Reddening Sequences for Banff Quivers and the Class $\mathcal{P}$

Reddening Sequences for Banff Quivers
and the Class $\boldsymbol{\mathcal{P}}$††This paper is a contribution to the
Special Issue on Cluster Algebras. The full collection is available at
https://www.emis.de/journals/SIGMA/cluster-algebras.html

Eric BUCHER † and John MACHACEK ‡

E. Bucher and J. Machacek

† Department of Mathematics, Xavier University, Cincinnati, Ohio 45207, USA
<EMAIL_ADDRESS>
‡ Department of Mathematics and Statistics, York University,
‡ Toronto, Ontario M3J 1P3, Canada<EMAIL_ADDRESS>
Received June 15, 2019, in final form May 23, 2020; Published online June 08,
2020

We show that a reddening sequence exists for any quiver which is Banff. Our
proof is combinatorial and relies on the triangular extension construction for
quivers. The other facts needed are that the existence of a reddening sequence
is mutation invariant and passes to induced subquivers. Banff quivers define
locally acyclic cluster algebras which are known to coincide with their upper
cluster algebras. The existence of reddening sequences for these quivers is
consistent with a conjectural relationship between the existence of a
reddening sequence and a cluster algebra’s equality with its upper cluster
algebra. Our result completes a verification of the conjecture for Banff
quivers. We also prove that a certain subclass of quivers within the class
$\mathcal{P}$ define locally acyclic cluster algebras.

cluster algebras; quiver mutation; reddening sequences

13F60; 16G20

##  1 Introduction

Cluster algebras are commutative algebras where generators can be explicitly
described through a process known as quiver mutation. It is natural to ask
what influence the combinatorics of the quiver has on the algebra. It has been
observed that the existence of a maximal green or reddening sequence for a
quiver seems to correspond to when the cluster algebra defined by the quiver
equals its upper cluster algebra. We provide further evidence for this
relationship and aim to clarify why the two notions seem to coincide. Our
methods look at combinatorial properties of quiver and use them to either
produce reddening sequences or show the cluster algebra equals its upper
cluster algebra. Moreover, we find that the same combinatorial construction,
which is known as a triangular extension, is essential to our results on both
reddening sequences and upper cluster algebras.

For a semifield $\mathbb{P}$ with group algebra $\mathbb{ZP}$ we choose some
ground ring $\mathbb{A}$ inside the field of fractions of $\mathbb{ZP}$. We
will restrict our attention to skew-symmetric cluster algebras. In [10] Fomin
and Zelevinsky define a cluster algebra $\mathcal{A}$ as a certain
$\mathbb{A}$-algebra inside an ambient field with generators determined by a
quiver $Q$. In [2] Bernstein, Fomin, and Zelevinsky define the upper cluster
algebra $\mathcal{U}$ which satisfies $\mathcal{A}\subseteq\mathcal{U}$. A
fundamental problem in cluster algebra theory is to determine when there is
the equality $\mathcal{A}=\mathcal{U}$. In general this can be a difficult
problem. Typically the ground ring of choice is
$\mathbb{A}=\mathbb{Z}\mathbb{P}$. However, there has been recent attention
paid to the choice of ground ring in the $\mathcal{A}=\mathcal{U}$ question.
There can be a delicate dependence on the ground ring $\mathbb{A}$ as
demonstrated by the authors with M. Shapiro [5]. Goodearl and Yakimov have
developed techniques for showing $\mathcal{A}=\mathcal{U}$ for ground rings
$\mathbb{A}\subseteq\mathbb{Z}\mathbb{P}$ [14].

In the case that ground ring is $\mathbb{A}=\mathbb{ZP}$, Muller’s theory of
cluster localization and locally acyclic cluster algebras provides a means of
showing $\mathcal{A}=\mathcal{U}$ [24, 25]. The Banff algorithm presented by
G. Muller, [24], is one way of showing that a cluster algebra is locally
acyclic, and quivers for which the Banff algorithm produces a positive output
are called Banff. In particular if a quiver is Banff, then the cluster algebra
it defines over $\mathbb{ZP}$ satisfies $\mathcal{A}=\mathcal{U}$.

Keller [16, 17] introduced certain sequences of quiver mutations, which are
now known as maximal green sequences and reddening sequences, as a
combinatorial way to study Kontsevich and Soibelman’s Donaldson-Thomas
transformations [18]. The existence of both maximal green sequences and
reddening sequences are important in cluster algebra theory and has been
thought to be related to the equality of the cluster algebra and upper cluster
algebra.111In [23] Mills states “The conjecture that the cluster algebra and
upper cluster algebra coincide if and only if a maximal green sequence exists
arose from a discussion led by Arkady Berenstein and Christof Geiss at the
_Hall and cluster algebras_ conference at Centre de Recherches Mathématiques,
Montréal in 2014”. Canakci, Lee, and Schiffler [7] had observed existence of a
maximal green sequence coincided with $\mathcal{A}=\mathcal{U}$ in known cases
at the time. Mills [23] offers an explicit conjecture on the potential
relationship of maximal green sequences, the equality
$\mathcal{A}=\mathcal{U}$, locally acyclicity, and choice of ground ring. Our
work here is progress in understanding this relationship.

In Theorem 3.2 we show that Banff quivers admit reddening sequences. Theorem
3.3 gives reddening sequences for any quiver in the class $\mathcal{P}$. These
theorems cover many examples which have been the subject of previous research.
Ford and Serhiyenko [11] have shown that the quivers arising from Postnikov’s
reduced plabic graphs [28] admit reddening sequences. Theorem 3.2, combined
with the work of Muller and Speyer’s [27] which says such quivers are Banff,
gives another proof of this fact. Many quivers from marked surfaces are known
to be Banff [24, Theorem 10.6] [7, Proposition 12]. Thus, Theorem 3.2 gives
reddening sequences for these quivers. For surfaces covered by these cases
maximal green sequences have been previously constructed [1, 4, 6, 13].
Quivers associated to surfaces belong to an important class of quivers known
as mutation finite quivers. For mutation finite quivers there is a complete
classification of the existence of reddening and maximal green sequences [22].
Also, minimal mutation infinite quivers are further examples of Banff quivers
for which Lawson and Mills have shown have maximal green sequences [21].

Morally, the proposed conjecture (with some dependence on ground ring) is that
for a given quiver the following are equivalent:

  1. (i)

The quiver admits a reddening sequence.

  2. (ii)

The associated cluster algebra equals its upper cluster algebra.

  3. (iii)

The associated cluster algebra is locally acyclic.

Mills offers a more precise conjecture [23, Conjecture 2] and verifies the
conjecture for mutation finite quivers [23, Theorem 1.2]. Theorem 3.2
completes a proof that all three conditions are equivalent for Banff quivers.
Theorem 3.3 suggests the class $\mathcal{P}$ as a next step in verifying the
conjecture. To the knowledge of the authors every known Banff quiver is also
in the class $\mathcal{P}$ (in fact inside a more restrictive class we denote
$\mathcal{P}^{\prime}$). We ask Questions 3.5 and 3.6 in attempt to better
understand the relationship between Banff quivers and the class $\mathcal{P}$.
In Theorem 4.6 we make some progress toward the equivalence of (i), (ii), and
(iii) for the class $\mathcal{P}$ by showing the conditions are equivalent for
a certain subfamily of quivers.

##  2 Quiver mutation background

In this section we will briefly establish some of the basic definitions that
we will use to prove our main results on reddening sequences. A _quiver_ ,
$Q$, is a directed graph whose edge set contains no loops or 2-cycles. The
_framed quiver_ associated to $Q$, denoted $\widehat{Q}$, is the quiver whose
vertex set and edge set are the following:

$\displaystyle V\big{(}\widehat{Q}\big{)}:=V(Q)\sqcup\\{i^{\prime}\,|\,i\in
V(Q)\\},$ $\displaystyle E\big{(}\widehat{Q}\big{)}:=E(Q)\sqcup\\{i\rightarrow
i^{\prime}\,|\,i\in V(Q)\\}.$

The _coframed quiver_ associated to $Q$, denoted $\widecheck{Q}$, is the
quiver whose vertex set and edge set are the following:

$\displaystyle V\big{(}\widecheck{Q}\big{)}:=V(Q)\sqcup\\{i^{\prime}\,|\,i\in
V(Q)\\},$ $\displaystyle
E\big{(}\widecheck{Q}\big{)}:=E(Q)\sqcup\\{i^{\prime}\rightarrow i\,|\,i\in
V(Q)\\}.$

The vertices $i\in V(Q)$ are the _mutable vertices_ and the vertices
$i^{\prime}$ are the _frozen vertices_. Mutation is not allowed at any frozen
vertex. The framed quiver corresponds to considering a cluster algebra with
principle coefficients. For any mutable vertex $i$, _mutation_ at the vertex
$i$ produces a new quiver denoted $\mu_{i}(Q)$ obtained from $Q$ by doing the
following:

  1. (1)

For each pair of arrows $k\to j$, $i\to k$ add an arrow $j\to k$.

  2. (2)

Reverse all arrows incident on $i$.

  3. (3)

Delete a maximal collection of disjoint $2$-cycles and any arrows between two
frozen vertices.

Two quivers are said to be _mutation equivalent_ if one can be reached from
the other by a sequence of mutations.

Given any quiver $Q$ and $A\subseteq V(Q)$ we let $Q|_{A}$ denote the _induced
subquiver_ which has

$\displaystyle V(Q|_{A})=A,$ $\displaystyle
E(Q|_{A})=\big{\\{}i\overset{\alpha}{\to}j\in E(Q)\colon i,j\in A\big{\\}}$

and is a natural restriction of $Q$ to $A$. We will use $Q\setminus A$ to
denote $Q|_{V(Q)\setminus A}$.

A mutable vertex is _green_ if there are no incident incoming arrows from
frozen vertices. Similarly, a mutable vertex is _red_ if there are no incident
outgoing arrows to frozen vertices. If we start with an initial quiver $Q$ and
preform mutations at mutable vertices of the framed quiver $\widehat{Q}$, then
any mutable vertex will always be either green or red. The result is known as
_sign-coherence_ and was established by Derksen, Weyman, and Zelevinsky [9].
Notice also that all vertices are initially green when starting with
$\widehat{Q}$. Keller [16, 17] has introduced the following types of sequences
of mutations which will our main interest. A sequence of mutations is called a
_reddening sequence_ if after preforming this sequences of mutations all
mutable vertices are red. A _maximal green sequence_ is a reddening sequence
where each mutation occurs at a green vertex. By [3, Proposition 2.10], after
preforming a reddening sequence starting with $\widehat{Q}$ the resulting
quiver will be isomorphic to $\widecheck{Q}$. By definition all maximal green
sequences are reddening sequences. There are quivers for which a maximal green
sequence does not exist, but a reddening sequence does [26]. Furthermore there
are quivers for which no reddening sequence exists [29], green or otherwise.
If a reddening (maximal green) sequence exists for a quiver, we will say this
quiver admits a reddening (maximal green) sequence.

Now we recall two results of Muller which will be needed for the proofs of our
main results. Both these results were first shown in [26] and their proofs
make use scattering diagrams which have been connected with cluster algebras
by Gross, Hacking, Keel, and Kontsevich [15]. A version of Lemma 2.1 also
holds for maximal green sequences, but we will not need it. However, Lemma 2.2
is false for maximal green sequences. The mutation invariance of reddening is
essential to our results.

######  Lemma 2.1 ([26, Theorem 17]).

If a quiver $Q$ admits a reddening sequence, then any induced subquiver of $Q$
also admits a reddening sequence.

######  Lemma 2.2 ([26, Corollary 19]).

If a quiver $Q$ admits a reddening sequence, then any quiver mutation
equivalent to $Q$ also admits a reddening sequence.

$1$$2$$3$$4$$5$$6$ Figure 1: A triangular extension of quivers.

We now consider a result which states the triangular extension construction
preserves the existence of reddening sequences and maximal green sequences.
Let $Q_{1}$ and $Q_{2}$ be quivers. A  _triangular extension_ of $Q_{1}$ and
$Q_{2}$ is any quiver $Q$ with

$\displaystyle V(Q)=V(Q_{1})\sqcup V(Q_{2}),$ $\displaystyle
E(Q)=E(Q_{1})\sqcup E(Q_{2})\sqcup E,$

where $E$ is any set of arrows such that either we have either

for any $i\to j\in E$ implies $i\in V(Q_{1})$ and $j\in V(Q_{2})$

or else

for any $i\to j\in E$ implies $i\in V(Q_{2})$ and $j\in V(Q_{1})$.

That is, a triangular extension of quivers simply takes the disjoint union of
the two quivers then adds additional arrows between the quivers with the
condition that all arrows are directed from one quiver to the other. An
example of a triangular extension of quivers $Q_{1}$ and $Q_{2}$ where
$V(Q_{1})=\\{1,2,3\\}$ and $V(Q_{2})=\\{4,5,6\\}$ is given in Fig. 1.

We shortly will state a result that the triangular extension construction
preserves the existence of reddening sequences and maximal green sequences.
Garver and Musiker show an analogous result for maximal green sequences in a
restricted case needed for their work which they call a $t$-colored direct sum
[12, Theorem 3.12] and suggest the result holds in greater generality [12,
Remark 3.13]. Cao and Li show the result (stated for maximal green sequences,
but remark on reddening sequences [8, Remark 4.6]) for any triangular
extension in the generality of skew-symmetrizable matrices [8, Theorem 4.5].
In sections following this one we will emphasize the results for reddening
sequences. The fact that the existence of reddening sequences is mutation
invariant will allow applications of the following lemma to Banff quivers and
quivers in class $\mathcal{P}$ since both these classes are defined up to
mutation equivalence.

######  Lemma 2.3 ([8, Theorem 4.5, Remark 4.6]).

If $Q_{1}$ and $Q_{2}$ are any two quivers which both admit reddening
$($maximal green$)$ sequences, then any triangular extension of $Q_{1}$ and
$Q_{2}$ admits a reddening $($maximal green$)$ sequence.

##  3 Existence of reddening sequences

In this section we define Banff quivers and the class $\mathcal{P}$. We show
that Banff quivers and quivers in the class $\mathcal{P}$ admit reddening
sequences.

###  3.1 Banff quivers

A _bi-infinite path_ in a quiver $Q$ is a sequence $(i_{a})_{a\in\mathbb{Z}}$
of mutable vertices such that $i_{a}\to i_{a+1}$ is a arrow for each
$a\in\mathbb{Z}$. A pair of vertices $(i,j)$ is a _covering pair_ if $i\to j$
is an arrow which is not part of any bi-infinite path. Muller’s class of Banff
quivers is the smallest class of quivers such that

  * •

any acyclic quiver is Banff,

  * •

any quiver mutation equivalent to a Banff quiver is Banff,

  * •

and any quiver $Q$ with a covering pair $(i,j)$ where both $Q\setminus\\{i\\}$
and $Q\setminus\\{j\\}$ are Banff is a Banff quiver.

A demonstration that a quiver is Banff is shown in Fig. 2.

######  Remark 3.1.

It is possible to replace “any acyclic quiver is Banff” with “any quiver which
is a collection of isolated vertices is Banff” to give closer analogy to the
class $\mathcal{P}$ define in Section 3.2.

$1$$2$$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}3$$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}4$$1$$2$$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}4$Acyclic$1$$2$$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}3$Mutation
acyclic Figure 2: Showing a quiver is Banff using a covering pair consisting
of vertices $3$ and $4$.

######  Theorem 3.2.

Let $Q$ be a Banff quiver, then $Q$ admits a reddening sequence.

###### Proof.

We will induct on the number of vertices of $Q$. If $Q$ has a single vertex,
the result is immediate. The result is also immediate if $Q$ a collection of
isolated vertices. Now assume $Q$ is a non-isolated quiver with more than one
vertex. Since $Q$ is a Banff quiver there is covering pair in either $Q$ or
some quiver mutation equivalent to $Q$. We may assume that $Q$ has this
covering pair because existence of a reddening sequence is mutation invariant
by Lemma 2.2.

Let $(i,j)$ be a covering pair used in determining that $Q$ is Banff. Let $B$
be the set of vertices $k$ such that there exists a directed path from $j$ to
$k$ in $Q$. Here we count the path consisting of no arrows, and hence $j\in
B$. Let $A$ be the complement of $B$ in $V(Q)$. We have $i\in A$ since $(i,j)$
is a covering pair. If $i\not\in A$, then $i\in B$ and there would be a cycle
containing the arrow $i\to j$ which would contradict the fact that $(i,j)$ is
a covering pair. It follows by the definition of the sets $A$ and $B$ that $Q$
is a triangular extension of $Q|_{A}$ and $Q|_{B}$ since if there was an arrow
$k\to\ell$ with $k\in B$ then $\ell\in B$. That is, there are no arrows
$k\to\ell$ with $k\in B$ and $\ell\in A$.

Now $Q\setminus\\{i\\}$ and $Q\setminus\\{j\\}$ are both Banff quivers with
strictly fewer vertices than $Q$. Hence, $Q\setminus\\{i\\}$ and
$Q\setminus\\{j\\}$ admit reddening sequences by induction. The quiver
$Q|_{A}$ is an induced subquiver of $Q\setminus\\{j\\}$ and $Q|_{B}$ is an
induced subquiver of $Q\setminus\\{i\\}$. Thus, $Q|_{A}$ and $Q|_{B}$ both
admit reddening sequences by Lemma 2.1. We then conclude that $Q$ admits a
reddening sequence by Lemma 2.3. ∎

###  3.2 The class $\boldsymbol{\mathcal{P}}$

The following three properties define _the class $\mathcal{P}$_:

  * •

The quiver with one vertex is in the class $\mathcal{P}$.

  * •

The class $\mathcal{P}$ is closed under quiver mutation.

  * •

The class $\mathcal{P}$ is closed under taking any triangular extension of two
quivers in the class $\mathcal{P}$.

This class of quivers was introduced by Kontsevich and Soibelman [18, Section
8.4]. The nest theorem follows immediately from results in the literature.
This theorem will be used as a starting point for exploration of classes of
quivers defined by similar properties.

######  Theorem 3.3.

Let $Q$ be in the class $\mathcal{P}$, then $Q$ admits a reddening sequence.

###### Proof.

The quiver with one vertex admits a reddening sequence. The theorem then
follows from Lemmas 2.2 and 2.3 along with the definition of the class
$\mathcal{P}$. ∎

We now discuss the relationship between the class $\mathcal{P}$ and the class
of Banff quivers. Ladkani [19, Remark 4.21] also considers the class
$\mathcal{P}^{\prime}$ of quivers defined by:

  * •

The quiver with one vertex is in the class $\mathcal{P}^{\prime}$.

  * •

the class $\mathcal{P}^{\prime}$ is closed under quiver mutation.

  * •

The class $\mathcal{P}^{\prime}$ is closed under taking triangular extensions
of quivers where one is in the class $\mathcal{P}^{\prime}$ and the other is
the quiver with one vertex.

Now let $\mathcal{B}^{\prime}$ denote the class of quivers defined by:

  * •

Any quiver without arrows is in $\mathcal{B}^{\prime}$.

  * •

Any quiver mutation equivalent to a quiver in $\mathcal{B}^{\prime}$ is in
$\mathcal{B}^{\prime}$.

  * •

Any quiver $Q$ with an arrow $i\to j$ such that $i$ is a source or $j$ is a
sink, and both $Q\setminus\\{i\\}$ and $Q\setminus\\{j\\}$ are in
$\mathcal{B}^{\prime}$ is in $\mathcal{B}^{\prime}$.

Notice $\mathcal{P}^{\prime}$ and $\mathcal{B}^{\prime}$ are subclasses of
$\mathcal{P}$ and $\mathcal{B}$ respectively. These classes make use of
sources (or equivalently sinks) which can be helpful. In practice this is how
quivers are often shown to be Banff. For example, quivers from many marked
surfaces [24, Section 10] and quivers for Grassmannians [27] are in
$\mathcal{B}^{\prime}$ since covering pairs are always taken to use a source
(or sink). The next proposition follows immediately from the definitions of
$\mathcal{P}^{\prime}$ and $\mathcal{B}^{\prime}$.

######  Proposition 3.4.

If a quiver is in $\mathcal{B}^{\prime}$, then the quiver is in
$\mathcal{P}^{\prime}$.

######  Question 3.5.

Does there exist a Banff quiver which is not in the class $\mathcal{P}$?

######  Question 3.6.

Does there exist a Banff quiver which is not in $\mathcal{B}^{\prime}$?

A quiver has a covering pair if and only if it has a source or sink [24,
Proposition 8.1]. So, at any step which covering pairs are available there is
the option to choose a covering pair which contains a source or sink. A quiver
giving a positive answer to Question 3.6 must fail to be Banff whenever using
only source and sink covering pairs, but succeed when making use of a covering
pair not containing a source nor a sink.

Lam and Speyer’s class of Louise quivers [20] is another class of quivers for
which similar questions could be asked. It would be interesting to better
understand the relationship of Louise quivers to the classes of quivers
defined in this section. Any Louise quiver is Banff, and to the authors’
knowledge there is no known example of a Banff quiver which is not Louise. We
will not define or work further with Louise quivers here.

The results in this section readily generalize to the following system for
producing quivers with reddening sequences. We can choose a property of a
quiver for which we know any quiver with this property admits a reddening
sequence. We will say a quiver is of type $T$ if it has this chosen property.
We then create a collection $\mathcal{C}$ of quivers by:

  * •

Any quiver of type $T$ is in $\mathcal{C}$.

  * •

Any quiver mutation equivalent to a quiver in $\mathcal{C}$ is in
$\mathcal{C}$.

  * •

The triangular extension of any two quivers in $\mathcal{C}$ is in
$\mathcal{C}$.

If a quiver is of type $T$ only if it has a single vertex, then $\mathcal{C}$
is just the class $\mathcal{P}$. Theorem 3.2 says we can take a quiver to be
of type $T$ if it is Banff. If Question 3.5 has a positive answer then this
will be a class of quivers with reddening sequences which is strictly larger
than the class $\mathcal{P}$. This raises the following fairly ambitious
question:

######  Question 3.7.

What is a minimal $T$ for which $\mathcal{C}$ consists of all quivers which
admit reddening sequences?

The goal would be to classify all quivers which admit a reddening sequence by
looking at finding the essential generating quivers up to triangular extension
and mutation. As one can see from the class $\mathcal{P}$, a small set of
generating quivers can produce a large and interesting class. Also, the
collection of quivers which admit a reddening sequence is strictly larger than
the class $\mathcal{P}$. The quivers associated to a triangulation of a torus
with one boundary component and one marked point on the boundary give examples
of quivers not in the class $\mathcal{P}$ which admit reddening sequences.
Such a quiver is shown in Fig. 3 along with a maximal green (reddening)
sequence for it. Up to isomorphism the quiver shown in the figure is the only
quiver for the torus with one boundary component and one marked point on the
boundary. Hence, it is readily checked that the quiver in Fig. 3 is not Banff
either. Furthermore, quivers from closed surfaces with more than one puncture
admit reddening sequence [6], but they are neither Banff [24, Theorem 10.9]
nor do they belong to class $\mathcal{P}$ [19, Theorem 4.11].

$1$$2$$3$$4$ Figure 3: Quiver for torus with one boundary component and one
marked point. A maximal green sequence for this quiver is $(1,3,4,2,1,3)$.

##  4 Locally acyclic cluster algebras

Let $\mathbb{P}$ be a semifield. Let $\mathcal{F}$ be a field which contains
$\mathbb{Z}\mathbb{P}$. A _seed_ of rank $n$ in $\mathcal{F}$ is a triple
$(\mathbf{x},\mathbf{y},Q)$. The _cluster_
$\mathbf{x}=\\{x_{1},x_{2},\dots,x_{n}\\}$ is an $n$-tuple in $\mathcal{F}$
which freely generates $\mathcal{F}$ as a field over the fraction field of
$\mathbb{Z}\mathbb{P}$. The _coefficients_
$\mathbf{y}=\\{y_{1},y_{2},\dots,y_{n}\\}$ consist of elements of
$\mathbb{P}$. Here $Q$ a is quiver on vertex set $V(Q)=\\{1,2,\dots,n\\}$.

A seed $(\mathbf{x},\mathbf{y},Q)$ may be mutated at any index $1\leq i\leq
n$, to produce a new seed
$(\mu_{i}(\mathbf{x}),\allowbreak\mu_{i}(\mathbf{y}),\mu_{i}(Q))$. Quiver
mutation works exactly as defined in Section 2. Let $Q_{ij}$ denote the
difference between the number of arrows $i\to j$ and the number of arrows
$j\to i$ in $Q$. Note that then skew symmetry will follow. In other words
$Q_{ji}=-Q_{ij}$. The cluster mutates as
$\mu_{i}(\mathbf{x}):=\\{x_{1},x_{2},\dots,x_{i-1},x_{i}^{\prime},x_{i+1},\dots,x_{n}\\}$,
where

$x_{i}x_{i}^{\prime}:=\frac{y_{i}}{y_{i}\oplus
1}\prod_{Q_{ij}>0}x_{j}^{Q_{ij}}+\frac{1}{y_{i}\oplus
1}\prod_{Q_{ji}>0}x_{j}^{Q_{ji}}.$

The coefficients mutate by
$\mu_{i}(\mathbf{y}):=\\{y_{1}^{\prime},y_{2}^{\prime},\dots,y_{n}^{\prime}\\},$
where

$y^{\prime}_{k}:=\begin{cases}y_{k}^{-1}&\text{ if }k=i,\\\ y_{k}(y_{i}\oplus
1)^{Q_{ik}}&\text{ if }k\neq i\text{ and }Q_{ik}\geq 0,\\\
y_{k}\left(\dfrac{y_{i}}{y_{i}\oplus 1}\right)^{Q_{ki}}&\text{ if }k\neq
i\text{ and }Q_{ki}\geq 0.\end{cases}$

Two seeds are call _mutation equivalent_ if one can be obtained from the other
(up to permuting the indices) by a sequence of mutations.

Given a seed $(\mathbf{x},\mathbf{y},Q)$ we will call the union of all
$\mathbf{x}^{\prime}$ from any seed which is mutation equivalent to
$(\mathbf{x},\mathbf{y},Q)$ the set of _cluster variables_. The _cluster
algebra_ , $\mathcal{A}=\mathcal{A}(\mathbf{x},\mathbf{y},Q)$, is the unital
$\mathbb{ZP}$-subalgebra of $\mathcal{F}$ generated by the cluster variables.
Here we have restricted our attention to the ground ring
$\mathbb{Z}\mathbb{P}$ since we will make use of the theory of cluster
localization. Notice that since we are allowed to freely mutate when
generating the cluster variables, that two mutation equivalent seeds will
generate the same cluster algebra.

The Laurent phenomenon [10, Theorem 3.1] states that $\mathcal{A}$ is a
subalgebra of $\mathbb{ZP}\big{[}x_{1}^{\pm 1},x_{2}^{\pm 1},\dots,\allowbreak
x_{n}^{\pm 1}\big{]}$. The _upper cluster_ algebra is denoted by $\mathcal{U}$
or $\mathcal{U}(\mathbf{x},\mathbf{y},Q)$ and defined by

$\mathcal{U}:=\bigcap_{(\mathbf{x},\mathbf{y},Q)}\mathbb{ZP}\big{[}x_{1}^{\pm
1},x_{2}^{\pm 1},\dots,x_{n}^{\pm 1}\big{]},$

where the intersection is taken over all seeds. The Laurent phenomenon gives
an inclusion $\mathcal{A}\subseteq\mathcal{U}$. We will be interested in
conditions on the quiver $Q$ which imply $\mathcal{A}=\mathcal{U}$.

To show $\mathcal{A}=\mathcal{U}$ we will use Muller’s theory of locally
acyclic cluster algebras [24]. Let $(\mathbf{x},\mathbf{y},Q)$ be a seed of
rank $n$. The _freezing_ of $\mathcal{A}$ at $x_{n}\in\mathbf{x}$ is the
cluster algebra
$\mathcal{A}^{\dagger}=\mathcal{A}\big{(}\mathbf{x}^{\dagger},\mathbf{y}^{\dagger},Q^{\dagger}\big{)}$
defined as follows

  * •

The new semifield is $\mathbb{P}^{\dagger}=\mathbb{P}\times\mathbb{Z}$ with
$x_{n}$ as the generator of the free abelian group $\mathbb{Z}$. The auxiliary
addition is extended as

$\big{(}p_{1}x_{n}^{a}\big{)}\oplus\big{(}p_{2}x_{n}^{b}\big{)}=(p_{1}\oplus
p_{2})x_{n}^{\min(a,b)}.$

  * •

The new ambient field is
$\mathcal{F}^{\dagger}=\mathbb{Q}\big{(}\mathbb{P}^{\dagger},x_{1},x_{2},\dots,x_{n-1}\big{)}$
and the new cluster is $\mathbf{x}^{\dagger}=(x_{1},x_{2},\dots,x_{n-1})$.

  * •

The new coefficients are
$\mathbf{y}^{\dagger}=\big{(}y_{1}^{\dagger},y_{2}^{\dagger},\dots,y_{n-1}^{\dagger}\big{)}$
where $y_{i}^{\dagger}=y_{i}x_{n}^{Q_{in}}$.

  * •

The new quiver $Q^{\dagger}$ is obtained from $Q$ by deleting the vertex $n$.

We will denote the new upper cluster algebra by $\mathcal{U}^{\dagger}$. By
permuting indices we can freeze at any $x_{i}\in\mathbf{x}$. Freezing at a
subset of cluster variables can be done iteratively and is independent of the
order of freezing. When the freezing $\mathcal{A}^{\dagger}$ of $\mathcal{A}$
at $\\{x_{i_{1}},x_{i_{2}},\dots,x_{i_{m}}\\}\in\mathbf{x}$ satisfies
$\mathcal{A}^{\dagger}=\mathcal{A}\big{[}(x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}})^{-1}\big{]}$ we then call $\mathcal{A}^{\dagger}$ a _cluster
localization_. A  _cover_ of $\mathcal{A}$ is a collection
$\\{\mathcal{A}_{i}\\}_{i\in I}$ of cluster localizations such that there
exists $i\in I$ where $\mathcal{A}_{i}P\subsetneq\mathcal{A}_{i}$ for any
prime ideal $P\subseteq\mathcal{A}$. A cluster algebra is called _acyclic_ if
it has a seed $(\mathbf{x},\mathbf{y},Q)$ where $Q$ is an acyclic quiver. A
_locally acyclic_ cluster algebra is a cluster algebra for which there exists
a cover by acyclic cluster algebras. Being a cover is a transitive property.
Thus to show a cluster algebra is locally acyclic it suffices to produce a
cover by cluster algebras known to be locally acyclic. Another key property of
covers is that equality of the cluster algebra with its upper cluster algebra
can be checked locally.

######  Lemma 4.1 ([25, Lemma 2]).

Let $\\{\mathcal{A}_{i}\\}_{i\in I}$ be a cover of $\mathcal{A}$. If
$\mathcal{A}_{i}=\mathcal{U}_{i}$ for all $i\in I$, then
$\mathcal{A}=\mathcal{U}$.

We also will use a property which guarantees that a freezing is a cluster
localization.

######  Lemma 4.2 ([25, Lemma 1]).

If $\mathcal{A}^{\dagger}$ is a freezing of a cluster algebra $\mathcal{A}$ at
cluster variables $\\{x_{i_{1}},x_{i_{2}},\dots,x_{i_{m}}\\}$ and
$\mathcal{A}^{\dagger}=\mathcal{U}^{\dagger}$, then
$\mathcal{A}^{\dagger}=\mathcal{A}\big{[}(x_{i_{1}},x_{i_{2}},\dots,x_{i_{m}})^{-1}\big{]}$
is a cluster localization.

We now prove a lemma that will give us a cover. Observe in the situation of
Lemma 4.3 the Banff algorithm would freeze $i$ and some $j$ with $i\to j$. The
only change we are making is freezing at potentially more vertices.

######  Lemma 4.3.

Let $(\mathbf{x},\mathbf{y},Q)$ be a seed for a cluster algebra $\mathcal{A}$
so that

$V(Q)=\\{i\\}\sqcup A\sqcup B,$

where $i$ is a source and there exists an arrow $i\to j$ for all $j\in A$.
Furthermore, let $\mathcal{A}^{\dagger}$ denote the freezing at $\\{i\\}$ and
$\mathcal{A}^{\dagger\dagger}$ the freezing at $A$. If both
$\mathcal{A}^{\dagger\dagger}$ and $\mathcal{A}^{\dagger}$ are cluster
localizations, then
$\big{\\{}\mathcal{A}^{\dagger},\mathcal{A}^{\dagger\dagger}\big{\\}}$ is a
cover for $\mathcal{A}$.

###### Proof.

Under the hypothesis of the lemma it suffices to check for any prime ideal
$P\subseteq\mathcal{A}$ either
$\mathcal{A}^{\dagger}P\subsetneq\mathcal{A}^{\dagger}$ or
$\mathcal{A}^{\dagger\dagger}P\subsetneq\mathcal{A}^{\dagger\dagger}$.
Consider a prime ideal $P$ and the mutation relation at $i$ which gives

$x_{i}x^{\prime}_{i}=\frac{y_{i}M}{y_{i}\oplus 1}\prod_{j\in
A}x_{j}^{Q_{ij}}+\frac{1}{y_{i}\oplus 1},$

where $M$ is a monomial in cluster variables. Since $1/(y_{i}\oplus 1)$ is
invertible in $\mathcal{A}$ and $Q_{ij}>0$ for all $j\in A$ it follows that
$1\in\mathcal{A}x_{i}+\mathcal{A}\big{(}\prod_{j\in A}x_{j}\big{)}$. Since $P$
is prime, it is a proper ideal. So, $x_{i}\in P$ implies $x_{j}\not\in P$ for
all $j\in A$. If $x_{i}\not\in P$, then
$\mathcal{A}^{\dagger}P\subsetneq\mathcal{A}^{\dagger}$. If $x_{j}\not\in P$
for all $j\in A$, then
$\mathcal{A}^{\dagger\dagger}P\subsetneq\mathcal{A}^{\dagger\dagger}$. ∎

Let us now consider a class of quivers we call $\mathcal{P}^{\prime}_{m}$
defined by:

  * •

The quiver with one vertex is in the class $\mathcal{P}^{\prime}_{m}$.

  * •

The class $\mathcal{P}^{\prime}_{m}$ is closed under quiver mutation.

  * •

If $Q$ is a quiver in $\mathcal{P}^{\prime}_{m}$, then
$\mathcal{P}^{\prime}_{m}$ contains any triangular extension of $Q$ and a
quiver with a single vertex $v$ such that the number of vertices in $Q$ not
connected to $v$ by an arrow is less than or equal to $m$.

######  Remark 4.4.

It is clear that $\mathcal{P}^{\prime}_{m}\subseteq\mathcal{P}^{\prime}$ for
any $m$. One should note that it is indeed the case that
$\mathcal{P^{\prime}}_{m}\subsetneq\mathcal{P}^{\prime}$. Ladkani [19] has
classified which finite mutation type quivers are in $\mathcal{P}^{\prime}$.
Using this classification we can choose a quiver $Q$ in $\mathcal{P}^{\prime}$
coming from a triangulation of a surface which has $N$ vertices for some
$N\geq 8$. Since any arc in a triangulation will be in one quadrilateral a
quiver associated to a surface will only have vertices of degree at most $4$.
It follows that $Q$ is not in $\mathcal{P}^{\prime}_{m}$ when $m<N-5$.

We will focus on $m=3$ and make use of the fact that quivers on $3$ vertices
admit reddening sequences if and only if they are acyclic. We record this as
the following lemma which is implied by a result of Seven [29, Theorem 1.4] on
$c$-vectors which holds in skew-symmetrizable generality.

######  Lemma 4.5 ([29, Theorem 1.4]).

Let $Q$ be a quiver on three or fewer vertices. The quiver $Q$ admits a
reddening sequence if and only if $Q$ is mutation equivalent to an acyclic
quiver.

We are now ready to prove our main theorem on locally acyclic cluster
algebras.

######  Theorem 4.6.

If $Q$ is a quiver in the class $\mathcal{P}^{\prime}_{3}$, then for any seed
$(\mathbf{x},\mathbf{y},Q)$ containing the quiver $Q$ the cluster algebra
$\mathcal{A}(\mathbf{x},\mathbf{y},Q)$ is locally acyclic.

###### Proof.

We will induct on the number of vertices. The theorem holds when $Q$ has one
vertex. Now assume the theorem is true for quivers in the class
$\mathcal{P}^{\prime}_{3}$ which have $n$ vertices. Take a quiver in the class
$\mathcal{P}^{\prime}_{3}$ which has $n$ vertices and consider a triangular
extension with a new vertex $i_{0}$. Let this quiver be denoted $Q$ so that

$V(Q)=\\{i_{0}\\}\sqcup A\sqcup B,$

where $i_{0}$ is a source, there exists an arrow $i_{0}\to j$ for all $j\in
A$, and $|B|\leq 3$ such that there are no arrows from $i_{0}$ to any vertex
in $B$.

Let $\mathcal{A}=\mathcal{A}(\mathbf{x},\mathbf{y},Q)$ and denote the
freezings at $\\{i_{0}\\}$ and $A$ by $\mathcal{A}^{\dagger}$ and
$\mathcal{A}^{\dagger\dagger}$ respectively. Now $\mathcal{A}^{\dagger}$ is
defined by a seed with quiver $Q\setminus\\{i_{0}\\}$ which is in the class
$\mathcal{P}^{\prime}_{3}$. Hence, $\mathcal{A}^{\dagger}$ is locally acyclic
by induction. This implies $\mathcal{A}^{\dagger}$ equals its upper cluster
algebra and this is a cluster localization by Lemma 4.2. The freezing
$\mathcal{A}^{\dagger\dagger}$ is defined by a seed with a quiver which is the
disjoint union of the vertex $i_{0}$ and $Q|_{B}$. Since $Q$ is in
$\mathcal{P}_{3}^{\prime}$ it admits a reddening sequence by Theorem 3.3. Thus
$Q|_{B}$ admits a reddening sequence by Lemma 2.1. It follows by Lemma 4.5
that $Q|_{B}$ is mutation acyclic since $|B|\leq 3$. Thus
$\mathcal{A}^{\dagger\dagger}$ agrees with its upper cluster algebra and is a
cluster localization by Lemma 4.2.

We can now apply Lemma 4.3 to conclude that
$\big{\\{}\mathcal{A}^{\dagger},\mathcal{A}^{\dagger\dagger}\big{\\}}$ is a
cover of $\mathcal{A}$. Therefore $\mathcal{A}$ is locally acyclic as desired
since $\mathcal{A}^{\dagger}$ is locally acyclic and
$\mathcal{A}^{\dagger\dagger}$ is acyclic. ∎

We believe the condition of connecting to all but at most three vertices to be
artificial. We do offer the following conjecture of a more natural result.

######  Conjecture 4.7.

If $Q$ is in the class $\mathcal{P}^{\prime}$, then $\mathcal{A}=\mathcal{U}$
$($over $\mathbb{ZP})$ for any cluster algebra defined by a seed with the
quiver $Q$.

There are obstacles to extending the result in Theorem 4.6 to Conjecture 4.7.
The main problem is that being locally acyclic over $\mathbb{Z}\mathbb{P}$
does not pass to induced subquivers in general [24, Remark 3.11].

######  Conjecture 4.8.

If $Q$ is in the class $\mathcal{P}$, then $\mathcal{A}=\mathcal{U}$ over some
ground ring $($over $\mathbb{ZP}?)$ for the cluster algebra defined by $Q$.

### Acknowledgements

The authors wish to thank the anonymous referees for their feedback which has
improved this paper.

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