Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Sortable Elements for Quivers with Cycles. | Sortable Elements for Quivers with Cycles Nathan Reading David E Speyeh Department of Mathematics Department of Mathematics North Carolina State University USA Massachusetts Institute of Technology USA nathan_reading@ speyer@ Submitted Sep 22 2009 Accepted Jun 8 2010 Published Jun 14 2010 Mathematics Subject Classification 20F55 Abstract Each Coxeter element c of a Coxeter group W defines a subset of W called the c-sortable elements. The choice of a Coxeter element of W is equivalent to the choice of an acyclic orientation of the Coxeter diagram of W. In this paper we define a more general notion of Q-sortable elements where Q is an arbitrary orientation of the diagram and show that the key properties of c-sortable elements carry over to the Q-sortable elements. The proofs of these properties rely on reduction to the acyclic case but the reductions are nontrivial in particular the proofs rely on a subtle combinatorial property of the weak order as it relates to orientations of the Coxeter diagram. The c-sortable elements are closely tied to the combinatorics of cluster algebras with an acyclic seed the ultimate motivation behind this paper is to extend this connection beyond the acyclic case. 1 Introduction The results of this paper are purely combinatorial but are motivated by questions in the theory of cluster algebras. To define a cluster algebra one requires the input data of a skew-symmetrizable integer matrix that is to say an n X n integer matrix B and a vector of positive integers Ố1 . ỏn such that ỏiBij ỏjBji. For the experts we are discussing cluster algebras without coefficients. This input data defines a recursion which produces among other things a set of cluster variables. Each cluster variable is a rational function in x1 . xn and the cluster variables are grouped into overlapping sets of size n called clusters. The cluster algebra is the algebra generated as a ring by the cluster variables. Partially supported by NSA grant .