Reddening sequences for Banff quivers and the class $\mathcal{P}$

We show that a reddening sequence exists for any quiver which is Banff or in the class $\mathcal{P}$. Our proofs are combinatorial and rely on the direct sum 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 either a maximal green or reddening sequence and a cluster algebra's equality with its upper cluster algebra. Moreover, this 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.


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 direct sum of quivers, is essential to our results on both reddening sequences and upper cluster algebras.
For a semifield P with group algebra ZP we choose some ground ring A inside the field of fractions of ZP. We will restrict our attention to skew-symmetric cluster algebras. In [FZ02] Fomin and Zelevinsky define a cluster algebra A as a certain A-algebra inside an ambient field with generators determined by a quiver Q. In [BFZ05] Bernstein, Fomin, and Zelevinsky define the upper cluster algebra U which satisfies A Ď U . A fundamental problem in cluster algebra theory is to determine when there is the equality A " U . In general this can be a difficult problem. Typically the ground of choice is A " ZP. However, there has been recent attention paid to the choice of ground ring in the A " U question. There can be a delicate dependence on the ground ring A as demonstrated by the authors with M. Shapiro [BMS]. Goodearl and Yakimov have developed techniques for showing A " U for ground rings A Ď ZP [GY].
In the case that ground ring is A " ZP, Muller's theory of cluster localization and locally acyclic cluster algebras provides a means of showing A " U [Mul13,Mul14]. The Banff algorithm is one way of showing that a cluster algebra is locally acyclic, and 2010 Mathematics Subject Classification. Primary 13F60; Secondary 16G20. 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 ZP satisfies A " U .
Keller [Kel11,Kel12] 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 [KS]. 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 1 . Canakci, Lee, and Schiffler [CLS15] had observed existence of a maximal green sequence coincided with A " U in know cases at the time. Mills [Mil] offers an explicit conjecture on the potential relationship of maximal green sequences, the equality A " U , locally acyclicity, and choice of ground ring. Our work here is progress understanding this relationship.
In Theorem 3.1 we show that Banff quivers admit reddening sequences. Theorem 3.2 gives reddening sequences for any quiver in the class P. These theorems cover many examples which have been subject to previous research. Ford and Serhiyenko [FS18] have shown that quivers arising from Postnikov's reduced plabic graphs [Pos] Buc16,BM18]. 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 [Mil17]. Also, minimal mutation infinite quivers are further examples of Banff quivers for which Lawson and Mills have shown have maximal green sequences [LM18].
Morally, the proposed conjecture (with some dependence on ground ring) is that for a given quiver the following are equivalent: (i) The quiver admits a reddening sequence. (ii) The associated cluster algebra equals its upper cluster algebra. (iii) The associated cluster algebra is locally acyclic.
Mills offers a more precise conjecture [Mil, Conjecture 1] and verifies the conjecture for mutation finite quivers [Mil, Theorem 1.2]. Theorem 3.1 completes a proof that all three conditions are equivalent for Banff quivers. Theorem 3.2 suggests the class P as a next step in verifying the conjecture. To the knowledge of the authors every known Banff quiver is also in the class P (in fact inside a more restrictive class we denote P 1 ). We ask Questions 3.4 and 3.5 in attempt to better understand the relationship between Banff quivers and the class P. In Theorem 4.5 we make some progress toward the equivalence of (i), (ii), and (iii) for the class P by showing the conditions are equivalent for a certain subfamily of quivers.

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 p Q, is the quiver whose vertex set and edge set are the following: The coframed quiver associated to Q, denoted q Q, is the quiver whose vertex set and edge set are the following: The vertices i P V pQq are the mutable vertices and the vertices i 1 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 µ i pQq obtained from Q by doing the following: (1) For each pair of arrows j Ñ i, i Ñ k such that not both i and j are frozen add an arrow j Ñ k.
(2) Reverse all arrows incident on i.
(3) Delete a maximal collection of disjoint 2-cycles. 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 Ď V pQq we let Q| A denote the induced subquiver which has and is a natural restriction of Q to A. We will use QzA to denote Q| V pQqzA . A mutable vertex is green if it there are no incident incoming arrows from frozen vertices. Similarly, a mutable vetex 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 p 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 [DWZ10]. Notice also that all vertices are initially green when starting with p Q. Keller [Kel11,Kel12] has introduced the following types of sequences of mutations which will our main interest. A sequence 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 [BDP14, Proposition 2.10] after preforming an reddening sequence starting with p Q will end with a quiver isomorphic to q Q. By definition all maximal green sequences are reddening sequences. There quivers for which a maximal green sequence does not exist, but a reddening sequence does. Furthermore there are quivers for which no reddening sequence exists. If a reddening (maximal green) sequence exists of a quiver, we will say this quivers 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 [Mul16] and their proofs make use scattering diagrams which have been connected with cluster algebras by Gross, Hacking, Keel, and Kontsevich [GHKK18]. 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.
Theorem 17] If a quiver Q admits a reddening sequence, then any induced subquiver of Q also admits a reddening sequence.
Corollary 19] If a quiver Q admits a reddening sequence, then any quiver mutation equivalent to Q also admits a reddening sequence.
We now consider a result which states the direct sum construction respects both the existence of reddening sequences and maximal green sequences. Let Q 1 and Q 2 be quivers. A direct sum of Q 1 and Q 2 is any quiver Q with where E is any set of arrows such that either we have either for any i Ñ j P E implies i P V pQ 1 q and j P V pQ 2 q or else for any i Ñ j P E implies i P V pQ 2 q and j P V pQ 1 q.
That is, a direct sum 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 direct sum of quivers Q 1 and Q 2 where V pQ 1 q " t1, 2, 3u and V pQ 2 q " t4, 5, 6u is given in Figure 1.
We shortly will state a result that the direct sum construction respects reddening sequences and maximal green sequences. Garver and Musiker show as analogous result for maximal green sequences in a restricted case needed for their work which they call a t-colored direct sum [GM17, Theorem 3.12] and suggest the result holds in greater generality [GM17, Remark 3.13]. Cao and Li show the result (stated for maximal green sequences, but remark on reddening sequences [CL,Remark 4.6]) for any direct sum in the generality of skew-symmetrizable matrices [CL,Theorem 4.5]. In sections following this one we will emphasize reddening sequences, and the mutation invariance of reddening sequences will allow applications of the following lemma to Banff quivers and quivers in the class P.  ). If Q 1 and Q 2 are any two quivers which both admit a reddening (maximal green) sequences, then any direct sum of Q 1 and Q 2 admits a reddening (maximal green) sequence.

Existence of reddening sequences
In this section we define Banff quivers and the class P. We show Banff quivers and quivers in the class P admit reddening sequences.
3.1. Banff quivers. A bi-infinite path in a quiver Q is a sequence pi a q aPZ of mutable vertices such that i a Ñ i a`1 is a arrow for each a P Z. A pair of vertices pi, jq is a covering pair if i Ñ j is a arrow which is not part of any bi-infinite path. Muller's (reduced) Banff algorithm is a nondeterministic algorithm which takes as input a quiver Q and performs the following steps: (1) If Q is mutation equivalent to an acyclic quiver, then stop.
(2) Choose a covering pair pi, jq in some quiver Q 1 mutation equivalent to Q. Otherwise if this is impossible, then the algorithm fails. For a given quiver there may be many branches in the Banff algorithm from making various choices in step (2). A quiver Q is called a Banff quiver if there is some branch of the Banff algorithm which does not fail. Alternatively, we can characterize Banff quivers as 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 pi, jq where both Qztiu and Qztju are Banff is a Banff quiver.
Example branches of the Banff algorithm are shown in Figure 2.
Theorem 3.1. 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 is an isolated quiver. 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 which is the first step in a terminating branch of the Banff algorithm. We may assume that Q has this covering pair because existence of a reddening sequence is mutation invariant by Lemma 2.2. Let pi, jq be a covering pair used in a terminating branch of the Banff algorithm. 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 P B. Let A be the complement of B in V pQq. We have i P A since pi, jq is a covering pair. If i R A, then i P B and there would be a cycle containing the arrow i Ñ j which would contradict the fact that pi, jq is a covering pair. It follows by the definition of the sets A and B that Q is a direct sum of Q| A and Q| B since if there was an arrow k Ñ ℓ with k P B then ℓ P B. That is, there are no arrows k Ñ ℓ with k P B and ℓ P A.
Now Qztiu and Qztju are both Banff quivers with strictly fewer vertices than Q. Hence, Qztiu and Qztju admit reddening sequences by induction. The quiver Q| A is an induced subquiver of Qztju and Q| B is an induced subquiver of Qztiu. 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 P. The following three properties define the class P: ‚ The quiver with one vertex is in the class P. ‚ The class P is closed under quiver mutation. ‚ The class P is closed under taking any direct sum of two quivers in the class P.

This class of quivers was introduced by Kontsevich and Soibelman [KS, Section 8.4].
Theorem 3.2. Let Q be in the class 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 P.
We now discuss the relationship between the class P and the class of Banff quivers. It has been observed [Lad,Remark 4.20] that quivers associated to a triangulation of a surface of genus g ą 1 with exactly one boundary component and a single marked point on the boundary give examples of quivers in the class P that are not Banff. Ladkani [Lad,Remark 4.21] also considers the class P 1 of quivers defined by: ‚ The quiver with one vertex is in the class P 1 . ‚ the class P 1 is closed under quiver mutation. ‚ The class P 1 is closed under taking direct sums of quivers where one summand is in the class P 1 and the other summand is the quiver with one vertex. Now let B 1 denote the class of quivers which have successful terminating branches of the Banff algorithm where the covering pair used always involves a source or sink. That is, the class B 1 is defined by: ‚ Any quiver without arrows is in B 1 . ‚ Any quiver mutation equivalent to a quiver in B 1 is in B 1 . ‚ Any quiver Q with an arrow i Ñ j such that i is or source or j is a sink where both Qztiu and Qztju are in B 1 is in B 1 . In practice this is how quivers are often shown to be Banff. For example, quivers from many marked surfaces [Mul13, Section 10] and quivers for Grassmannians [MS16] are in B 1 . The next proposition follows immediately from the definitions of P 1 and B 1 . Proposition 3.3. If a quiver is in B 1 , then the quiver is in P 1 .
Question 3.4. Does there exist a Banff quiver which is not in the class P?
Question 3.5. Does there exist a Banff quiver which is not in B 1 ?
If Question 3.5 has a negative answer, then by Proposition 3.3 the answer to Question 3.4 is also negative. A quiver has a covering pair if and only if it has a source or sink [Mul13,Proposition 8.1]. So, at any step which the Banff algorithm can proceed there is the option to choose a covering pair which contains a source or sink. A quiver giving a positive answer to Question 3.5 must fail whenever a branch uses only source and sink covering pairs, but succeed on some branch which makes use of a covering pair not containing a source nor a sink.
Lam and Speyer's class of Louise quivers [LS] is another class of quivers for which similar questions could be asked. It would be interesting to better understand the relationship of Louise quivers with 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 C of quivers by: ‚ Any quiver of type T is in C. ‚ Any quiver mutation equivalent to a quiver in C is in C. ‚ The direct sum of any two quivers in C is in C. If a quiver is of type T only if it has a single vertex, then C is just the class P. Theorem 3.1 says we can take a quiver to be of type T if it is Banff. If Question 3.4 has a positive answer then this will be a class of quivers with reddening sequences which is strictly larger than the class P. This raises the following fairly ambitious question: Question 3.6. What is a minimal T , for which C consists of exactly 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 direct sum and mutation. As one can see from the class 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 P. The quiver associated to a triangulation of a torus with one boundary component at one marked point on the boundary gives a quiver not in the class P which admits a reddening sequence. This quiver is shown in Figure 3 along with a maximal green (reddening) sequence for it.

Locally acyclic cluster algebras
Let P be a semifield. Let F be a field which contains ZP. A seed of rank n in F is a triple px, y, Qq. The cluster x " tx 1 , x 2 , . . . , x n u is an n-tuple in F which freely generates F as a field over the fraction field of ZP. The coefficients y " ty 1 , y 2 , . . . , y n u consist of elements of P. Here Q a is quiver on vertex set V pQq " t1, 2, . . . , nu.
A seed px, y, Qq may be mutated at any index 1 ď i ď n, to produce a new seed pµ i pxq, µ i pyq, µ i pQqq. Quiver mutation works exactly as defined in Section 2. Let Q ij 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 p1, 3, 4, 2, 1, 3q.
denote the number of arrows i Ñ j in Q. We also let Q ji "´Q ij . The cluster mutates as µ i pxq :" tx 1 , x 2 , . . . , x i´1 , x 1 i , x i`1 , . . . , x n u, where x i x 1 i :" The coefficients mutate by µ i pyq :" ty 1 1 , y 1 2 , . . . y 1 n u, where 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 px, y, Qq we will call the union of all x 1 from any seed which is mutation equivalent to px, y, Qq the set of cluster variables. The cluster algebra, A " Apx, y, Qq, is the unital ZP-subalgebra of F generated by the cluster variables. Here we have restricted our attention to the ground ring ZP 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 [FZ02, Theorem 3.1] states that A is a subalgebra of ZPrx˘1 1 , x˘1 2 , . . . , x˘1 n s. The upper cluster algebra is denoted by U or U px, y, Qq and defined by U :" č px,y,Qq ZPrx˘1 1 , x˘1 2 , . . . , x˘1 n s where the intersection is taken over all seeds. The Laurent phenomenon gives an inclusion A Ď U . We will be interested in conditions on the quiver Q which imply A " U .
To show A " U we will use Muller's theory of locally acyclic cluster algebras [Mul13]. Let px, y, Qq be a seed of rank n. The freezing of A at x n P x is the cluster algebra A : " Apx : , y : , Q : q defined as follows ‚ The new semifield is P : " PˆZ with x n as the generator of the free abelian group Z. The auxiliary addition is extended as pp 1 x a n q ' pp 2 x b n q " pp 1 ' p 2 qx minpa,bq n .
‚ The new ambient field is F : " QpP : , x 1 , x 2 , . . . , x n´1 q and the new cluster is x : " px 1 , x 2 , . . . , x n´1 q. ‚ The new coefficients are y : " py : 1 , y : 2 , . . . , y : n´1 q where y : i " y i x Q in n . ‚ The new quiver Q : is obtained from Q by deleting the vertex n.
We will denote the new upper cluster algebra by U : . By permuting indices we can freeze at any x i P x. Freezing at a subset of cluster variables can be done iteratively and is independent of the order of freezing. When the freezing A : of A at tx i 1 , x i 2 , . . . , x im u P x satisfies A : " Arpx i 1 x i 2¨¨¨x im q´1s we then call A : a cluster localization. A cover of A is a collection tA i u iPI of cluster localizations such that there exists i P I where A i P A i for any prime ideal P Ď A. A cluster algebra is called acyclic if it has a seed px, y, Qq 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. We also will use a property which garantees that a freezing is a cluster localization.
Lemma 4.2 ([Mul14, Lemma 1]). If A : is a freezing of a cluster algebra A at cluster variables tx i 1 , x i 2 , . . . , x im u and A : " U : , then A : " Arpx i 1 , x i 2 , . . . , x im q´1s 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 Ñ j. The only change we are making is freezing at potentially more vertices.
Lemma 4.3. Let px, y, Qq be a seed for a cluster algebra A so that where i is a source and there exists an arrow i Ñ j for all j P A. Furthermore, let A : denote the freezing at tiu and A :: the freezing at A. If both A :: and A : are cluster localizations, then tA : , A :: u is a cover for A.
Proof. Under the hypothesis of the lemma it suffices to check for any prime ideal P Ď A either A : P A : or A :: P A :: . Consider a prime ideal P and the mutation relation at i which gives where M is a monomial in cluster variables. Since 1{py i ' 1q is invertible in A and Q ij ą 0 for all j P A it follows that 1 P Ax i 0`A´śjPA x j¯. Since P is prime, it is a proper ideal. So, x i 0 P P implies x j R P for all j P A. If x i 0 R P , then A : P A : . If x j R P for all j P A, then A :: P A :: .
Let us now consider a class of quivers we call P 1 m defined by: ‚ The quiver with one vertex is in the class P 1 m . ‚ The class P 1 m is closed under quiver mutation. ‚ The class P 1 m is closed under taking direct sums of quivers where one summand is in the class P 1 m , the other summand is the quiver with one vertex, and the direct sum connects the quiver with one vertex to all but at most m vertices of the other direct summand.
It is clear that P 1 m Ď P 1 for any m. We will focus on m " 3 and make use of the fact the 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 [Sev14, Theorem 1.4] on c-vectors which holds in skew-symmetrizable generality.
Lemma 4.4 ([Sev14, 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 4.5. If Q is a quiver in the class P 1 3 , then for any seed px, y, Qq containing the quiver Q the cluster algebra Apx, y, Qq 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 P 1 3 which have n vertices. Take a quiver in the class P 1 3 which has n vertices and consider a direct sum with a new vertex i 0 . Let this quiver be denoted Q so that where i 0 is a source, there exists an arrow i 0 Ñ j for all j P A, and |B| ď 3.
Let A " Apx, y, Qq and denote the freezings at ti 0 u and A by A : and A :: respectively. Now A : is defined by a seed with quiver Qzti 0 u which is in the class P 1 3 . Hence, A : is locally acyclic by induction. This implies A : equals its upper cluster algebra and this is a cluster localization by Lemma 4.2. The freezing A :: 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 P 1 3 it admits a reddening sequence by Theorem 3.2. Thus Q| B admits a reddening sequence by Lemma 2.1. It follows by Lemma 4.4 that Q| B is mutation acyclic since |B| ď 3. Thus A :: 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 tA : , A :: u is a cover of A. Therefore A is locally acyclic as desired since A : is locally acyclic and A :: 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 4.6. If Q is in the class P 1 , then A " U (over ZP) for any cluster algebra defined by a seed with the quiver Q.
There are obstacles to extending the result in Theorem 4.5 to Conjecture 4.6. The main problem is that that being locally acyclic over ZP does not pass to induced subquivers in general [Mul13, Remark 3.11].