# Some consequences of categorification

Dylan Rupel  University of Notre Dame, Department of Mathematics, Notre Dame,
IN 46556, USA<EMAIL_ADDRESS>and  Salvatore Stella  University of Haifa,
Departments of Mathematics, Haifa, Mount Carmel, 31905, Israel
<EMAIL_ADDRESS>
###### Abstract.

Several conjectures on acyclic skew-symmetrizable cluster algebras are proven
as direct consequences of their categorification via valued quivers. These
include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella
related to $\mathbf{d}$-vectors, $\mathbf{g}$-vectors, and $F$-polynomials.

##  1\. Introduction

The categorification of skew-symmetric cluster algebras using representations
of quivers was initiated by Marsh, Reineke, and Zelevinsky [MRZ03] for Dynkin
quivers. With the advent of cluster characters [CC06], the subject has
exploded as an industry all its own, leading to the publication of numerous
articles including [CCS06, BMR+06, DWZ08, GLS05, CK08, CK06, Pla11, Pal08,
Rup11, Qin12, Rup15], just to name a few.

The main idea is to understand the combinatorics of cluster mutations in terms
of a category of representations of a quiver, and in particular to obtain an
interpretation of Laurent expansions of non-initial cluster variables as
cluster characters. These are generating functions for certain geometric
invariants (e.g. Euler characteristics, point counts over finite fields,
Poincaré polynomials, etc.) of varieties of subrepresentations in an
associated quiver representation.

Using categorification, many structural conjectures on skew-symmetric cluster
algebras, their cluster variables, and the associated combinatorics of
mutations have been established. The main goal of this note is to observe that
the categorification of acyclic skew-symmetrizable (quantum) cluster algebras
[Rup11, Rup15] naturally leads to proofs of many of the same conjectures.

Alternative approaches to categorification of skew-symmetrizable cluster
algebras using quivers with automorphism or group species with potential have
been introduced by Demonet [Dem11, Dem10]. Another approach using species with
potential has been pursued by Labardini-Fragoso and Zelevinsky [LFZ16], see
Section 15 there for a broader overview of current developments. These
formalisms have allowed to prove many conjectures on cluster algebras that we
thus omit here even though they should be deducible in our setting.

Since the appearance of these works, new conjectures on cluster algebras,
particularly ones related to denominator vectors, have been made and our goal
here is to resolve several of these. In some cases our proofs will follow by
direct translation of arguments given in the skew-symmetric case.

######  Theorem 1.

The following conjectures on cluster algebras are true in the acyclic case:
Conjectures 4.14(3-4) from [FZ03]; Conjectures 6.11, 7.4(1-2), 7.5, 7.6, and
7.17 from [FZ07]; Conjectures 3.10 and 3.21 from [RSp16]; Conjectures 2.7,
2.8, and 2.9 from [RSt18].

Acknowledgments We are grateful to Giovanni Cerulli Irelli for his help with
references. D.R. was partially supported by an AMS-Simons Travel Grant; S.S.
was supported by ISF grant 1144/16.

##  2\. Quantum Cluster Algebras and Quantum Cluster Characters

In the interest of brevity, we will assume that the reader is familiar with
the basic notions in the theory of cluster algebras and refer to [FZ07] for
background material.

Let $B=(b_{ij})$ be a $n\times n$ skew-symmetrizable integer matrix; this
means that there exist positive integers $d_{i}$ such that
$d_{j}b_{ji}=-d_{i}b_{ij}$ for all $i,j\in[1,n]$. We write $D$ for the
diagonal matrix whose $i$-th diagonal entry is $d_{i}$ and fix such a matrix
once and for all. We assume throughout the paper that $B$ is _acyclic_ , i.e.
up to a simultaneous permutation of rows and columns, $B$ has only non-
negative entries above the diagonal. Let $\widetilde{B}$ be the matrix
obtained by stacking a $n\times n$ identity matrix below $B$ and write
$\mathcal{A}(\mathbf{x},\widetilde{B})$ for the associated cluster algebra; it
is a cluster algebra with _principal coefficients_.

Since $\widetilde{B}$ has full rank, the algebra
$\mathcal{A}(\mathbf{x},\widetilde{B})$ admits a log-canonical Poisson
structure compatible with mutations [GSV10] and one may obtain a quantum
cluster algebra via a remarkably simple deformation quantization [BZ05].

Intuitively, there are only few points to keep in mind. Fix an indetermiate
$q$, then the definition of the quantum cluster algebra
$\mathcal{A}_{q}(\mathbf{X},\widetilde{B})$ can be obtained from that of
$\mathcal{A}(\mathbf{x},\widetilde{B})$ via the following modifications:

  * •

The initial cluster $\mathbf{X}=(X_{i})$ consists of $2n$ variables which
$q$-commute, i.e. there exists a skew-symmetric $2n\times 2n$ matrix
$\Lambda=(\lambda_{ij})$ so that $X_{i}X_{j}=q^{\lambda_{ij}}X_{j}X_{i}$.

  * •

Let $\mathcal{T}_{\Lambda}$ be the quantum torus algebra generated by
$X_{i}^{\pm 1}$, $i\in[1,2n]$, over the ring
$\mathbb{Z}[q^{\pm{\frac{1}{2}}}]$. It admits an anti-involution (called the
_bar-involution_) fixing each $X_{i}$ and interchanging $q$ with $q^{-1}$.
Every non-initial cluster variable should also be “bar-invariant” and this
uniquely determines the power of $q^{\frac{1}{2}}$ by which to multiply each
monomial in the exchange relations. More precisely, using the bar-invariant
monomial basis $X^{\mathbf{a}}$, $\mathbf{a}\in\mathbb{Z}^{2n}$, of
$\mathcal{T}_{\Lambda}$ given by

$X^{\mathbf{a}}=q^{-{\frac{1}{2}}\sum\limits_{i<j}\lambda_{ij}a_{i}a_{j}}X_{1}^{a_{1}}\cdots
X_{2n}^{a_{2n}},$

we may write the mutated variables as
$X^{\prime}_{k}=X^{\mathbf{b}_{+}^{k}-\varepsilon_{k}}+X^{\mathbf{b}_{-}^{k}-\varepsilon_{k}}$,
where $\varepsilon_{k}$ denotes the $k$-th standard basis vector of
$\mathbb{Z}^{2n}$ and the $k$-th column of $\widetilde{B}$ decomposes as
$\mathbf{b}^{k}=\mathbf{b}^{k}_{+}-\mathbf{b}^{k}_{-}$ for minimal non-
negative vectors $\mathbf{b}^{k}_{+},\mathbf{b}^{k}_{-}\in\mathbb{Z}_{\geq
0}^{2n}$.

  * •

Each cluster $\mathbf{X}^{\prime}$ obtainable from $\mathbf{X}$ by a sequence
of mutations should again generate a quantum torus, i.e. consist of
$q$-commuting variables. This forces a compatibility condition between
$\widetilde{B}$ and $\Lambda$. That is the matrix $\Lambda$ has to have the
following form for some skew-symmetric $n\times n$ matrix $\Lambda_{0}$:

$\Lambda=\left[\begin{array}[]{cc}\Lambda_{0}&-\Lambda_{0}B-D\\\
-B^{T}\Lambda_{0}+D&B^{T}\Lambda_{0}B+B^{T}D\end{array}\right]$

and thus $\widetilde{B}^{T}\Lambda=\big{[}\,D\ \boldsymbol{0}\,\big{]}$. An
easy calculation shows that this condition naturally reproduces under
mutations. Note that this compatibility condition is identical to the one
required to define a compatible Poisson structure.

Let $\mathcal{F}_{\Lambda}$ denote the skew-field of fractions of
$\mathcal{T}_{\Lambda}$. The _quantum cluster algebra_
$\mathcal{A}_{q}(\mathbf{X},\widetilde{B})$ is the
$\mathbb{Z}[q^{\pm{\frac{1}{2}}}]$-subalgebra of $\mathcal{F}_{\Lambda}$
generated by all quantum cluster variables obtainable from the initial cluster
$\mathbf{X}$ by a sequence of mutations. By work of Berenstein and Zelevinsky
[BZ05], the famous Laurent phenomenon holds in this adapted setting:
$\mathcal{A}_{q}(\mathbf{X},\widetilde{B})$ is a subalgebra of
$\mathcal{T}_{\Lambda}$. Similar to the case of classical cluster algebras,
there are formulas to describe these quantum Laurent expansions in terms of
the representation theory of valued quivers that we recall below.

To the matrix $B$ is associated a quiver $Q$ with vertices $\\{1,\ldots,n\\}$
and $\gcd(b_{ij},b_{ji})$ arrows $i\longrightarrow j$ whenever $b_{ij}<0$; by
our assumption this quiver is acyclic. Write $\widetilde{Q}$ for the quiver
obtained from $Q$ by attaching to each vertex $i$ an additional vertex $n+i$
via a single arrow $i\to n+i$. Let $\widetilde{D}=\mathrm{diag}(D,D)$. We will
be interested in the representation theories of the valued quivers $(Q,D)$ and
$(\widetilde{Q},\widetilde{D})$.

Let $\mathbb{F}$ be a finite field and fix an algebraic closure
$\overline{\mathbb{F}}$ of $\mathbb{F}$. Write $\mathbb{F}^{\langle d\rangle}$
for the degree $d$ extension of $\mathbb{F}$ inside $\overline{\mathbb{F}}$.
Note that this provides a canonical identification of $\mathbb{F}^{\langle
d\rangle}$ as a vector space over $\mathbb{F}^{\langle g\rangle}$ whenever
$g|d$.

A representation $V=(V_{i},V_{a})$ of the valued quiver $(Q,D)$ consists of an
$\mathbb{F}^{\langle d_{i}\rangle}$-vector space $V_{i}$ for each vertex $i$
and an $\mathbb{F}^{\langle\gcd(d_{i},d_{j})\rangle}$-linear map
$V_{a}:V_{i}\to V_{j}$ for each arrow $a:i\to j$. The finite-dimensional
representations of $(Q,D)$ form a hereditary abelian category denoted by
$\operatorname{rep}_{\mathbb{F}}(Q,D)$. The category
$\operatorname{rep}_{\mathbb{F}}(Q,D)$ naturally embeds into the category
$\operatorname{rep}_{\mathbb{F}}(\widetilde{Q},\widetilde{D})$ and we will
always identify $\operatorname{rep}_{\mathbb{F}}(Q,D)$ as this full
subcategory.

Write $\widetilde{\mathcal{Q}}$ for the Grothendieck group of
$\operatorname{rep}_{\mathbb{F}}(\widetilde{Q},\widetilde{D})$ and let
$\mathcal{Q}\subset\widetilde{\mathcal{Q}}$ denote the Grothendieck group of
$\operatorname{rep}_{\mathbb{F}}(Q,D)$. Since $\widetilde{Q}$ is acyclic,
there is a natural identification
$\widetilde{\mathcal{Q}}\cong\mathbb{Z}^{2n}$ by taking classes
$\alpha_{i}:=[S_{i}]$ of the vertex-simple representations as a basis. The
Euler-Ringel form

$\langle
V,W\rangle:=\dim_{\mathbb{F}}\operatorname{Hom}(V,W)-\dim_{\mathbb{F}}\operatorname{Ext}(V,W)$

on pairs of representations $V$ and $W$ only depends on their classes in
$\widetilde{\mathcal{Q}}$. More precisely, it may be computed in the basis of
vertex-simples by

$\langle\alpha_{i},\alpha_{j}\rangle=\begin{cases}d_{i}&\text{if $i=j$;}\\\
d_{i}b_{ij}&\text{if $b_{ij}<0$;}\\\ -d_{i}&\text{if $j=n+i$;}\\\
0&\text{otherwise.}\end{cases}$

For $\mathbf{e}\in\mathcal{Q}$, define
${}^{*}\mathbf{e}=\sum\limits_{i=1}^{n}\frac{1}{d_{i}}\langle\alpha_{i},\mathbf{e}\rangle\alpha_{i}\in\mathcal{Q}$
and
$\mathbf{e}^{*}=\sum\limits_{j=1}^{2n}\frac{1}{d_{j}}\langle\mathbf{e},\alpha_{j}\rangle\alpha_{j}\in\widetilde{\mathcal{Q}}$,
noting that these do not depend on the choice of symmetrizing matrix $D$. Then
the _quantum cluster character_ of a representation
$V\in\operatorname{rep}_{\mathbb{F}}(Q,D)$ with $[V]=:\mathbf{v}$ is the
element of $\mathcal{T}_{\Lambda}$ given by

$X_{V}:=\sum\limits_{\mathbf{e}\in\mathcal{Q}}|\mathbb{F}|^{-{\frac{1}{2}}\langle\mathbf{e},\mathbf{v}-\mathbf{e}\rangle}\big{|}\\!\operatorname{Gr}_{\mathbf{e}}(V)\big{|}X^{\widetilde{B}\mathbf{e}-{}^{*}\mathbf{v}},$

where $\operatorname{Gr}_{\mathbf{e}}(V)$ denotes the set of all
subrepresentations $E\subset V$ with isomorphism class $[E]=\mathbf{e}$. Note
that $\widetilde{B}\mathbf{e}={}^{*}\mathbf{e}-\mathbf{e}^{*}$ and, for
$j\in[1,n]$, we have
${}^{*}\alpha_{j}=\alpha_{j}+\sum\limits_{i=1}^{n}\min(b_{ij},0)\alpha_{i}$.

A representation $V\in\operatorname{rep}_{\mathbb{F}}(Q,D)$ is _rigid_ if
$\operatorname{Ext}^{1}(V,V)=0$. If there exists a rigid representation with
dimension vector $\mathbf{v}\in\mathcal{Q}$ for the finite field $\mathbb{F}$,
then there exists a rigid representation of $(Q,D)$ with dimension vector
$\mathbf{v}$ over any finite field and thus we refer to the dimension vector
$\mathbf{v}$ as being _rigid_ in this case.

######  Theorem 2.

[Rup11, Rup15] Let $B$ be any acyclic skew-symmetrizable matrix with
symmetrizer $D$ and let $(Q,D)$ be the associated valued quiver.

  1. (a)

For each rigid representation $V\in\operatorname{rep}_{\mathbb{F}}(Q,D)$, the
quantum cluster character $X_{V}$ is a quantum cluster monomial of
$\mathcal{A}_{|\mathbb{F}|}(\mathbf{X},\widetilde{B})$. Moreover, every
quantum cluster monomial of
$\mathcal{A}_{|\mathbb{F}|}(\mathbf{X},\widetilde{B})$ not involving initial
cluster variables arises in this way, the decomposition of $V$ into
indecomposables mirroring the factorization of the quantum cluster monomial
into quantum cluster variables.

  2. (b)

For all dimension vectors $\mathbf{v},\mathbf{e}\in\mathcal{Q}$ with
$\mathbf{v}$ rigid, there exists a polynomial $P_{\mathbf{v},\mathbf{e}}(q)$
so that for any finite field $\mathbb{F}$ and rigid representation
$V\in\operatorname{rep}_{\mathbb{F}}(Q,D)$ of dimension vector $\mathbf{v}$,
we have
$\big{|}Gr_{\mathbf{e}}(V)\big{|}=P_{\mathbf{v},\mathbf{e}}\big{(}|\mathbb{F}|\big{)}$.
These polynomials give a “generic” quantum cluster character

$X_{\mathbf{v}}=\sum\limits_{\mathbf{e}\in\mathcal{Q}}q^{-{\frac{1}{2}}\langle\mathbf{e},\mathbf{v}-\mathbf{e}\rangle}P_{\mathbf{v},\mathbf{e}}(q)X^{\widetilde{B}\mathbf{e}-{}^{*}\mathbf{v}}$

which computes the quantum cluster monomials not involving initial quantum
cluster variables of $\mathcal{A}_{q}(\mathbf{X},\widetilde{B})$ with
arbitrary parameter $q$.

An analogous result was obtained for acyclic skew-symmetric quantum cluster
algebras by Qin [Qin12] using representations of acyclic quivers and with
counting polynomials replaced by Poincaré polynomials. Setting $q=1$ in the
formula for generic quantum cluster characters from Theorem 2 gives the
following corollary from which we will deduce the main results of this note.

######  Corollary 3.

All cluster monomials of the cluster algebra
$\mathcal{A}(\mathbf{x},\widetilde{B})$ not involving initial cluster
variables are computed by the cluster characters

$x_{\mathbf{v}}=x^{-{}^{*}\mathbf{v}}\sum\limits_{\mathbf{e}\in\mathcal{Q}}P_{\mathbf{v},\mathbf{e}}(1)x^{\widetilde{B}\mathbf{e}}$

as $\mathbf{v}$ ranges over all rigid dimension vectors in $\mathcal{Q}$. In
particular, the cluster variable $x_{\mathbf{v}}$ has $\mathbf{g}$-vector
given by $-{}^{*}\mathbf{v}$ and its $F$-polynomial is
$F_{\mathbf{v}}(y)=\sum\limits_{\mathbf{e}\in\mathcal{Q}}P_{\mathbf{v},\mathbf{e}}(1)y^{\mathbf{e}}$.

The connection between the representation theory of $(Q,D)$ and the cluster
algebra is actually much stronger. For a source (resp. sink) vertex $k$ in
$Q$, write
$\Sigma_{k}^{-}:\operatorname{rep}_{\mathbb{F}}(Q,D)\to\operatorname{rep}_{\mathbb{F}}(\mu_{k}Q,D)$
(resp.
$\Sigma_{k}^{+}:\operatorname{rep}_{\mathbb{F}}(Q,D)\to\operatorname{rep}_{\mathbb{F}}(\mu_{k}Q,D)$)
for the reflection functor as defined in [DR76, Sec. 2]. In what follows, we
will drop the adornment and simply write $\Sigma_{k}$ for both reflection
functors, which one to apply should be clear from context. Write
$\operatorname{rep}_{\mathbb{F}}^{\langle
k\rangle}(Q,D)\subset\operatorname{rep}_{\mathbb{F}}(Q,D)$ for the full
subcategory consisting of representations which contain no summands isomorphic
to the simple $S_{k}$.

######  Theorem 4.

[Rup11] Let $k$ be a sink or a source in $Q$ and let $\mathbf{X}^{\prime}$ be
the cluster obtained by mutating the initial cluster $\mathbf{X}$ in direction
$k$. For any representation $V\in\operatorname{rep}_{\mathbb{F}}^{\langle
k\rangle}(Q,D)$, we have $X_{V}=X^{\prime}_{\Sigma_{k}V}$, where
$X^{\prime}_{\Sigma_{k}V}$ denotes the quantum cluster character of
$\Sigma_{k}V\in\operatorname{rep}_{\mathbb{F}}(\mu_{k}Q,D)$ in the variables
$\mathbf{X}^{\prime}$.

##  3\. Deducing the Conjectures

In this section, we apply the results of Section 2 to deduce the conjectures
in Theorem 1.

######  Proposition 5.

For any rigid dimension vector $\mathbf{v}\in\mathcal{Q}$, the denominator
vector of $x_{\mathbf{v}}$ is precisely $\mathbf{v}$.

###### Proof.

The proof is identical to that of [CK08, Sec. 4, Cor. 2] with appropriate
modifications in the valued quiver setting, we recall the details here for
convenience of the reader.

First note that for any representation $W$ with $[W]=\mathbf{w}$, we have
$\frac{1}{d_{i}}\langle W,I_{i}\rangle=w_{i}=\frac{1}{d_{i}}\langle
P_{i},W\rangle$, where $I_{i}$ and $P_{i}$ denote respectively the injective
hull and projective cover of the vertex simple $S_{i}$. Next using the
injective resolution of $S_{i}$ we see that $\langle W,S_{i}\rangle\leq\langle
W,I_{i}\rangle$ while using the projective resolution gives $\langle
S_{i},W\rangle\leq\langle P_{i},W\rangle$. Now consider a subrepresentation
$E\subset V$ with $[E]=\mathbf{e}$, applying the above considerations we see
that the $i$-th component of $\mathbf{e}^{*}+{}^{*}(\mathbf{v}-\mathbf{e})$ is
bounded by the $i$-th component of $\mathbf{v}$:

$\frac{1}{d_{i}}\langle E,S_{i}\rangle+\frac{1}{d_{i}}\langle
S_{i},V/E\rangle\leq\frac{1}{d_{i}}\langle
E,I_{i}\rangle+\frac{1}{d_{i}}\langle
P_{i},V/E\rangle=e_{i}+(v_{i}-e_{i})=v_{i}.$

To finish the proof, for each $i\in[1,n]$ we must exhibit a subrepresentation
$E\subset V$ which realizes this bound. To construct such a subrepresentation,
let $J_{i}$ be the set of all vertices $j$ in $Q$ for which there exists a
path (possibly trivial) beginning at $i$ and ending at $j$. Now set
$E_{j}=V_{j}$ for $j\in J_{i}$ and $E_{j}=0$ for $j\notin J_{i}$. Recall that
in the injective coresolution $0\longrightarrow S_{i}\longrightarrow
I_{i}\longrightarrow I\longrightarrow 0$ the injective representation $I$ has
nonzero components only at vertices $j$ which admit a nontrivial path _to_
vertex $i$, while in the projective resolution $0\longrightarrow
P\longrightarrow P_{i}\longrightarrow S_{i}\longrightarrow 0$ the projective
representation $P$ has nonzero components only at vertices $j$ which admit a
nontrivial path _from_ vertex $i$. It follows that $\langle
E,S_{i}\rangle=\langle E,I_{i}\rangle-\langle E,I\rangle=\langle
E,I_{i}\rangle$ and $\langle S_{i},V/E\rangle=\langle P_{i},V/E\rangle-\langle
P,V/E\rangle=\langle P_{i},V/E\rangle$ and thus

$\frac{1}{d_{i}}\langle E,S_{i}\rangle+\frac{1}{d_{i}}\langle
S_{i},V/E\rangle=\frac{1}{d_{i}}\langle E,I_{i}\rangle+\frac{1}{d_{i}}\langle
P_{i},V/E\rangle=e_{i}+(v_{i}-e_{i})=v_{i}$

as desired. ∎

######  Corollary 6 ([FZ07, Conjecture 7.17] for acyclic initial exchange
matrices).

For a rigid dimension vector $\mathbf{v}\in\mathcal{Q}$, the denominator
vector of $x_{\mathbf{v}}$ is the exponent vector of the monomial obtained by
tropically evaluating the corresponding $F$-polynomial at
$y_{1}^{-1},\dots,y_{n}^{-1}$.

###### Proof.

It is enough to observe that the $F$-polynomial has a unique monomial of
maximal degree and this monomial is divisible by all other monomials. Indeed,
this is the monomial corresponding to the full subrepresentation and hence it
has exponent vector $\mathbf{v}$ in $F_{\mathbf{v}}$. ∎

This immediately implies [FZ07, Conjecture 6.11] using [FZ07, Proposition
7.16]. From Proposition 5, we deduce this weakening of [FZ07, Conjecture 7.4]
in the acyclic case.

######  Corollary 7.

Let $\mathbf{v}$ be the denominator vector of a non-initial cluster variable
$x_{\mathbf{v}}$ in $\mathcal{A}(\mathbf{x},\widetilde{B})$. Then:

  1. (1)

all entries in $\mathbf{v}$ are non-negative;

  2. (2)

$v_{i}=0$ if and only if there is a seed containing both $x_{\mathbf{v}}$ and
the initial cluster variable $x_{i}$;

  3. (3)

$v_{i}$ depends only on $x_{\mathbf{v}}$ and $x_{i}$ but not on the seed
containing $x_{\mathbf{v}}$.

###### Proof.

By Proposition 5, the denominator vector of any non-initial cluster variable
is a dimension vector and therefore it has non-negative entries. Moreover, its
coordinates are independent of the seed containing the cluster variable. This
proves points (1) and (3).

By the categorification construction [Rup15], seeds in
$\mathcal{A}(\mathbf{x},\widetilde{B})$ are in bijection with local tilting
representations $T$. The seed corresponding to $T$ contains the $i$-th initial
cluster variable precisely when vertex $i$ is not in the support of $T$. Any
rigid indecomposable representation can be completed to a local tilting
representation with the same support and this establish part (2). ∎

The key difference in the above corollary compared to the original statement
of the conjecture is that in point (3) we do not claim independence from the
seed containing $x_{i}$. Nonetheless the corollary immediately implies the
weaker statements [FZ07, Conjecture 7.5] and [RSt18, Conjecture 2.9] for
acyclic initial exchange matrices.

######  Corollary 8 ([FZ07, Conjecture 7.6] for acyclic initial exchange
matrices).

Different cluster monomials in $\mathcal{A}(\mathbf{x},\widetilde{B})$ have
different denominator vectors. Moreover, the denominator vectors of the
cluster variables of any given cluster form a $\mathbb{Z}$-basis of
$\mathcal{Q}$.

###### Proof.

By Corollary 7, if a cluster monomial contains initial cluster variables this
can be detected in the denominator vector. Therefore we can assume that we are
dealing with cluster monomials not containing initial cluster variables. By
Theorem 2, these are parametrized by rigid representations. If the two cluster
monomials have equal denominator vectors, then the dimension vectors of the
two representations they correspond to are equal. By rigidity, the orbits of
each of these representations inside the moduli space of representations with
this dimension vector are both dense (see e.g. [Bri08, Corollary 2.2.5]). But
then the representations must be isomorphic and thus they correspond to the
same cluster monomial.

Each seed of $\mathcal{A}(\mathbf{x},\widetilde{B})$ corresponds to a local
tilting representation $T$. By [HR82, Lemma 4.3 and Theorem 4.5], the
dimension vectors of the summands of $T$ provide a $\mathbb{Z}$-basis for the
sublattice on which they are supported. Negative simples corresponding to
vertices outside the support complete this basis to a $\mathbb{Z}$-basis of
$\mathcal{Q}$. ∎

Let $E=(e_{ij})$ be the matrix given by

$e_{ij}=\begin{cases}1&\text{if $i=j$;}\\\ \min(b_{ij},0)&\text{if $i\neq
j$.}\end{cases}$

Combining Corollary 3 and Proposition 5 we immediately get the following.

######  Proposition 9 ([RSp16, Conjecture 3.21]).

Let $\mathbf{v}\in\mathcal{Q}$ be a rigid dimension vector. Then the
$\mathbf{g}$-vector of $x_{\mathbf{v}}$ is $-E\mathbf{v}$.

###### Proof.

Since the $\mathbf{g}$-vector of the cluster monomial $x_{\mathbf{v}}$ is
given by $-{}^{*}\mathbf{v}$, the result follows immediately from the
definitions of $E$ and the operator ${}^{*}(-)$. ∎

Replacing $E$ with its piecewise linear counterpart $\nu_{c}$ defined in
[RSt17], the last result extends to cover all cluster variables in
$\mathcal{A}(\mathbf{x},\widetilde{B})$.

######  Corollary 10 ([RSt18, Conjecture 2.8] for acyclic initial exchange
matrices).

The mutation of the initial cluster of $\mathcal{A}(\mathbf{x},\widetilde{B})$
at a sink or a source vertex transforms denominator vectors of non-initial
cluster variables according to the simple reflection associated to that
vertex.

###### Proof.

By Theorem 4, the initial cluster mutation at a sink or a source vertex
transforms cluster characters according to the associated reflection functor
on $\operatorname{rep}_{\mathbb{F}}(Q,D)$. The result then follows from [DR76,
Proposition 2.1] and Proposition 5. ∎

From this corollary, following [RSt18, Proposition 2.10], we also get [RSt18,
Conjecture 2.7] for acyclic initial exchange matrices. The following technical
result is needed in [RSW17].

######  Proposition 11.

Assume vertex $k$ is a source for $B$. Let $x_{\mathbf{v}}$ be any non-initial
cluster variable in $\mathcal{A}(\mathbf{x},\widetilde{B})$ and write
$F^{\prime}_{\mathbf{v}}$ for the tropical evaluation of its $F$-polynomial at
$y_{j}^{\prime}=y_{j}y_{k}^{-b_{kj}-2\delta_{jk}}$. Then
$F^{\prime}_{\mathbf{v}}=1$ unless $\mathbf{v}=[S_{k}]$ in which case
$F^{\prime}_{\mathbf{v}}=y_{k}^{-1}$.

###### Proof.

Let $V$ be a rigid indecomposable representation with $[V]=\mathbf{v}$. The
terms of the $F$-polynomial $F_{\mathbf{v}}(y)$ are labeled by
subrepresentations of $[V]$ and the claim is that each of these monomials
produces only non-negative exponents when evaluated at the given
$y_{j}^{\prime}$.

Since vertex $k$ is a source in $B$, we have $b_{kj}\leq 0$ for $j\in[1,n]$
and therefore only the exponent of $y_{k}$ could end up being negative in the
tropical evaluation. However, observe that the total exponent of any monomial
in $F_{\mathbf{v}}(y^{\prime})$ is given by applying the simple reflection
$s_{k}$ to the dimension vector of the subrepresentation it corresponds to. In
particular, this implies that the only possibility to have a negative exponent
is that $S_{k}$ is a subrepresentation of $V$.

On the other hand, vertex $k$ is a source so that the only rigid
indecomposable representation admitting $S_{k}$ as a subrepresentation is
$S_{k}$ itself and the result follows. ∎

We turn now our attention towards results dealing with the exchange graph of
$\mathcal{A}(\mathbf{x},\widetilde{B})$.

######  Proposition 12 ([RSp16, Conjecture 3.10] in the acyclic case).

Given a cluster algebra having an acyclic seed, the induced subgraph of its
exchange graph consisting of all the seeds containing any fixed collection of
cluster variables is connected.

###### Proof.

Since the exchange graph of a cluster algebra with principal coefficients
covers the exchange graph of any cluster algebra with the same mutation
pattern (cf. [FZ07]), it suffices to establish the claim for
$\mathcal{A}(\mathbf{x},\widetilde{B})$. The proof given in [CK06, Corollary
3] for the weaker statement [FZ03, Conjecture 4.14(3)] is written in terms of
arbitrary hereditary abelian categories. In particular, their proof applies in
our situation. More generally, one may replace the rigid indecomposable
representation appearing in [CK06, Section 5.4] by an arbitrary partial
tilting representation (cf. [HRS88, Proposition 3]). Then running a similar
argument as in the proof of [CK06, Theorem 6] proves the general case. These
constructions are closely related to the Iyama-Yoshino reduction of
triangulated categories (cf. [IY08, Section 4] or [Kel10, Section 7.2]). ∎

######  Proposition 13 ([FZ03, Conjecture 4.14(4)]).

The full subgraph of the exchange graph consisting of all seeds having acyclic
exchange matrix is connected.

###### Proof.

The proof of the analogous statement for skew-symmetric cluster algebras given
in [CK06, Corollary 4] is also written in terms of arbitrary hereditary
abelian categories and thus it applies in our situation as well. ∎

## References

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