Mathematical Research Letters

Volume 15, Issue 2, March 2008 pp. 321-330.

On the properties of the exchange graph of a cluster algebra

Authors: Michael Gekhtman (1), Michael Shapiro (2), and Alek Vainshtein (3)
Author institution: University of Notre Dame (1), University of Notre Dame (2), and Michigan State University (3)

Summary: 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.

Contents Full-Text PDF