_{3})}$$\scriptstyle{x_{k}>0}$$\scriptstyle{x_{k}<0}$$\textstyle{g(T_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

###  6.2 $c$-vectors and Frozen Vertices

Recall the notion of quiver mutation from Definition 2.2. We can provide an
analogous definition of mutation in terms of the exchange matrix, which can be
found in [6], as follows.

######  Definition 6.1.

Let $B$ be the exchange matrix for a quiver $Q$. Then for any $1\leq k\leq n$,
the mutation of $B$ in the direction $k$ is the matrix $\mu_{k}(B)=[b_{ij}]$
given by

$b_{ij}=\begin{cases}-b_{ij}&i=k\,\text{or}\,j=k\\\
b_{ij}+\text{max}(b_{ik},0)\text{max}(b_{kj},0)-\text{min}(b_{ik},0)\text{min}(b_{kj},0)&\text{otherwise}\end{cases}$

Notice that $\mu_{k}(B)$ is the exchange matrix of $\mu_{k}(Q)$, hence is
skew-symmetric. We can perform mutations of the exchange matrix in two steps
as in the following example.

######  Example 6.2.

Recall the exchange matrix from Example 2.2. Then we can mutate at 2 using the
following two step process. Since we are mutating at two, we highlight both
the second row and column.

$\begin{bmatrix}0&1&-2\\\ -1&0&1\\\ 2&-1&0\end{bmatrix}$Step 1Compute the
entriesoutside the ovalsStep 2Negate the entriesinside the
ovals$\begin{bmatrix}0&1&-1\\\ -1&0&1\\\
1&-1&0\end{bmatrix}$$\begin{bmatrix}0&-1&-1\\\ 1&0&-1\\\ 1&1&0\end{bmatrix}$

We will now extend the exchange matrix of $Q$ by adding the identity matrix to
the bottom of $B$. This new extended exchange matrix of $B$ will be denoted
$\tilde{B}={B\over C}$ where $C$ is the identity matrix. We can provide a
purely combinatorial description of how mutation for this new extended
exchange matrix works; however, it may be useful to introduce the notion of
frozen vertices which can also be found in [6].

######  Definition 6.2.

To any quiver
$Q=(\\{1,2,\dots,n\\},\\{\alpha_{1},\alpha_{2},\dots\alpha_{k}\\})$, we denote
by $\check{Q}$ the quiver with
$\check{Q}_{0}=Q_{0}\cup\\{1^{\prime},2^{\prime},\dots,n^{\prime}\\}$ and
$\check{Q}_{1}=Q_{1}\cup\\{\alpha^{\prime}_{i}:i^{\prime}\rightarrow
i:i\in\\{1,2,\dots,n\\}\\}$. The vertices not in $Q_{0}$ are called frozen
vertices since we can’t mutate $\check{Q}$ at these vertices.

######  Remark 6.1.

Sometimes the notation $\hat{Q}$ is used. The quiver
$\hat{Q}=(\check{Q}_{0},Q_{1}\cup\\{\alpha^{\prime}_{i}:i^{\prime}\leftarrow
i:i\in\\{1,2,\dots,n\\}\\})$.

We can use this extended quiver to define mutation of the extended exchange
matrix. In particular the mutated extended exchange matrix is just the
exchange matrix of the mutated extended quiver.

######  Example 6.3.

Let $Q=1\rightarrow 2$. Then the extended quiver along with a mutation at
vertex 1 is depicted below. Note that the frozen vertices are blue.

$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

The extended exchange matrix along with its mutation at one is depicted below.
For $i,j\leq|Q_{0}|$, the $ij$th entry denotes the number of arrows from $i$
to $j$ minus the number of arrows from $j$ to $i$. For $i>|Q_{0}|$, the $ij$th
entry denotes the number of arrows from $(i-|Q_{0}|)^{\prime}$ to $j$ minus
the number of arrows from $j$ to $(i-|Q_{0}|)^{\prime}$. For example in the
mutated matrix, entry 31 is -1 since in the mutated extended quiver there are
no arrows from $1^{\prime}$ to 1 and there is one arrow from 1 to
$1^{\prime}$.

$\begin{bmatrix}0&1\\\ -1&0\\\ \hline\cr 1&0\\\ 0&1\end{bmatrix}$ | $\overset{\mu_{1}}{\rightarrow}$ | $\begin{bmatrix}0&-1\\\ 1&0\\\ \hline\cr-1&1\\\ 0&1\end{bmatrix}$
---|---|---

######  Definition 6.3.

Let $\tilde{B}={B\over C}$ be the extended exchange matrix associated to the
cluster algebra with initial seed $(Q,x_{*})$. Then the set of $c$-vectors is
the set consisting of the columns of $C$ along with the columns of any
$C^{\prime}$ that arises from a mutation of $\tilde{B}$. The matrix $C$ along
with any mutation of it is called a C-matrix.

###  6.3 Maximal Green Sequences

We are now ready to define maximal green sequences. As a remark, these
sequences are not named after a person, but the color green. Keller is
credited with coming up with the green/red traffic light coloring system.

######  Definition 6.4.

A maximal green sequence (MGS) is a sequence of green mutations starting with
$\tilde{B}={B\over I}$ and ending with all negative $c$-vectors. A mutation
$\mu_{k}$ is said to be green if the $c$-vector $c_{k}$ is positive. If
$c_{k}$ is negative, $\mu_{k}$ is said to be a red mutation.

######  Remark 6.2.

The definition of a MGS is dependent on the fact that $c$-vectors can’t be
both red and green at the same time; that is, they are sign-coherent. This was
proven first by Derksen, Weymen, and Zelevinski for quivers with potential in
[15] then later by Gross, Hacking, Keel, and Kontsevich for general cluster
algebras in [22]. In [19], Fu defined the $c$-vectors in terms of $g$-vectors
then proved the statement for all finite dimensional algebras. Fu used a fact
that we will see later; that is, the sign coherence of $c$-vectors follows
from the fact that they are the dimension vectors of certain indecomposable
modules. In [42], Treffinger classifies precisely the modules whose dimension
vectors give positive $c$-vectors.

######  Conjecture 6.1.

For $Q$ a quiver with no oriented cycles, there are only finitely many MGS.

This is known to be true for tame, affine, and wild quivers with at most three
vertices and was proven by Brüstle, Dupont, and Perotin in [6], but unknown in
general.

######  Example 6.4.

Let $Q$ be the quiver $1\rightarrow 2$. Then we have a maximal green sequence
given by

$\begin{bmatrix}0&1\\\ -1&0\\\ \hline\cr{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}1}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}0}\\\ {\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}0}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}1}\end{bmatrix}$ | $\overset{\mu_{1}}{\rightarrow}$ | $\begin{bmatrix}0&-1\\\ 1&0\\\ \hline\cr{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}1}\\\ {\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}1}\end{bmatrix}$ | $\overset{\mu_{2}}{\rightarrow}$ | $\begin{bmatrix}0&1\\\ -1&0\\\ \hline\cr{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1}\\\ {\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}1}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1}\end{bmatrix}$ | $\overset{\mu_{1}}{\rightarrow}$ | $\begin{bmatrix}0&-1\\\ 1&0\\\ \hline\cr{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1}\\\ {\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}\end{bmatrix}$
---|---|---|---|---|---|---

Notice that the $c$-vectors are the dimension vectors of $S_{1},P_{1}$, and
$S_{2}$ or their negatives. We can also visualize this same maximal green
sequence using frozen vertices as follows.

$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1^{\prime}}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1^{\prime}}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2^{\prime}}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

A third way to visualize a MGS is through so-called green paths in the
stability picture, which were introduced by Igusa and Todorov in [31] and
further explored by Igusa in [27]. The walls indicate which $c$-vectors to
mutate. For instance in the lower path, which corresponds to the MGS shown
earlier in this example, we mutate the $c$-vector that gives the dimension
vectors of $S_{1},P_{2},$ and $S_{2}$ in that order. This is precisely
$\mu_{1}\circ\mu_{2}\circ\mu_{1}$ as seen above. Moreover, the below picture
shows that there are only two paths from the all green region to the all red
region. Therefore, there are only two maximal green sequences in this example;
the lower path corresponds to the MGS shown earlier in this example and the
MGS corresponding to the upper path is shown below the stability picture.

$D(S_{1})$$D(S_{2})$$D(P_{2})$All GreenAll Red

$\begin{bmatrix}0&1\\\ -1&0\\\ \hline\cr{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}1}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}0}\\\ {\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}0}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}1}\end{bmatrix}$ | $\overset{\mu_{2}}{\rightarrow}$ | $\begin{bmatrix}0&-1\\\ 1&0\\\ \hline\cr{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}1}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}\\\ {\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1}\end{bmatrix}$ | $\overset{\mu_{1}}{\rightarrow}$ | $\begin{bmatrix}0&1\\\ -1&0\\\ \hline\cr{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}\\\ {\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1}\end{bmatrix}$
---|---|---|---|---

###  6.4 Relationship Between $c$-vectors and $g$-vectors

As we have seen in the previous examples, the $c$-vectors are dimension
vectors of certain indecomposable modules. In particular, this means they
correspond to the walls in the stability picture and the $g$-vectors are
vectors that lie in these walls. The following theorem formalizes these two
ideas. The first part of the following theorem was done by Nakanishi and
Zelevinski in [37] where the latter part was done by Fu in [19].

######  Theorem 6.4.

Let $T=\oplus T_{i}$ be a silting module. Then the following hold.

  1. 1.

The $g$-vectors $g(T_{i})$ and $c$-vectors $c_{j}$ are related by the
Nakanishi-Zelevinsky formula:

$g(T_{i})\cdot c_{j}=-\delta_{ij}.$

  2. 2.

$c_{j}=\pm\text{dim}M_{j}$ where $g(T_{i})\in D(M_{j})$ for all $i\neq j$.

###### Sketch of Part of Proof.

We will take $j=1$ and assume the Nakanishi-Zelevinsky formula. By Lemma 6.1,
$(T_{2},\dots,T_{n})$ forms an exceptional sequence. By properties of
exceptional sequences, there exists a unique module $M_{1}$ such that
$(M_{1},T_{2},\dots,T_{n})$ is exceptional. We conclude that
Hom$(T_{j},M_{1})=0=\text{Ext}(T_{j},M_{1})$ for all $j$. Therefore,
$g(T_{j})\in D(M_{1})\subset(\text{dim}M_{1})^{\perp}$. This implies that
$g(T_{j})$ is orthogonal to the dimension vector of $M_{1}$. By the Nakanishi-
Zelevinsky formula, $c_{1}=\pm\text{dim}M_{1}$. ∎

It is not the common notation to negate the Kronecker delta in the Nakanishi-
Zelevinsky formula. We will finish this section with a brief explanation of
why the negation of the Kronecker delta makes sense from a representation-
theoretic perspective.

The initial cluster in a cluster algebra is $(x_{1},x_{2},\dots,x_{n})$ and
this corresponds to the silting pair $(0,P_{1}[1]\oplus
P_{2}[1]\oplus\dots\oplus P_{n}[1])$ where $P_{i}[1]$ has $g$-vector $-e_{i}$.
By definition of a MGS, the initial $c$-vectors are $c_{i}=e_{i}$. Therefore,
the initial $g$-vectors dotted with the initial $c$-vectors satisfy the
formula $g(T_{i})\cdot c_{j}=-\delta_{ij}$. This relation is preserved under
mutations since clusters are defined to be those rational functions attained
from the initial cluster by a finite sequence of mutations, hence the negation
of the Kronecker delta.

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