On the properties of the exchange graph of a cluster algebra
By Michael I. Gekhtman, Michael Shapiro, A. Vainshtein. 2007
59 citations recorded by the source corpus.
We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra $\A$ in two cases: when $\A$ is of geometric type and when $\A$ is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of $\A$. Both conjectures were formulated recently by Fomin and Zelevinsky.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/17c8d8d94cd8f65c005271ea57f585abfdc6f29e
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Algebra and Number Theory
- M. Kontsevich
- P. Hacking
- S. Keel
- Mark Gross
- B. Keller
- Alexander Kuznetsov
- I. Reiten
- Pierre-Guy Plamondon
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Cluster algebras and derived categories (2012, background): “It has rapidly turned out that the combinatorics of cluster algebras also appear in many other subjects, for example in • Poisson geometry [60] [61] [62] [63] [10] . . . ; • discrete dynamical systems [27] [47] [74] [79] [84] [81] [91] . . . ; • higher Teichmüller spaces [35] [36] [37] [38] [39] .…”
- Canonical bases for cluster algebras (2014, result): “ theorem follows. We close this section by proving part of [FZ03], Conjecture 4.14, to which we refer for the definition of the Fomin-Zelevinsky exchange graph. These results were originally proved in [GSV07], but we include our arguments here as the result follows quite easily from our techniques. T”