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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 ..."
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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 X7. In this paper we show that the quiver X7 does not have any maximal green sequences. We also generalize the idea of the proof to give sufficient conditions for the nonexistence of maximal green sequences for an arbitrary quiver. Key words: skewsymmetrizable matrices; maximal green sequences; mutation classes 2010 Mathematics Subject Classification: 15B36; 05C50 1 Introduction and main results Maximal green sequences are particular sequences of mutations of quivers. They were used in [9] to study the refined Donaldson–Thomas invariants and quantum dilogarithm identities. Moreover, the same sequences appeared in theoretical physics where they yield the complete spectrum of a BPS particle, see [5, Section 4.2]. The existence of maximal green sequences for