Mutation Classes of Skew-Symmetric 3 × 3-Matrices

Abstract

In this article, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3 × 3-matrices and the set of triples of integers (a, b, c) such that 2 ≤ a ≤ b ≤ c and ab ≥ c. We also give an algorithm allowing to verify whether a matrix is mutation-cyclic or not. We prove that given a, b, the two cases depend on whether c is large enough or not.

Key Words:

• Cluster algebras
• Cluster-tilted algebras
• Mutations of quivers and matrices

2000 Mathematics Subject Classification:

• Primary 16S99
• Secondary 16G20

ACKNOWLEDGMENT

Ibrahim Assem is partially supported by NSERC of Canada and the Université de Sherbrooke. Martin Blais and Audrey Samson were working under a summer research fellowship of NSERC. Thomas Brüstle is partially supported by NSERC, by Bishop's University and the Université de Sherbrooke.

Notes

Communicated by D. Zacharia.

Dedicated to the memory of A. V. Roiter.