TAILIEUCHUNG - Báo cáo toán học: "Sortable Elements for Quivers with Cycles"

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 .

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