Vladimir I. Danilov ; Alexander V. Karzanov ; Gleb A. Koshevoy - Coherent fans in the space of flows in framed graphs

dmtcs:3056 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) - https://doi.org/10.46298/dmtcs.3056
Coherent fans in the space of flows in framed graphsArticle

Authors: Vladimir I. Danilov 1; Alexander V. Karzanov 2; Gleb A. Koshevoy 1

  • 1 Central Institute of Economics and Mathematics of the RAS
  • 2 Institute for System Analysis of the RAS

Let $G=(V,E)$ be a finite acyclic directed graph. Being motivated by a study of certain aspects of cluster algebras, we are interested in a class of triangulations of the cone of non-negative flows in $G, \mathcal F_+(G)$. To construct a triangulation, we fix a raming at each inner vertex $v$ of $G$, which consists of two linear orders: one on the set of incoming edges, and the other on the set of outgoing edges of $v$. A digraph $G$ endowed with a framing at each inner vertex is called $framed$. Given a framing on $G$, we define a reflexive and symmetric binary relation on the set of extreme rays of $\mathcal F_+ (G)$. We prove that that the complex of cliques formed by this binary relation is a pure simplicial complex, and that the cones spanned by cliques constitute a unimodular simplicial regular fan $Σ (G)$ covering the entire $\mathcal F_+(G)$.


Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)
Section: Proceedings
Published on: January 1, 2012
Imported on: January 31, 2017
Keywords: Simplicial fan, regular fan, Plücker cluster algebra, coherence binary relation, weakly-separated sets,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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