Representation Theory Seminar

ABSTRACT: The exchange graph of a quiver is the graph whose vertices are mutation-equivalent quivers and whose edges correspond to single mutations connecting two quivers. The exchange graph admits a natural acyclic orientation called the oriented exchange graph. Its maximal directed paths of finite length are in bijection with maximal green sequences, which are of interest to representation theorists and string theorists. Perhaps surprisingly, each oriented exchange graph is isomorphic to the poset of functorially finite torsion classes of a Jacobian algebra. Using this isomorphism, we show that oriented exchange graphs with finitely many vertices are semidistributive lattices and that certain oriented exchange graphs with finitely many vertices are obtained by a lattice quotient of the lattice of biclosed subcategories, which we will introduce in this talk. This is joint work with Thomas McConville.