Mathematical Research Letters

Volume 15 (2008)

Number 2

On the properties of the exchange graph of a cluster algebra

Pages: 321 – 330

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n2.a10

Authors

Michael Gekhtman (University of Notre Dame)

Michael Shapiro (University of Notre Dame)

Alek Vainshtein (Michigan State University)

Abstract

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.

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