Quivers of Finite Mutation Type and Skew-symmetric Matrices

Quivers of finite mutation type have been classified recently in [4]. Main examples of these quivers are the quivers associated with triangulations of surfaces as introduced in [5]. They are also closely related to the representation theory of algebras [1]. In this paper, we study structural properties of finite mutation type quivers. We determine a class of subquivers, which we call basic quivers, and show that they have a natural algebraic interpretation in terms of the corresponding skew-symmetric matrices. In particular, we obtain a characterization of finite mutation type quivers that are associated with triangulations of surfaces and give a new numerical invariant for their mutation classes. We also prove a statement (Lemma 3.3) which was obtained in [4] using a computer program. To state our results, we need some terminology. Formally, a quiver is a pair Q = (Q0, Q1) where Q0 is a finite set of vertices and Q1 is a set of arrows between them. It is represented as a directed graph with the set of vertices Q0 and a directed edge for each arrow. In this paper, we are more concerned with the number of arrows between the vertices rather than the arrows themselves, so by a quiver we mean a directed graph Q, with no loops or 2-cycles, whose edges are weighted with positive integers. If the weight of an edge is 1, we do not specify it in the picture and call it a single edge; if an edge has weight 2 we call it a double edge for convenience. If all edges of Q are single edges, we call Q ”simply-laced”. By a subquiver of Q, we always mean a quiver obtained from Q by taking an induced (full) directed subgraph on a subset of vertices and keeping all its edge weights the same as in Q. For a quiver Q with vertices 1, ..., n, there is the uniquely associated skewsymmetric matrix B = B defined as follows: for each edge {i, j} directed from i to j, the entry Bi,j is the corresponding weight; if i and j are not connected to each other by an edge then Bi,j = 0. Recall from [6] that for each vertex k, the mutation of the quiver Q in direction k transforms Q to the quiver Q = μk(Q) whose corresponding skew-symmetric matrix B = B ′ is the following: B i,j = −Bi,j if i = k or j = k; otherwise B i,j = Bi,j + sgn(Bi,k)[Bi,kBk,j ]+ (where we use the notation [x]+ = max{x, 0} and sgn(x) = x/|x| with sgn(0) = 0). The operation μk is involutive, so it defines a mutation-equivalence relation on quivers (or equivalently on skew-symmetric matrices). A quiver Q is said to be of ”finite mutation type” if its mutation-equivalence class is finite. It is well known that any edge in a finite mutation type quiver with at least three vertices is a single edge or a double edge; any subquiver is also of finite mutation type. The most basic examples of finite