Maximal Green Sequences of Exceptional Finite Mutation Type Quivers

Abstract

Maximal green sequences are particular sequences of mutations of quivers 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. The existence of maximal green sequences for exceptional finite mutation type quivers has been shown by Alim-Cecotti-Córdova-Espahbodi-Rastogi-Vafa except for the quiver $X_7$. In this paper we show that the quiver $X_7$ does not have any maximal green sequences. We also generalize the idea of the proof to give sufficient conditions for the non-existence of maximal green sequences for an arbitrary quiver.

References in corpus (3)

• Braids, Walls, and Mirrors
• New Graphs of Finite Mutation Type
• Maximal green sequences of skew-symmetrizable 3x3 matrices

Cited by in corpus (7)

• Sequence-to-Sequence Neural Net Models for Grapheme-to-Phoneme Conversion
• On Maximal Green Sequences For Type A Quivers
• On the relationship between green-to-red sequences, local-acyclicity, and upper cluster algebras
• Acyclic cluster algebras with dense $g$-vector fans
• Building maximal green sequences via component preserving mutations
• Non-degenerate potentials on the quiver $X_7$
• Red sizes of quivers