### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 089, 5 pages      arXiv:1406.1072      https://doi.org/10.3842/SIGMA.2014.089
Contribution to the Special Issue on New Directions in Lie Theory

### Maximal Green Sequences of Exceptional Finite Mutation Type Quivers

Ahmet I. Seven
Middle East Technical University, Department of Mathematics, 06800, Ankara, Turkey

Received June 18, 2014, in final form August 15, 2014; Published online August 19, 2014

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.

Key words: skew-symmetrizable matrices; maximal green sequences; mutation classes.

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