The Number of Elements in the Mutation Class of a Quiver of Type Dn

We show that the number of quivers in the mutation class of a quiver of Dynkin type Dn is given by ∑ d|n φ(n/d) ( 2d d ) /(2n) for n ≥ 5. To obtain this formula, we give a correspondence between the quivers in the mutation class and certain rooted trees. Introduction Quiver mutation is an important ingredient in the definition of cluster algebras [FZ1]. It is an operation on quivers, which induces an equivalence relation on the set of quivers. The mutation class M of a quiver Q consists of all quivers mutation equivalent to Q. If Q is a Dynkin quiver, then M is finite. In [T] an excplicit formula for |M| is given for Dynkin type An. Here we give an explicit formula for the number of quivers in the mutation class of a quiver of Dynkin type Dn. The formula is given by d(n) = { ∑ d|n φ(n/d) ( 2d d ) /(2n) if n ≥ 5, 6 if n = 4, where φ is the Euler function. The proof for this formula consists of two parts. The first part shows that the mutation class of type Dn is in 1–1 correspondence with the triangulations (with tagged edges) of the electronic journal of combinatorics 16 (2009), #R49 1 a punctured n-gon, up to rotation and inversion of tags. This is a generalization of the method used in [T] to count the number of elements in the mutation class of quivers of Dynkin type An. Here we are strongly using the ideas in [FST] and [S]. In the second part we count the number of (equivalence classes of) triangulations of a punctured n-gon, by describing an explicit correspondence to a certain class of rooted trees. A tree in this class is constructed by taking a family of full binary trees T1, . . . , Ts such that the total number of leaves is n, and then adding a node S and an edge from this node to the root of Ti for each i, such that S becomes a root (Figure 21 displays all such trees for n = 5). When these rooted trees are considered up to rotation at the root, they are in 1– 1 correspondence with the above mentioned equivalence classes of triangulations of the punctured n-gon. To count these rooted trees we use a simple adaption of a known formula found in [I] and [St, exercise 7.112 b]. We also point out a mutation operation on these rooted trees, corresponding to the other mutation operations involved (on triangulations and on quivers). Our formula and the bijection to triangulations of the punctured n-gon were presented at the ICRA in Torun, August 2007 [T2]. After completing our work, we learnt about the paper [GLZ]. They also generalize the methods in [T] to prove the bijection from the mutation class of Dn to triangulations of the punctured n-gon. However, their method of counting triangulations is very different from ours. They use the classification of quivers of mutation type Dn, recently given in [V]. The authors of [GLZ] end up with a very different formula than ours. In particular, their formula is not explicit, and it seems they get a different output than we get, e.g. for n = 6. We are grateful to Hugh Thomas for several useful discussions and for the idea of making use of binary trees as an alternative to rooted planar trees. We would also like to thank Dagfinn Vatne for useful discussions. 1 Quiver mutation Let Q be a quiver with no multiple arrows, no loops and no oriented cycles of length two. Mutation of Q at the vertex k gives a quiver Q obtained from Q in the following way. 1. Add a vertex k. 2. If there is a path i → k → j, then if there is an arrow from j to i, remove this arrow. If there is no arrow from j to i, add an arrow from i to j. 3. For any vertex i replace all arrows from i to k with arrows from k to i, and replace all arrows from k to i with arrows from i to k. 4. Remove the vertex k. the electronic journal of combinatorics 16 (2009), #R49 2 It is easy to see that mutating Q twice at k gives Q. We say that two quivers Q and Q are mutation equivalent if Q can be obtained from Q by a finite number of mutations. The mutation class of Q consists of all quivers mutation equivalent to Q. Figure 1 gives all quivers in the mutation class of D4, up to isomorphism. •4 •1 // •2 // •3 •4 •1 // •2 •3 oo •4 •1 •2 oo // •3 •4