On Maximal Green Sequences

Maximal green sequences are particular sequences of quiver mutations appearing in the context of quantum dilogarithm identities and supersymmetric gauge theory. Interpreting maximal green sequences as paths in various natural posets arising in representation theory, we prove the finiteness of the number of maximal green sequences for cluster finite quivers, affine quivers, and acyclic quivers with at most three vertices. We also give results concerning the possible numbers and lengths of these maximal green sequences.