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1	Fadenmoduln über Ãn und Cluster-Kombinatorik / String modules over Ãn and cluster combinatorics Warkentin, Matthias 22 August 2012 (has links) (PDF) Inspired by work of Hubery [Hub] and Fomin, Shapiro and Thurston [FST06] related to cluster algebras, we construct a bijection between certain curves on a cylinder and the string modules over a path algebra of type Ãn. We show that under this bijection irreducible maps and the Auslander-Reiten translation have a geometric interpretation. Furthermore we prove that the dimension of extension groups can be expressed in terms of intersection numbers. Finally we explain the connection to cluster algebras and apply our results to describe the exchange graph in type Ãn. / Angeregt durch Arbeiten zu Cluster-Algebren von Hubery [Hub] und Fomin, Shapiro und Thurston [FST06] konstruieren wir eine Bijektion zwischen gewissen Kurven auf einem Zylinder und den Fadenmoduln über einer Wege-Algebra vom Typ Ãn. Wir zeigen, daß unter dieser Bijektion sowohl irreduzible Abbildungen als auch die Auslander-Reiten-Verschiebung eine geometrische Interpretation haben. Weiterhin beweisen wir, daß sich die Dimension der Erweiterungsgruppen mittels Anzahlen von Schnittpunkten ausdrücken läßt. Schließlich erklären wir die Verbindung zu Cluster-Algebren und verwenden unsere Ergebnisse um den Austauschgraph im Typ Ãn zu beschreiben. Cluster-Algebra Triangulation Kreisring orientierte Kurven Schnittpunktzahlen Fadenmodul Kippmodul Austauschgraph cluster algebra triangulation annulus oriented curves intersection numbers string module tilting module exchange graph ddc:512 Darstellungstheorie Triangulation Kreisring
2	Fadenmoduln über Ãn und Cluster-Kombinatorik / String modules over Ãn and cluster combinatorics Warkentin, Matthias 22 December 2008 (has links) Inspired by work of Hubery [Hub] and Fomin, Shapiro and Thurston [FST06] related to cluster algebras, we construct a bijection between certain curves on a cylinder and the string modules over a path algebra of type Ãn. We show that under this bijection irreducible maps and the Auslander-Reiten translation have a geometric interpretation. Furthermore we prove that the dimension of extension groups can be expressed in terms of intersection numbers. Finally we explain the connection to cluster algebras and apply our results to describe the exchange graph in type Ãn. / Angeregt durch Arbeiten zu Cluster-Algebren von Hubery [Hub] und Fomin, Shapiro und Thurston [FST06] konstruieren wir eine Bijektion zwischen gewissen Kurven auf einem Zylinder und den Fadenmoduln über einer Wege-Algebra vom Typ Ãn. Wir zeigen, daß unter dieser Bijektion sowohl irreduzible Abbildungen als auch die Auslander-Reiten-Verschiebung eine geometrische Interpretation haben. Weiterhin beweisen wir, daß sich die Dimension der Erweiterungsgruppen mittels Anzahlen von Schnittpunkten ausdrücken läßt. Schließlich erklären wir die Verbindung zu Cluster-Algebren und verwenden unsere Ergebnisse um den Austauschgraph im Typ Ãn zu beschreiben. info:eu-repo/classification/ddc/512 ddc:512
3	Exchange Graphs via Quiver Mutation Warkentin, Matthias 02 October 2014 (has links) (PDF) Inspired by Happel's question, whether the exchange graph and the simplicial complex of tilting modules over a quiver algebra are independent from the multiplicities of multiple arrows in the quiver, we study quantitative aspects of Fomin and Zelevinsky's quiver mutation rule. Our results turn out to be very useful in the mutation-infinite case for understanding combinatorial structures as the cluster exchange graph or the simplicial complex of tilting modules, which are governed by quiver mutation. Using a class of quivers we call forks we can show that any such quiver yields a tree in the exchange graph. This allows us to provide a good global description of the exchange graphs of arbitrary mutation-infinite quivers. In particular we show that the exchange graph of an acyclic quiver is a tree if (and in fact only if) any two vertices are connected by at least two arrows. Furthermore we give classification results for the simplicial complexes and thereby obtain a partial positive answer to Happel's question. Another consequence of our findings is a confirmation of Unger's conjecture about the infinite number of components of the tilting exchange graph in all but finitely many cases. Finally we generalise and conceptualise our results by introducing what we call "polynomial quivers", stating several conjectures about "polynomial quiver mutation", and giving proofs in special cases. Köchermutation Austauschgraph Baum Cluster-Algebra Cluster-Komplex Kippmodul simplizialer Komplex Polynomköcher Astgabel mutations-unendlich Mehrfachpfeil quiver mutation exchange graph tree cluster algebra cluster complex tilting module simplicial complex polynomial quiver fork mutation-infinite multiple arrow ddc:512 ddc:514 Kombinatorik Darstellungstheorie Simplizialer Komplex Graph Baum
4	Exchange Graphs via Quiver Mutation Warkentin, Matthias 11 June 2014 (has links) Inspired by Happel's question, whether the exchange graph and the simplicial complex of tilting modules over a quiver algebra are independent from the multiplicities of multiple arrows in the quiver, we study quantitative aspects of Fomin and Zelevinsky's quiver mutation rule. Our results turn out to be very useful in the mutation-infinite case for understanding combinatorial structures as the cluster exchange graph or the simplicial complex of tilting modules, which are governed by quiver mutation. Using a class of quivers we call forks we can show that any such quiver yields a tree in the exchange graph. This allows us to provide a good global description of the exchange graphs of arbitrary mutation-infinite quivers. In particular we show that the exchange graph of an acyclic quiver is a tree if (and in fact only if) any two vertices are connected by at least two arrows. Furthermore we give classification results for the simplicial complexes and thereby obtain a partial positive answer to Happel's question. Another consequence of our findings is a confirmation of Unger's conjecture about the infinite number of components of the tilting exchange graph in all but finitely many cases. Finally we generalise and conceptualise our results by introducing what we call "polynomial quivers", stating several conjectures about "polynomial quiver mutation", and giving proofs in special cases. info:eu-repo/classification/ddc/512 ddc:512 info:eu-repo/classification/ddc/514 ddc:514

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