New Graphs of Finite Mutation Type

To a directed graph without loops or 2-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context 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 with marked points. We will call such graphs “of geometric type”. Besides graphs with 2 vertices, and graphs of geometric type, there are only 9 other “exceptional” mutation classes that are known to be finite. In this paper we introduce 2 new exceptional finite mutation classes. Cluster algebras were introduced by Fomin and Zelevinsky in [5, 6] to create an algebraic framework for total positivity and canonical bases in semisimple algebraic groups. An n × n matrix B = (bi,j) is called skew symmetrizable if there exists nonzero d1, d2, . . . , dn such that dibi,j = −djbj,i for all i, j. An exchange matrix is a skewsymmetrizable matrix with integer entries. A seed is a pair (x, B) where B is an exchange matrix and x = {x1, x2, . . . , xn} is a set of n algebraically independent elements. For any k with 1 ≤ k ≤ n we define another ∗This first author is partially supported by NSF grant DMS 0349019. This grant also supported the REU research project of the second author on which this paper is based. the electronic journal of combinatorics 15 (2008), #R139 1 seed (x, B) = μk(x, B) as follows. The matrix B ′ = (b′i,j) is given by b′i,j = { −bi,j if i = k or j = k; bi,j + [bi,k]+[bk,j]+ − [−bi,k]+[−bk,j]+ otherwise. Here, [z]+ = max{z, 0} denotes the positive part of a real number z. Define x′ = {x1, x2, . . . , xk−1, x ′ k, xk+1, . . . , xn} where x′k is given by x′k = ∏n i=1 x [bi,k]+ i + ∏n i=1 x [−bi,k]+ i xk . Note that μk is an involution. Starting with an initial seed (x, B) one can construct many seeds by applying sequences of mutations. If (x, B) is obtained from the initial seed (x, B) by a sequence of mutations, then x is called a cluster, and the elements of x are called cluster variables. The cluster algebra is the commutative subalgebra of Q(x1, x2, . . . , xn) 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 were classified in [6]. Example 1 (Cluster algebra of type A1) If we start with the initial seed (x, B) where x = {x1, x2} and B = ( 0 −1 1 0 ) Using mutations we get {x1, x2}, ( 0 1 −1 0 ) ↔ { 1 + x2 x1 , x2}, ( 0 −1 1 0 ) ↔ { 1 + x2 x1 , 1 + x1 + x2 x1, x2 }, ( 0 1 −1 0 ) ↔ ↔ { 1 + x1 x2 , 1 + x1 + x2 x1, x2 }, ( 0 −1 1 0 ) ↔ { 1 + x1 x2 , x1} ( 0 1 −1 0 ) ↔ {x2, x1}, ( 0 −1 1 0 ) The last seed {x2, x1}, ( 0 −1 1 0 ) is considered the same as the initial seed. We just need to exchange x1 and x2 (and accordingly swap the 2 rows and swap the 2 columns in the exchange matrix) to get the initial seed. A cluster algebra is called mutation-finite if only finitely many exchange matrices appear in the seeds. Obviously a cluster algebra of finite type is mutation-finite. But the converse is not true. For example, the exchange matrix B = ( 0 −2 2 0 ) the electronic journal of combinatorics 15 (2008), #R139 2 gives a cluster algebra that is not of finite type. However, the only exchange matrix that appears is B (and −B, but −B is the same as B after swapping the 2 rows and swapping the 2 columns). In this paper we will only consider exchange matrices that are already skew-symmetric. To a skew-symmetric n × n matrix B = (bi,j) we can associate a directed graph Γ(B) as follows. The vertices of the graph are labeled by 1, 2, . . . , n. If bi,j > 0, draw bi,j arrows from j to i. Any finite directed graph without loops or 2 cycles can be obtained from a skew-symmetric exchange matrix in this way. We can understand mutations in terms of the graph. If Γ = Γ(B) then μkΓ := Γ(μkB) is obtained from Γ as follows. Start with Γ. For every incoming arrow a : i → k at k and every outgoing arrow b : k → j, draw a new composition arrow ba : i → j. Then, revert every arrow that starts or ends at k. The graph now may have 2-cycles. Discard 2-cycles until there are now more 2-cycles left. The resulting graph is μkΓ. Two 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 will always mean a full subgraph, i.e., for every two vertices x, y in the subgraph, the subgraph also will contain all arrows from x to y. 1 Known 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 will discuss all known examples of graphs of finite mutation type. 1.1 Connected graphs with 2 vertices Let Θ(m) be the graph with two vertices 1, 2 and m ≥ 1 arrows 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 exchange matrix of a cluster algebra of finite type is mutation finite. The cluster algebras of finite type were classified in [6]. This classification goes parallel to the classification of simple Lie algebras, there are types An,Bn,Cn,Dn,E6,E7,E8,F4,G2. the electronic journal of combinatorics 15 (2008), #R139 3 The types with a skew-symmetric exchange graph correspond to the simply laced Dynkin diagrams An,Dn,E6,E7,E8: An : • // • // · · · // • Dn : • • // • // · · · // • E6 : • • // • // • • oo • oo