New Graphs of Finite Mutation Type

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1 Ne Graphs of Finite Mutation Type Harm Derksen Department of Mathematics University of Michigan Theodore Oen Department of Mathematics Ioa State University Submitted: Apr 21, 2008; Accepted: Nov 3, 2008; Published: Nov 14, 2008 Mathematics Subject Classification: 05E99 Abstract To a directed graph ithout loops or 2-cycles, e can associate a ske-symmetric matri ith integer entries. Mutations of such ske-symmetric matrices, and more generally ske-symmetrizable matrices, have been defined in the contet of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph Γ is the set of all isomorphism classes of graphs that can be obtained from Γ by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces ith marked points. We ill call such graphs of geometric type. Besides graphs ith 2 vertices, and graphs of geometric type, there are only 9 other eceptional mutation classes that are knon to be finite. In this paper e introduce 2 ne eceptional finite mutation classes. Cluster algebras ere introduced by Fomin and Zelevinsky in [5, 6] to create an algebraic frameork for total positivity and canonical bases in semisimple algebraic groups. An n n matri B = (b i,j ) is called ske symmetrizable if there eists nonzero d 1, d 2,..., d n such that d i b i,j = d j b j,i for all i, j. An echange matri is a skesymmetrizable matri ith integer entries. A seed is a pair (, B) here B is an echange matri and = { 1, 2,..., n } is a set of n algebraically independent elements. For any k ith 1 k n e define another This first author is partially supported by NSF grant DMS This grant also supported the REU research project of the second author on hich this paper is based. the electronic journal of combinatorics 15 (2008), #R139 1

2 seed (, B ) = µ k (, B) as follos. The matri B = (b i,j) is given by { b i,j = bi,j if i = k or j = k; b i,j + [b i,k ] + [b k,j ] + [ b i,k ] + [ b k,j ] + otherise. Here, [z] + = ma{z, 0} denotes the positive part of a real number z. Define here k is given by = { 1, 2,..., k 1, k, k+1,..., n } k = n i=1 [b i,k] + i + n i=1 [ b i,k] + i. k Note that µ k is an involution. Starting ith an initial seed (, B) one can construct many seeds by applying sequences of mutations. If (, B ) is obtained from the initial seed (, B) by a sequence of mutations, then is called a cluster, and the elements of are called cluster variables. The cluster algebra is the commutative subalgebra of Q( 1, 2,..., n ) generated by all cluster variables. A cluster algebra is called of finite type if there are only finitely many seeds that can be obtained from the initial seed by sequences of mutations. Cluster algebras of finite type ere classified in [6]. Eample 1 (Cluster algebra of type A 1 ) If e start ith the initial seed (, B) here = { 1, 2 } and ( ) 0 1 B = 1 0 Using mutations e get ( ) 0 1 { 1, 2 }, 1 0 { , , 2 }, The last seed { , 2 }, ( ) 0 1 { ( ) 0 1 { 1 + 1, } 2 { 2, 1 }, ( ) , }, 1 1, 2 ( ) 0 1 { 1 0 2, 1 }, ( ) ( ) is considered the same as the initial seed. We just need to echange 1 and 2 (and accordingly sap the 2 ros and sap the 2 columns in the echange matri) to get the initial seed. A cluster algebra is called mutation-finite if only finitely many echange matrices appear in the seeds. Obviously a cluster algebra of finite type is mutation-finite. But the converse is not true. For eample, the echange matri ( ) 0 2 B = 2 0 the electronic journal of combinatorics 15 (2008), #R139 2

3 gives a cluster algebra that is not of finite type. Hoever, the only echange matri that appears is B (and B, but B is the same as B after sapping the 2 ros and sapping the 2 columns). In this paper e ill only consider echange matrices that are already ske-symmetric. To a ske-symmetric n n matri B = (b i,j ) e can associate a directed graph Γ(B) as follos. The vertices of the graph are labeled by 1, 2,..., n. If b i,j > 0, dra b i,j arros from j to i. Any finite directed graph ithout loops or 2 cycles can be obtained from a ske-symmetric echange matri in this ay. We can understand mutations in terms of the graph. If Γ = Γ(B) then µ k Γ := Γ(µ k B) is obtained from Γ as follos. Start ith Γ. For every incoming arro a : i k at k and every outgoing arro b : k j, dra a ne composition arro ba : i j. Then, revert every arro that starts or ends at k. The graph no may have 2-cycles. Discard 2-cycles until there are no more 2-cycles left. The resulting graph is µ k Γ. To graphs are called mutation-equivalent, if one is obtained from the other by a sequence of mutations and relabeling of the vertices. The mutation class of a graph Γ is the set of all isomorphism classes of graphs that are mutation equivalent to Γ. A graph is mutation-finite if its mutation class is finite. Convention 2 In this paper, a subgraph of a directed graph Γ ill alays mean a full subgraph, i.e., for every to vertices, y in the subgraph, the subgraph also ill contain all arros from to y. 1 Knon mutation-finite connected graphs It is easy to see that a graph Γ is mutation-finite if and only if each of its connected components is mutation finite. We ill discuss all knon eamples of graphs of finite mutation type. 1.1 Connected graphs ith 2 vertices Let Θ(m) be the graph ith to vertices 1, 2 and m 1 arros from 1 to 2. The mutation class of Θ(m) is just the isomorphism class of Θ(m) itself. So Θ(m) is mutation-finite. Θ(3) : 1.2 Graphs from cluster algebras of finite type. An echange matri of a cluster algebra of finite type is mutation finite. The cluster algebras of finite type ere classified in [6]. This classification goes parallel to the classification of simple Lie algebras, there are types A n, B n, C n, D n, E 6, E 7, E 8, F 4, G 2. the electronic journal of combinatorics 15 (2008), #R139 3

4 The types ith a ske-symmetric echange graph correspond to the simply laced Dynkin diagrams A n, D n, E 6, E 7, E 8 : A n : D n : E 6 : E 7 : E 8 : The orientation of the arros here ere chosen somehat arbitrarily. For each diagram, a different choice of the orientation ill still give the same mutation class. 1.3 Graphs from etended Dynkin diagrams In [2] it as shon that a connected directed graph ithout oriented cycles is of finite mutation type if and only if it has at 2 vertices (the graphs Θ(m), m 1) or the underlying undirected graph is an etended Dynkin diagrams. The type D and E etended Dynkin diagrams give rise to the folloing finite mutation classes: D n : Ê 6 : the electronic journal of combinatorics 15 (2008), #R139 4

5 Ê 7 : Ê 8 : Again, for these types, a different choice for the orientations of the arros still give the same mutation class. The diagram for Ân is an (n + 1)-gon. If all arros go clockise or all arros go counterclockise, then e get the mutation class of D n. Let Âp,q be the mutation class of the graph here p arros go counterclockise and q arros go clockise, here p q 1. For the mutation class it does not matter hich arros are chosen to be counterclockise and ith ones are chosen counterclockise. 1.4 Graphs coming from triangulations of surfaces In [4] the authors construct cluster algebras from bordered oriented surfaces ith marked points. These cluster algebras are alays of finite mutation type. The echange matrices for these types are ske-symmetric. The mutation-finite graphs that come from oriented bordered surfaces ith marked points ill be called of geometric type. In 13 of that paper, the authors give a description of the graphs of geometric type. A block is one of the diagrams belo: I: II: IIIa: IIIb: IV: V: Start ith a disjoint union of blocks (every type may appear several times) and choose a partial matching of the open vertices (). No verte should be matched to a verte of the same block. Then construct a ne graph by identifying the vertices that are matched to each other. If in the resulting graph there are to vertices and y ith an arro from to y and an arro from y to, then e omit both arros (they cancel each other out). A graph constructed in this ay is called block decomposable. Fomin, Shapiro and Thurston prove in [4, 13] that a graph is block decomposable if and only if the graph is of geometric type. the electronic journal of combinatorics 15 (2008), #R139 5

6 For eample gives of type D 6. It is easy to see that all graphs of type A n, D n, Âp,q, D n are block decomposable. The partial matching yields the block decomposable graph the electronic journal of combinatorics 15 (2008), #R139 6

7 1.5 Graphs of etended affine types The folloing graphs are also of finite mutation type: E (1,1) 6 : 7 : E (1,1) E (1,1) 8 : These graphs are orientations of the Dynkin diagrams of etended affine roots systems first described by Saito (see [10, Table 1]). The connection beteen etended affine root systems and cluster combinatorics as first noticed by Geiss, Leclerc, and Schroër in [8]. It as shon using that these graphs are of finite mutation type by an ehaustive computer search using the Java applet for matri mutations ritten by Bernhard Keller ([9]) and Lauren Williams. 1.6 Summary All the knon quivers of finite mutation type can be summarized as follos: 1. graphs of geometric type, 2. graphs ith 2 vertices, 3. graphs in the 9 eceptional mutation classes E 6, E 7, E 8, Ê6, Ê7, Ê8, E (1,1) 6, E (1,1) 7, E (1,1) 8. Fomin, Shapiro and Thurston asked to folloing question (see [4, Problem 12.10]) 1. Problem 3 Are these all connected graphs of finite mutation type? If not, are there only finitely many eceptional finite mutation classes? In the net section, e ill introduce 2 ne mutation classes of finite type. 1 In the statement of Problem in [4], the authors accidentally rote n 2 instead of n 3. the electronic journal of combinatorics 15 (2008), #R139 7

8 2 Ne eceptional graphs of finite mutation-type Proposition 4 The folloing to graphs are of finite mutation type: X 6 : X 7 : Proof. This is easy to verify by hand or by using the applet [9]. The mutation classes for X 6 and X 7 are surprisingly small. The mutation class of X 6 consists of the folloing 5 graphs: the electronic journal of combinatorics 15 (2008), #R139 8

9 The mutation class of X 7 consists of the folloing 2 graphs: Corollary 5 The graphs X 6 and X 7 are not mutation-equivalent to E 6, E 7, E 8, Ê6, Ê7, Ê8, E (1,1) 6, E (1,1) 7, E (1,1) 8. Proof. The reader easily verifies that none of these 9 graphs are in the mutation classes of X 6 or X 7. Proposition 6 The graphs X 6 and X 7 are not block decomposable. In particular, these graphs do not come from oriented surfaces ith marked points. Proof. Suppose that X 6 is block decomposable. None of the blocks ill be of type V, since the block V contains a 4-cycle hich ill not vanish after the matching, and X 6 does not contain a 4-cycle. We label the vertices of X 6 as follos: y 1 z 1 y 2 z 2 (1) At verte there are 2 arros going out and 3 arros coming in. To form a graph ith this property from the blocks, e must either glue blocks II and IV along, or glue blocks IIIa and IV along. If e glue blocks IIIa and IV e get the folloing graph: here the open verte () may be matched further ith other blocks. Even after further matching, ill have at least 2 neighbors hich are only connected to. the electronic journal of combinatorics 15 (2008), #R139 9

10 If blocks II and IV are matched to form a verte, then e get the folloing graph (2) Here the open vertices can be matched further. Hoever, they cannot be matched among themselves, because this ould change the number of incoming and outgoing arros at. In X 6, has incoming arros from, z 1, z 2. So in (2), one of the vertices marked ith has to correspond to z 1 or z 2. But it is clear that even after further matching, the vertices marked ith ill only have 1 incoming arro. Contradiction. This shos that X 6 is not block decomposable. Therefore, X 6 does not come from an a triangulation of an oriented surface ith marked points. Since X 6 is a subgraph of X 7, X 7 does not come from a triangulation of an oriented surface ith marked points either. 3 Mutation-finite graphs containing X 6 or X 7 The folloing result as proven in [1]. We include the short proof for the reader s convenience. Theorem 7 The finite mutation classes of connected quivers ith 3 vertices are: A 3 : Â 2 : Z 3 : Proof. Suppose that Γ is a connected graph of finite mutation type ith 3 vertices. Assume that Γ, among all the graphs in its mutation class, has the largest possible number of arros. Without loss of generality e may assume that Γ is of the form 2 p q 1 r 3 (3) the electronic journal of combinatorics 15 (2008), #R139 10

11 here p, q, r 0 denote the number of arros. If Γ is not of the form (3), then it is of the form 2. (4) p q 1 r 3 and mutation at verte 2 ill not decrease the number of arros, and e obtain a graph of the form (3). Without loss of generality e may assume that In particular, p, q r. (5) p, q 1 (6) otherise the graph ould not be connected. After mutation at verte 2, e get 2 p q 1 pq r 3 (7) Since Γ had the maimal number of arros, e have pq r r, so pq 2r. (8) From r 2 pq 2r follos that r = 0, 1, 2. If r = 0, then pq = 0 by (8) hich contradicts (6). If r = 1, then pq = 1 or pq = 2 by (8). If pq = 1, then p = q = 1 hich yields type A 3. If pq = 2 then p = 2, q = 1 or p = 1, q = 2. In either case e get type Â2. If r = 2, then (5) and (8) imply that p = q = 2 and e obtain type Z 3. Corollary 8 If Γ is a graph of finite mutation type ith 3 vertices Then the number of arros beteen any 2 vertices is at most 2. Proof. Suppose and y are vertices of Γ ith p 1 arros from to y. Since Γ is connected, there eists a verte z that is connected to or y.the subgraph ith vertices, y, z is also of finite mutation type. From the classification in Theorem 7 it is clear that p 2. Definition 9 An obstructive sequence for a graph Γ is a sequence of vertices 1,..., l such that the mutated graph µ l µ 2 µ 1 Γ has to vertices ith at least 3 arros beteen them. By Corollary 8, a graph for hich an obstructive sequence eists cannot be of finite mutation type. the electronic journal of combinatorics 15 (2008), #R139 11

12 Lemma 10 If Γ is a mutation-finite connected graph ith 4 vertices, then Γ cannot contain Z 3 as a subgraph. Proof. Suppose Γ is a mutation-finite connected graph ith 4 vertices containing Z 3. Then Γ has a mutation-finite connected subgraph ith eactly 4 vertices containing Z 3. Without loss of generality e may assume that Γ has 4 vertices. We label the vertices of Z 3 as follos Let y be the 4-th verte of Γ. No y is connected to at least one of the vertices of Z 3, say to 1. Because the subgraph ith vertices {y, 1, 2 } is of finite mutation type, all arros must go from y to 1 and not in the opposite direction by Theorem 7. But because the subgraph ith vertices {y, 1, 3 } is of finite mutation type, all arros must go from 1 to y. Contradiction. Corollary 11 If Γ is a mutation-finite graph ith 4 vertices, then every connected subgraph ith 3 vertices must be of type A 3 or Â2. Theorem 12 The only connected mutation-finite quiver ith 7 vertices containing X 6 is X 7. Proof. Suppose that Γ is a mutation finite quiver containing X 6. We label the vertices of X 6 as shon in (1), and denote the other verte by u. Since Γ is connected, u is connected to at least 1 other verte. Case 1: Suppose that u is connected to 1 verte of X 6. If u is connected to y 1 or z 1 then the subgraph ith vertices {u, y 1, z 1 } is not of type A 3 or Â2, contradicting Corollary 11. Similarly u is not connected to y 2 or z 2. Suppose u is connected to. After mutation at u e may assume that arros go from u to. From Corollary 11 applied to the subgraph ith vertices {u,, } follos that there can be only 1 arro. No,, y 1, z 1,, u is a obstructive sequence. Suppose that u is connected to. Without loss of generality e may assume that arros go from u to. By Corollary 11 applied to the subgraph ith vertices {u,, } there is at most 1 arro. No,, y 1, y 2, z 1, z 2,,, y 2, is a obstructive sequence. attached to X 6 by only one set of arros. Case 2: Suppose that u is connected to 2 vertices. If u is connected to y 1 or z 1, then the only possibility is that there is one arro from u 1 to y 1 and one arro from z 1 to u 1. An obstructive sequence is u,, y 1, y 2.,,, z 2, y 1. Similarly, u cannot be connected to y 2 or z 2. Therefore, u must be connected to and. The only cases that avoid a connected subgraph ith 3 vertices not of type A 3 or Â2 are: (a) : u (b) : u (c) : u (d) : u the electronic journal of combinatorics 15 (2008), #R139 12

13 Case (a) reduces to Case 1 after mutation at verte u. Cases (b) and (c) give isomorphic graphs. Case (b) has an obstructive sequence,, u. Case (d) gives us the graph X 7. Case 3: 3 attachments Verte u must be attached to either vertices y 1 and z 1 or vertices y 2 and z 2. Without loss of generality, e may assume that u is connected to both y 1 and z 1. There is one arro from u to y 1 and one arro from z 1 to u. The verte u is also connected to either or. There are 4 cases: (a) : u (b) : u (c) : u (d) u Case (a) has the obstructive sequence u,, z 1. Case (b) has the obstructive sequence u,, z 1,,. In case (c) e have the obstructive sequence,, u, y 1. Case (d) has the obstructive sequence u,, z 2, y 1,. Case 4: Suppose that u is connected to 4 vertices. If u is connected to y 1, z 1, y 2, z 2 then there is an arro from u to y 1 and to y 2 and arros from z 1 and z 2 to u. An obstructive sequence is u,,, y 2, z 1. Otherise, u must be connected to either y 1 and z 1, or to y 2 and z 2, but not both. Without loss of generality e may assume that u is connected to y 1 and z 1. No u is also connected to and. The only cases that avoid a connected subgraph ith 3 vertices not of type A 3 or Â2 are: (a) : u (b) : u (c) : u (d) : u Case (a) has the obstructive sequence u,, y 1. Case (b) has the obstructive sequence u,, z 1. In case (c) e have the obstructive sequence u,. And case (d) is mutation equivalent to case (c) via the mutation at. Case 5: Suppose that u is connected to 5 of the vertices of X 6. Then u must be connected to y 1, z 1, y 2, z 2, ith arros going from u to y 1 and y 2 and arros going from z 1 and z 2 to u. No u must be connected to either or. There are 4 subcases: (a) : u (b) : u (c) : u (d) u Case (a) has the obstructive sequence u,. Case (b) has the obstructive sequence u,,. In case (c) e have the obstructive sequence u,. Case (d) has the obstructive sequence u,, y 1, y 2. Case 6: The verte u is connected to all 6 vertices of X 6. There must be arros from u to y 1 and y 2 and arros from z 1 and z 2 to u. The only possibilities to connect u to and that avoid a connected subgraph ith 3 vertices not of type A 3 or Â2 are: (a) : u (b) : u (c) : u (d) : u the electronic journal of combinatorics 15 (2008), #R139 13

14 Case (a) has the obstructive sequence, y 1, y 2. Case (b) has the obstructive sequence, z 1, z 2. Case 3 has the obstructive sequence, y 1, y 2. Case 4 has the obstructive sequence, z 1, z 2. Therefore the only connected mutation-finite quiver ith 7 vertices containing X 6 is X 7. Theorem 13 There is no connected mutation-finite quiver ith 8 vertices containing X 7. Proof. Suppose that Γ is a connected graph ith 8 vertices containing X 7. We ill sho that this ill lead to a contradiction. The graph Γ contains a connected subgraph ith eactly 8 vertices hich contains X 7. So ithout loss of generality e may assume that Γ has eactly 8 vertices. Let us label the vertices of X 7 as follos: z 1 y 1 z 3 y 2 z 2 Suppose that u is the 8-th verte of Γ. Because of the symmetry, e may assume, ithout loss of generality that u is connected to, y 1 or z 1. The subgraph ith vertices {, y 1, z 1, y 2, z 2, z 3, u} is connected and contains X 7. By Theorem 12 this graph must be isomorphic to X 7. This means that there must be 2 arros from u to z 3. No the subgraph ith vertices y 3, z 3, u cannot be of finite mutation type because of Theorem 7. Corollary 14 No graph of one of the 9 eceptional types contains X 6 or X 7 as a subgraph. 4 Conclusion We have ehibited to graphs hich are of finite mutation type, but hich are not of geometric type or isomorphic to the 9 knon eceptional cases. It is natural to ask hether there are any more eceptional graphs of finite mutation type. Problem 15 Is it true that every finite mutation class of graphs ith 3 vertices hich is not of geometric type is one of the folloing 11 mutation classes: E 6, E 7, E 8, Ê6, Ê7, Ê8, E (1,1) 6, E (1,1) 7, E (1,1) 8, X 6, X 7. Acknoledgements. The authors ould like to thank Ralf Spatzier for bringing them together in this REU project. The second author likes to thank Tracy Payne for getting him interested in the REU program, and Christine Betz Bolang for help getting into the REU program. y 3 the electronic journal of combinatorics 15 (2008), #R139 14

15 References [1] I. Assem, M. Blais, Th. Brüstle, A. Samson, Mutation classes of ske-symmetric 3 3 matrices, Comm. Algebra 36 (2008), no. 4, [2] A. B. Buan, I. Reiten, Acyclic quivers of finite mutation type, International Math. Research Notices 2006, Art. ID [3] A. Berenstein, S. Fomin, A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, [4] S. Fomin, M. Shapiro, D. Thurston, cluster algebras and triangulated surfaces part I: cluster complees, Acta Mathematica 201 (2008), no. 1, [5] S. Fomin, A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), [6] S. Fomin, A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), [7] S. Fomin, A. Zelevinsky, Cluster algebras IV: Coefficients, Compos. Math. 143 (2007), no. 2, [8] C. Geiss, B. Leclerc, J. Schröer, Semicanonical bases and preprojective algebras, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 2, [9] B. Keller, Quiver mutation in Java, [10] K. Saito, Etended affine root systems. I. Coeter transformations, Publ. Res. Inst. Math. Sci. 21 (1985), no. 1, the electronic journal of combinatorics 15 (2008), #R139 15