Maximal Green Sequences for Cluster Algebras Associated to the Orientable Surfaces of Genus n with Arbitrary Punctures
aa r X i v : . [ m a t h . C O ] S e p MAXIMAL GREEN SEQUENCES FOR CLUSTER ALGEBRAS ASSOCIATEDTO THE n -TORUS WITH ARBITRARY PUNCTURES ERIC BUCHER AND MATTHEW R. MILLS
Abstract.
It is well known that any triangulation of a marked surface produces a quiver. Inthis paper we will provide a triangulation for orientable surfaces of genus n with an arbitrarynumber interior marked points (called punctures) whose corresponding quiver has a maximal greensequence. Introduction
Cluster algebras were introduced by Fomin and Zelevinsky in [7]. Within a very short periodof time cluster algebras became an important tool in the study of phenomena in various areas ofmathematics and mathematical physics. They play an important role in the study of Teichm¨ullertheory, canonical bases, total positivity, Poisson Lie-groups, Calabi-Yau algebras, noncommutativeDonaldson-Thomas invariants, scattering amplitudes, and representations of finite dimensionalalgebras. For more information on the diverse scope of cluster algebras see the review paper byWilliams [14].A question that has arose that carries with it some ramifications is whether a quiver associatedto a cluster algebra has a maximal green sequence. The idea of maximal green sequences of clustermutations was introduced by Keller in [10]. He explored important applications of this notion,by utilizing it in the explicit computation of noncommutative Donaldson-Thomas invariants oftriangulated categories which were introduced by Kontsevich and Soibelman in [11]. If a quiver withpotential has a maximal green sequence, then its associated Jacobi algebra is finite dimensional,this fact follows from Theorem 5.4 [9] and Theorem 8.1 [3]. In the work of Amoit [2] there isgiven an explicit construction of a cluster category given a quiver with potential whose Jacobianalgebra is finite dimensional. This means that finding a maximal green sequence for a given quiverallows us to categorify the associated cluster algebra. Muller showed that in fact the existenceof a maximal green sequence corresponding to a quiver is not a mutation invariant [13], whichmakes it all the more difficult to show that a maximal green sequence exists. You must make astrategic choice of initial seed (or initial quiver) and then find the sequence. The iterative naturemutations means that exhaustive methods are not always effective when searching for a maximalgreen sequence. There has been some progress made, and in the final section of this paper we willdiscuss which cluster algebras are known to have or not have associated maximal green sequences.The main result in this paper focuses on cluster algebras that are associated to triangulationsof surfaces. This association is introduced by Gekhtman, Shapiro, and Vainshtein in [8] and ina more general setting by Fock and Goncharov in [5]. This construction is extremely importantbecause all but a few exceptional cases of cluster algebras of finite mutation type can be realizedas a cluster algebra which arises from a surface following this construction. For a more in depthlook into the procedure of creating a cluster algebra from a triangulated surface see the work byFomin, Shapiro, and Thurston [6].
Date : March, 2015.1991
Mathematics Subject Classification.
Surface Classification Exists a MGS ReferenceSurface with boundary Yes Alim et al., [1]Closed surface with p = 1 No S. Ladkani, [12]Sphere with p ≥ g = 1 and p ≥ g ≥ p = 2 Yes Bucher, [4]Closed surface with g ≥ p ≥ Table 1.
The different cases of surfaces, and whether or not they have a maximalgreen sequence. The letter g denotes the genus of the surface, and p denotes thenumber of punctures. By ”closed surface” we mean a surface with no boundarycomponent.Previous work by the first author in [4] constructs maximal green sequences for cluster algebraswhich arise from surfaces with no empty boundary component and two punctures. In this paperwe will push this result further, and construct maximal green sequences for cluster algebras arisingfrom orientable surfaces with arbitrary genus and three or more punctures. We have providedreferences for all known cases in Table 1. This brings us to the statement of the main theorem forour paper. Theorem 1.1.
For a surface with no boundary, genus at least two, and with three or more punc-tures there exists a triangulation whose associated quiver has a maximal green sequence.
The proof of Theorem 1.1 is given in section 4, but we will give a sketch of the proof here. Westart with an n -torus and then construct a specific triangulation of this surface for which we can finda maximal green sequence for the associated cluster algebra. This triangulation is a modificationof the triangulation that was used in [4], and will be given in section 3. After constructing thetriangulation, we look at its associated quiver. We take advantage of the symmetry of this quiver bybreaking it into smaller parts, finding maximal green sequences for these parts, and then carefullyputting these sequences together. By adding interior punctures to the structure we add a ladderstructure to our quiver. Increasing the number of punctures lengthens the ladder but its connectionto the rest of the quiver is unaffected. We then induct on the number of punctures to finish theproof.As an immediate corollary to our result we also have the following result. Theorem 1.2.
For every surface, it is known whether or not there exists a triangulation whoseassociated quiver has a maximal green sequence.Proof.
Surfaces are determined by the genus of the surface, the number of boundary components,the number of punctures, and the number of marked points on each boundary component. Checkingthe surfaces given in Table 1, we see that all possible cases are accounted for. (cid:3)
In section 2 we give background on quivers and maximal green sequences. In section 5 we brieflydiscuss future work to be done with maximal green sequences.2.
Preliminaries
We will follow the notation laid out by Br¨ustle, Dupont, and Perotin [3].
Definition 2.1. A quiver , Q , is a directed graph containing no 2-cycles or loops.The notation Q will denote the vertices of Q . Also, Q will denote the edges of Q which arereferred to as arrows . We will let Q = [ N ]. GS FOR THE N-TORUS 3
Definition 2.2. An ice quiver is a pair ( Q, F ) where Q is a quiver as described above and F ⊂ Q is a subset of vertices called frozen vertices; such that there are no arrows between them. Forsimplicity, we always assume that Q = { , , , . . . , n + m } and that F = { n + 1 , n + 2 , . . . , n + m } for some integers n, m ≥
0. If F is empty we write ( Q, ∅ ) for the ice quiver.In this paper we will be concerned with a process called mutation. Mutation is a process ofobtaining a new ice quiver from an existing one. Definition 2.3.
Let (
Q, F ) be an ice quiver and k ∈ Q a non-frozen vertex. The mutation of aquiver ( Q, F ) at a vertex k is denoted µ k , and produces a new ice quiver ( µ k ( Q ) , F ). The verticesof ( µ k ( Q ) , F ) are the same vertices from ( Q, F ). The arrows of the new quiver are obtained byperforming the following 3 steps:(1) For every 2-path i → k → j , adjoin a new arrow i → j .(2) Reverse the direction of all arrows incident to k .(3) Delete any 2-cycles created during the first two steps as well as any arrows created betweenfrozen vertices.It is important to note that we do not allow mutation at a frozen vertex. We will denote M ut ( Q )to be the set of all quivers who can be obtained from Q by a sequence of mutations.The ice quivers which are of concern in this paper have a very specific set of frozen vertices. Wewill be looking at what are referred to as the framed and coframed quivers associated to Q . Definition 2.4.
The framed quiver associated with Q is the quiver ˆ Q such that:ˆ Q = Q ⊔ { i ′ | i ∈ Q } ˆ Q = Q ⊔ { i → i ′ | i ∈ Q } The coframed quiver associated with Q is the quiver ˘ Q such that:˘ Q = Q ⊔ { i ′ | i ∈ Q } ˘ Q = Q ⊔ { i ′ → i | i ∈ Q } Both quivers ˆ Q and ˘ Q are naturally ice quivers whose frozen vertices are commonly written asˆ Q ′ and ˘ Q ′ . Next we will talk about what it means for a vertex to be green or red. Definition 2.5.
Let R ∈ M ut ( ˆ Q ). A non-frozen vertex i ∈ R is called green if { j ′ ∈ Q ′ | ∃ j ′ → i ∈ R } = ∅ . It is called red if { j ′ ∈ Q ′ | ∃ j ′ ← i ∈ R } = ∅ . In they show that every non-frozen vertex in R is either red or green. This idea is whatmotivates our work in this paper. It arises as a question of green sequences. Definition 2.6. A green sequence for Q is a sequence i = { i , . . . , i l } ⊂ Q such that i is greenin ˆ Q and for any 2 ≤ k ≤ l , the vertex i k is green in µ i k − ◦ · · · ◦ µ i ( ˆ Q ). The integer l is called thelength of the sequence i and is denoted by l ( i ).A green sequence i is called maximal if every non-frozen vertex in µ i ( ˆ Q ) is red where µ i = µ i l ◦ · · · ◦ µ i . We denote the set of all maximal green sequences for Q bygreen( Q ) = { i | i is a maximal green sequence for Q } . BUCHER AND MILLS b a b a b a b a b n a n b n a n e c d e d c e n d n c n f f f n − f n Figure 1.
A triangulation of an n -torus with two punctures. b a b a b a b a b n a n b n a n f f n − f n f n +2 g g g g f n +2 f n f n +1 g p − g p − g p − Figure 2.
A triangulation of n -torus with p punctures.In this paper we will construct a maximal green sequence for a specific infinite family of quiverswhich will be described in the following section. In essence what we want to show is that green( Q ) = ∅ for each quiver, Q , in this family. One important observation that the proof of our main resultrelies on is that if Q and Q are quivers such that Q has a maximal green sequence λ , and Q isa full subquiver of Q consisting of only green vertices, then λ is a green sequence for Q .3. Constructing the Triangulation
Following the work done by Fomin, Shapiro, and Thurston in [6] we construct a quiver Q pn associated to the genus n surface with no boundary and p ≥ p ≥ f n and replace it with a ( p − Q pn . The quiver associated to the 3-toruswith 7 punctures is given on the left of Figure 3. Also we define the quiver P p − for p ≥ Q pn consisting of the vertices { g , g , g , g , . . . , g p − , g p − , g p − } . P is given onthe right of Figure 3. Note that by increasing the genus of the surface the cycle containing the f vertices gets longer, and more handles are added. Increasing the number of punctures will increasethe number of rows in the P subquiver. The important thing to notice is that the fundamentalshape of the quiver Q pn doesn’t change. GS FOR THE N-TORUS 5 f f f f f g e e e g g g d a b c d a b c d a b c g g g g g g g g g ? ? ⑧⑧⑧ o o (cid:20) (cid:20) ✯✯✯✯✯ (cid:31) (cid:31) ❄❄❄ / / j j ❚❚❚❚❚ z z ttttt (cid:31) (cid:31) ❄❄❄ d d ❏❏❏❏❏ t t ❥❥❥❥❥ o o J J ✔✔✔✔✔ ? ? ⑧⑧⑧ : : ttttt $ $ ❏❏❏❏❏ o o (cid:15) (cid:15) (cid:15) (cid:15) : : ttttt (cid:31) (cid:31) ❄❄❄ ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ ? ? ⑧⑧⑧⑧ o o O O $ $ ❏❏❏❏❏ O O : : ttttt $ $ ❏❏❏❏❏ / / / / o o (cid:7) (cid:7) ✎✎✎✎✎✎✎✎ ♦♦ (cid:23) (cid:23) ✴✴✴✴✴✴✴✴ g g ❖❖ O O o o (cid:4) (cid:4) ✡✡✡✡✡ ♦♦ (cid:26) (cid:26) ✹✹✹✹✹✹ g g ❖❖ O O o o (cid:7) (cid:7) ✎✎✎✎✎✎✎✎ ♦♦ (cid:23) (cid:23) ✴✴✴✴✴✴✴✴ g g ❖❖ O O d d ❏❏❏❏❏ z z ttttt o o (cid:15) (cid:15) : : ttttt o o $ $ ❏❏❏❏❏ O O o o o o g g g g g g g g g g g g g (cid:31) (cid:31) ❄❄❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧⑧⑧ o o (cid:15) (cid:15) ? ? ⑧⑧⑧⑧⑧⑧ o o (cid:31) (cid:31) ❄❄❄❄❄❄ O O o o (cid:15) (cid:15) ? ? ⑧⑧⑧⑧⑧⑧ o o (cid:31) (cid:31) ❄❄❄❄❄❄ O O o o (cid:15) (cid:15) ? ? ⑧⑧⑧⑧⑧⑧ o o (cid:31) (cid:31) ❄❄❄❄❄❄ O O o o o o Figure 3.
The quivers Q (left) and P (right).4. Proof of Main Result
We begin by recalling a result from [4] that gives a maximal green sequence for oriented cycles.
Lemma 4.1. [4, Lemma 4.2]
Let C be a quiver that is an oriented n -cycle with vertices labeled c i i = 1 , . . . , n , with c i → c i − for ≤ i ≤ n and c → c n . Define a sequence γ ( c n , c n − , . . . , c , c ) = c n c n − · · · c c c c · · · c n − c n . Then γ is a maximal green sequence for C . Furthermore, after applying γ to C the resultant quiveris still an oriented cycle with c , and c interchanged. We will rephrase the statement of Theorem 1.1 to use the terminology we defined in the previoussection.
Theorem 4.2.
Let Q pn be the quiver obtained from our triangulation of a genus n surface with noboundary and p ≥ punctures. Then Q pn has a maximal green sequence given by γ ( f n +2 f n +1 · · · f ) σ n · · · σ α α · · · α p − γ ( f n +2 , f , f , f , f , . . . , f n ) f n +1 β p − τ τ · · · τ n , where γ is defined as in Lemma 4.1 and σ, τ, α, and β are defined as follows: σ i = e i d i b i c i a i b i d i e i c i a i b i ,τ i = e i b i a i c i e i d i b i a i e i ,α j = g j = 0 g g g g g j = 1 g g g g g g g g g j = 2 g j g j g j − g j − g j g j g j − g j − g j − g j − · · · g g g g g j ≥ BUCHER AND MILLS β j = ∅ j = 0 g g g j = 1 g g g g g g j = 2 g n − g j g n − g n g n − g n − g n − · · · g g g g g j ≥ Lemma 4.3. P n has a maximal green sequence given by α α · · · α n .Proof. The sequence is easily checked for n = 0 , ,
2. For n=3, apply α α α to P . We know thatthis is a green sequence for the P subquiver of P . The current state of the quiver is given in thefollowing diagram. g g g g g g g g g g g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ / / k k ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ o o | | ②②②②② ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛ (cid:12) (cid:12) ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ (cid:9) (cid:9) ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ _ _ ❄❄❄❄ (cid:3) (cid:3) ✞✞✞✞✞✞✞✞✞✞✞ e e ❑❑❑❑ + + ❲❲❲❲❲ ❥❥❥ (cid:15) (cid:15) O O O O | | ②②②②②②②②② / / o o o o ❧❧ % % ▲▲▲▲▲▲▲▲ ♥♥♥♥♥♥♥♥♥♥ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❧❧❧❧❧❧❧❧❧❧❧❧❧ m m ❬❬❬❬❬❬❬❬❬❬❬❬ (cid:5) (cid:5) ' ' PPPPPPPPPP * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (cid:15) (cid:15) (cid:30) (cid:30) ❂❂❂❂❂❂ " " ❊❊❊❊❊ We now apply the first four mutations of α to the quiver above. g g g g g g g g g g g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ o o o o ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ | | ②②②②② ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛ (cid:12) (cid:12) ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ (cid:9) (cid:9) ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ _ _ ❄❄❄❄ (cid:3) (cid:3) ✞✞✞✞✞✞✞✞✞✞✞ e e ❑❑❑❑ + + ❲❲❲❲❲ ❥❥❥ O O (cid:15) (cid:15) O O | | ②②②②②②②②② / / o o o o ❧❧ % % ▲▲▲▲▲▲▲▲ ♥♥♥♥♥♥♥♥♥♥ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❧❧❧❧❧❧❧❧❧❧❧❧❧ m m ❬❬❬❬❬❬❬❬❬❬❬❬ (cid:5) (cid:5) ' ' PPPPPPPPPP * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (cid:15) (cid:15) (cid:30) (cid:30) ❂❂❂❂❂❂ " " ❊❊❊❊❊ g g g g g g g g g g g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ o o o o ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ | | ②②②②② , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛ (cid:12) (cid:12) ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ (cid:9) (cid:9) ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ O O ✐✐✐✐✐✐✐✐✐✐ + + ❱❱❱❱ k k ❱❱❱❱ (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛ O O (cid:15) (cid:15) O O l l ❩❩❩❩❩❩❩❩❩❩ (cid:28) (cid:28) ✿✿✿ o o o o ❧❧ ❅❅❅❅❅❅❅❅❅❅❅❅❅ ♥♥♥♥♥♥♥♥♥♥ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ t t ✐✐✐✐✐✐✐✐✐✐ | | ②②②②② ♣♣♣♣♣ ' ' PPPPPPPPPP * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (cid:15) (cid:15) (cid:30) (cid:30) ❂❂❂❂❂❂ " " ❊❊❊❊❊ g g g g g g g g g g g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ o o o o ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ | | ②②②②② ✐✐✐✐✐✐✐✐✐✐ (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛ (cid:12) (cid:12) ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ (cid:9) (cid:9) ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ * * ❯❯❯❯❯❯❯❯❯❯ w w ♥♥♥♥♥♥♥♥♥♥ ✐✐✐✐✐✐✐✐✐✐ j j ❯❯❯❯❯❯❯❯❯❯ (cid:15) (cid:15) ♥♥♥♥♥♥♥♥♥♥ (cid:15) (cid:15) j j ❯❯❯❯❯❯❯❯❯❯ w w ♥♥♥♥♥♥♥♥♥♥ ❞❞❞❞❞❞❞❞❞ o o o o ❧❧ O O ♥♥♥♥♥♥♥♥♥♥ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜ (cid:15) (cid:15) (cid:29) (cid:29) ❁❁❁❁❁❁❁❁❁❁❁❁❁❁ ' ' PPPPPPPPPP * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (cid:15) (cid:15) (cid:30) (cid:30) ❂❂❂❂❂❂ " " ❊❊❊❊❊ g g g g g g g g g g g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ o o o o ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ | | ②②②②② ✐✐✐✐✐✐✐✐✐✐ (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛ (cid:12) (cid:12) ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ (cid:9) (cid:9) ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ * * ❯❯❯❯❯❯❯❯❯❯ w w ♥♥♥♥♥♥♥♥♥♥ ✐✐✐✐✐✐✐✐✐✐ j j ❯❯❯❯❯❯❯❯❯❯ (cid:15) (cid:15) v v ♥♥♥♥♥♥♥♥♥♥♥ O O (cid:15) (cid:15) * * ❯❯❯❯❯❯❯❯❯❯ ♦♦♦♦♦♦♦♦♦♦ r r ❡❡❡❡❡❡❡ o o o o ❧❧ O O ♥♥♥♥♥♥♥♥♥♥ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ (cid:29) (cid:29) ❁❁❁❁❁❁❁❁❁❁❁❁❁❁ ' ' PPPPPPPPPP * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (cid:15) (cid:15) (cid:30) (cid:30) ❂❂❂❂❂❂ " " ❊❊❊❊❊ GS FOR THE N-TORUS 7
After these four mutations g is the only remaining green vertex, and is the initial vertex in a 2-path through a frozen vertex for 6 vertices. However, the terminal vertex in these 2-paths form anequioriented affine subquiver with g being the sink for this subquiver. The remaining mutationsof α is just the mutation along the vertices of this subquiver. Note that at each step throughthis part of the sequence there is a unique green vertex with a unique edge with head at mutablevertex and tail at the green vertex. Rearranging the vertices from our previous picture we obtainthe following picture from which it is easy to see that the remaining mutations give a maximalgreen sequence. g g g g g g g g g g g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ O O (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ r r ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ o o t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ | | ③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③ w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ z z tttttttttttttttttttttttttttttttt l l ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ (cid:15) (cid:15) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ O O / / k k ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ (cid:15) (cid:15) ' ' ❖❖❖❖❖❖❖❖ w w ♦♦♦♦♦♦♦♦ (cid:15) (cid:15) O O O O / / / / O O / / O O / / (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ Thus α α α α is a maximal green sequence for P . Our claim follows from induction on n .Suppose for 1 ≤ k < n α · · · α k gives a maximal green sequence for α k . Note that Q n has asubquiver of P n − for which α α n − is a green sequence. Note that the local configuration of ”topnine” vertices g ji i = 1 , , j = n − , n − , n have the exact same configuration as the ”top nine”vertices of P and the first four mutations of α n exactly mimic that of n = 3 case. (Possibly needto show this in a lemma.) Therefore we get that after the green sequence α · · · α n − g j g j g j − g j − we have the quiver: g n g n g n g n − g n − g n − g n − g n − g n − g n − · · · g ... g g g g n ′ g n ′ g n ′ g n − ′ g n − ′ g n − ′ g n − ′ g n − ′ g n − ′ g n − ′ g n − ′ g ′ g ′ · · · ′ g ′ ... ′ O O (cid:2) (cid:2) ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ q q ❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝ o o s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ x x ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ } } ④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④④ y y tttttttttttttttttttttttttttttttttttt { { ①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①①① k k ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ (cid:15) (cid:15) * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ O O / / k k ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ (cid:15) (cid:15) * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ m m ❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬ ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ x x ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ (cid:15) (cid:15) O O O O / / / / o o , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ y y sssssssssssssssss (cid:15) (cid:15) O O + + ❱❱❱ O O / / / / | | ②②②②②② (cid:15) (cid:15) O O O O / / / / O O / / O O / / (cid:28) (cid:28) ✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾✾ Where again it is easy to check the remaining mutations will give us a maximal green sequence for P n . (cid:3) BUCHER AND MILLS
Theorem 4.4. [4, Theorem 4.1]
The quiver Q n has a maximal green sequence of γ ( f n , f n − , . . . , f ) σ n σ n − . . . σ γ ( f , f , f . . . f n ) τ n τ n − . . . τ . We now prove our main result. To assist in reading the proof we provide a running example ofthe maximal green sequence for Q . Proof of Theorem 4.2.
By Lemma 4.3 and the proof of Theorem 4.4 in [4] we know that γ ( f n +2 f n +1 · · · f ) σ n · · · σ α α · · · α p − is a green sequence. After performing this mutation sequence we have the that all of the verticesare red except for the vertices f · · · f n +2 . Furthermore, all of these vertices except for f n +1 forman ( n + 1)-cycle . f f f f f g e e e g g g d a b c d a b c b a d c g g g f ′ f ′ f ′ g ′ g ′ e ′ e ′ g ′ e ′ g ′ d ′ a ′ b ′ c ′ a ′ d ′ b ′ c ′ g ′ g ′ g ′ (cid:4) (cid:4) ✡✡ / / i i ❙❙❙❙❙❙❙❙❙❙ c c ●●●●●●● (cid:20) (cid:20) w w ♦♦♦♦♦♦ (cid:2) (cid:2) ✆✆ , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ o o Z Z ✹✹ (cid:26) (cid:26) ✹✹ , , ❨❨❨❨❨❨❨❨❨ , , ❨❨❨❨❨❨❨❨❨ ♦♦♦♦♦♦♦ \ \ ✾✾✾✾✾✾✾✾✾✾ ❡❡❡❡ r r ❡❡❡❡ ❦❦❦❦ u u ❦❦❦❦ ▼▼ f f ▼▼ ✩✩✩✩✩✩✩✩ Q Q ✩✩✩✩✩✩✩✩ ✰✰✰✰✰✰ U U ✰✰✰✰✰✰ { { ✇✇✇✇ s s ❣❣❣❣❣❣ , , ❨❨❨❨❨❨❨❨❨ ✇✇✇✇ ; ; ✇✇✇✇ ❙❙❙❙❙❙❙ ) ) ❙❙❙❙❙❙❙ ✄✄✄✄✄✄ A A ✄✄✄✄✄ ✍✍✍✍✍✍✍✍✍✍ F F ✍✍✍✍✍✍✍✍✍✍ ✡✡✡✡✡✡✡✡ D D ✡✡✡✡✡✡✡✡ n n ❭❭❭❭❭❭❭ (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦ ` ` ❢❢❢ g g ◆◆◆ (cid:4) (cid:4) . . ❭❭❭❭❭❭❭❭ ❵❵❵❵❵❵❵❵❵❵❵❵❵ Z Z ✹✹ ❦❦❦❦❦❦❦❦❦❦ (cid:10) (cid:10) ✔✔✔✔✔✔✔ O O ❦❦❦ ) ) ❙❙❙ ♦♦♦♦♦♦♦ g g PPPPPPP Z Z ✹✹ (cid:8) (cid:8) ✑✑✑✑ y y rrr ❞❞❞❞❞❞ U U ✰✰✰✰✰✰ t t r r ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ h h ❘❘ r r ❡❡❡❡❡❡❡❡❡ { { ✇✇✇✇ ) ) ❙❙❙ (cid:29) (cid:29) ❀❀❀❀ - - ❩❩❩❩❩❩❩❩❩❩❩❩ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ s s ❣❣❣❣❣❣ j j ❯❯❯❯ (cid:24) (cid:24) ✵✵✵✵✵✵✵✵✵✵ c c v v ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ O O l l ❨❨❨❨❨❨❨❨❨ y y ssssssssssssssssss (cid:26) (cid:26) ✹✹✹✹✹✹✹✹ o o i i ❙❙❙ (cid:4) (cid:4) ✡✡✡✡✡✡✡✡ O O ❦❦❦ k k ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ (cid:15) (cid:15) ●●●● (cid:29) (cid:29) ❀❀❀❀❀ s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ / / " " ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ : : ✈✈✈✈ O O (cid:27) (cid:27) ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ & & / / (cid:6) (cid:6) ☞☞☞☞☞☞☞☞ (cid:28) (cid:28) ✾✾✾✾✾✾✾✾✾✾ By Lemma 4.1 our next section of the maximal green sequence is a green sequence for this cycle.Performing this cycle will make the e , . . . , e n , and g green. f f f f f g e e e g g g d a b c d a b c d a b c g g g g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ o o (cid:7) (cid:7) ✎✎ / / ♦♦ h h ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ ' ' ❖❖ / / . . ❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭❭ i i ❚❚❚❚❚ _ _ ❄❄❄ o o ❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ●●●●●●●●● I I ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ ❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜ C C ✞✞✞✞✞✞✞✞ G G ✎✎✎✎✎✎✎✎✎✎✎✎✎ : : ttttt v v ♠♠♠♠♠♠♠ $ $ ■■■■■■■■■■■■■■ D D ✠✠✠✠ (cid:15) (cid:15) ✣✣✣✣✣ W W ✵✵✵✵✵✵✵ (cid:14) (cid:14) ✢✢ (cid:27) (cid:27) ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ $ $ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ * * ❯❯❯❯❯❯❯❯❯❯❯❯ G G ✎✎✎ $ $ ❏❏❏❏❏ ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ : : ttttt (cid:23) (cid:23) ✴✴✴ u u ❦❦❦❦❦ ●●● O O ✥✥✥ w w ♥♥♥♥♥♥♥ (cid:31) (cid:31) ❄❄❄❄❄❄ O O s s ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ O O j j ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ (cid:26) (cid:26) ✹✹✹✹✹✹ o o g g ❖❖ (cid:4) (cid:4) ✡✡✡✡✡✡ O O ♦♦ (cid:26) (cid:26) ✹✹✹✹✹✹ o o g g ❖❖ (cid:4) (cid:4) ✡✡✡✡✡✡ O O ♦♦ (cid:26) (cid:26) ✹✹✹✹✹✹ o o g g ❖❖ (cid:4) (cid:4) ✡✡✡✡✡✡ O O ♦♦ U U ✰✰✰✰✰✰✰✰✰✰✰✰ B B ✝✝✝✝✝✝✝✝✝✝✝ O O ✥✥ w w ♣♣♣♣♣♣♣ w w ♣♣♣♣♣♣♣ After mutating at f n +1 and g p − we have a similar situation as we did at the end of the proofof Lemma 4.3. Note that in our running example of Q that g p − = g , and is not shown to be GS FOR THE N-TORUS 9 green in any picture. The remaining vertices that we mutate along in the β subsequence form anequioriented affine subquiver. It is easy to follow that this is a green sequence that will make the P p − subquiver of Q pn red. f f f f f g e e e g g g d a b c d a b c d a b c g g g g ′ g ′ g ′ g ′ g ′ g ′ g ′ g ′ o o (cid:6) (cid:6) ✌✌ / / ♠♠ i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ( ( ◗◗ / / . . ❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪❪ j j ❯❯❯❯❯ a a ❇❇❇ o o ❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴ $ $ ■■■■■■■■ H H ✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒ ❣❣❣❣ B B ☎☎☎☎☎☎☎ F F ✌✌✌✌✌✌✌✌✌✌✌✌ : : ttttt / / ❴❴❴❴❴❴❴❴❴❴❴❴ p p ❵❵❵❵❵❵❵ u u ❦❦❦❦❦❦❦ e e ❑❑❑❑❑❑❑❑❑❑❑❑❑❑ O O ✣✣✣✣ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ N N ✜✜ (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ - - ❬❬❬❬❬❬❬❬❬❬❬❬ & & ▲▲▲▲▲▲▲▲▲▲▲▲▲ ! ! ❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇ * * ❯❯❯❯❯❯❯❯❯❯❯❯ F F ✌✌ % % ▲▲▲▲▲ = = ⑤⑤⑤ ! ! ❇❇❇ rrrrr (cid:24) (cid:24) ✶✶ t t ❥❥❥❥❥ d d ■■■ O O ✥✥ u u ❦❦❦❦❦❦❦ ! ! ❈❈❈❈❈❈ O O ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ O O k k ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ (cid:27) (cid:27) ✼✼✼✼✼ o o h h ◗◗ (cid:3) (cid:3) ✞✞✞✞✞ O O ♠♠ (cid:27) (cid:27) ✼✼✼✼✼ o o h h ◗◗ (cid:3) (cid:3) ✞✞✞✞✞ O O ♠♠ (cid:27) (cid:27) ✼✼✼✼✼ o o h h ◗◗ (cid:3) (cid:3) ✞✞✞✞✞ O O ♠♠ r r ❢❢❢❢❢❢❢❢❢❢❢❢ O O ✥✥ w w ♥♥♥♥♥♥♥ w w ♥♥♥♥♥♥♥ Finally, the only remaining green vertices are e i for i = 1 , . . . , n . By inspection of the localconfiguration of the frozen vertices we see that this is the exact same configuration as in the twicepunctured case. Therefore it follows from the proof of Theorem 4.4. That τ i is a green sequencefor our quiver, and concluding the proof that our green sequence is maximal. (cid:3) Future Interests
In light of Theorem 1.2 we know of particular triangulations of surfaces that have maximalgreen sequences. Muller in [13] gives an example of a quiver with a maximal green sequence anda mutation equivalent quiver which does not have a maximal green sequence. The question ofwhether every triangulation of a surface has a maximal green sequence still remains open in manycases. It is worthwhile to point out that the example given in [13] is not one of a quiver thatis associated to a surface. It is the belief of the authors that when dealing with cluster algebrasthat arise from surfaces that the existence of a maximal green sequence may in fact be a mutationinvariant.
Conjecture 5.1.
Let Q be a quiver associated to a surface. If Q exhibits a maximal green sequence,then any Q ′ mutation equivalent to Q also exhibits a maximal green sequence. The question of existence of maximal green sequences for quivers that are not associated tosurfaces is still largely open.
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Mathematics Department, Louisiana State University, Baton Rouge, Louisiana
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