On the Structure of Oriented Exchange Graphs

Abstract

The exchange graph of a quiver is the graph of mutation-equivalent quivers whose edges correspond to mutations. The exchange graph admits a natural acyclic orientation called the oriented exchange graph. Oriented exchange graphs arise in many areas of mathematics including representation theory, algebraic combinatorics, and noncommutative algebraic geometry. In representation theory, an oriented exchange graph is isomorphic to a poset of certain torsion classes of a finite dimensional algebra. Of particular interest to mathematicians and string theorists are the finite length maximal directed paths in oriented exchange graphs, which are known as maximal green sequences. Maximal green sequences were introduced to obtain quantum dilogarithm identities and combinatorial formulas for refined Donaldson-Thomas invariants. They were also used in supersymmetric gauge theory to compute the complete spectrum of BPS states. For quivers mutation-equivalent to an orientation of a type A Dynkin diagram, we show that the oriented exchange graphs can realized as quotients of other posets of representation theoretic objects. For the same class of quivers we also show how to explicitly construct some of their maximal green sequences.