Maximal Green Sequences of Exceptional Finite Mutation Type Quivers

TL;DR

This paper investigates the existence of maximal green sequences in exceptional finite mutation type quivers, proving that the specific quiver X_7 lacks such sequences and establishing general conditions for their non-existence.

Contribution

It demonstrates that the quiver X_7 does not have maximal green sequences and provides a generalized method to determine non-existence for other quivers.

Findings

X_7 quiver has no maximal green sequences

General sufficient conditions for non-existence established

Method applicable to other 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-C\'ordova-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-C\'ordova-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.