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

SIGMA 9 (2013), 005, 13 pages      arXiv:1301.4541      https://doi.org/10.3842/SIGMA.2013.005
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

### Upper Bounds for Mutations of Potentials

John Alexander Cruz Morales a and Sergey Galkin b, c, d, e
a) Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-037, Japan
b) Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
c) Independent University of Moscow, 11 Bolshoy Vlasyevskiy per., 119002, Moscow, Russia
d) Moscow Institute of Physics and Technology, 9 Institutskii per., Dolgoprudny, 141700, Moscow Region, Russia
e) Universität Wien, Fakultät für Mathematik, Garnisongasse 3/14, A-1090 Wien, Austria

Received May 31, 2012, in final form January 16, 2013; Published online January 19, 2013

Abstract
In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52].

Key words: cluster algebras; Laurent phenomenon; mutation of potentials; mirror symmetry.

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References

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