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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:20127 |
| Date | 11 June 2014 |
| Creators | Warkentin, Matthias |
| Contributors | Happel, Dieter, Schröer, Jan, Marsh, Robert, Martini, Horst, Technische Universität Chemnitz |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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