Lattice Properties of Oriented Exchange Graphs and Torsion Classes
نویسندگان
چکیده
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. Building on work of Iyama, Reiten, Thomas, and Todorov, we show that this directed graph is a semidistributive lattice by using the isomorphism to the lattice of functorially finite torsion classes of the corresponding cluster-tilted algebra when the exchange graph is finite. Furthermore, if the quiver is mutation-equivalent to a type A Dynkin diagram or is an oriented cycle, then the oriented exchange graph is a lattice quotient of a lattice of biclosed subcategories of modules over the cluster-tilted algebra, generalizing Reading’s Cambrian lattices in type A. We also apply our results to address a conjecture of Brüstle, Dupont, and Pérotin on the lengths of maximal green sequences.
منابع مشابه
Minimal Inclusions of Torsion Classes
Let Λ be a finite-dimensional associative algebra. The torsion classes of mod Λ form a lattice under containment, denoted by tors Λ. In this paper, we characterize the cover relations in tors Λ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection...
متن کاملLattices in graphs with polynomial growth
We present results on the structure of graphs with polynomial growth. For certain classes of these graphs we prove that they contain subgraphs which are similar to the lattice graph Ld. These results are then applied to investigate isoperimetric properties of certain classes of graphs with polynomial growth. In addition these structural results can also be applied to percolation problems.
متن کاملNew skew equienergetic oriented graphs
Let $S(G^{sigma})$ be the skew-adjacency matrix of the oriented graph $G^{sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $sigma$ to each of its edges. The skew energy of an oriented graph $G^{sigma}$ is defined as the sum of absolute values of all eigenvalues of $S(G^{sigma})$. Two oriented graphs are said to be skew equienergetic iftheir skew energies...
متن کاملOn varieties of orgraphs
In this paper we investigate varieties of orgraphs (that is, oriented graphs) as classes of orgraphs closed under isomorphic images, suborgraph identifications and induced suborgraphs, and we study the lattice of varieties of tournament-free orgraphs.
متن کاملIntersection graphs associated with semigroup acts
The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the fi...
متن کامل