Maximal green sequences of skew-symmetrizable 3×3 matrices

Maximal green sequences are particular sequences of mutations of skew-symmetrizable matrices which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti-Córdova-Vafa in the context of supersymmetric gauge theory. In this paper we study maximal green sequences of skew-symmetrizable 3×3 matrices. We show that such a matrix with a mutation-cyclic diagram does not have any maximal green sequence. We also obtain some basic properties of maximal green sequences of skew-symmetrizable matrices with mutation-acyclic diagrams.