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Geometry of mutation classes of rank 3 quivers

Felikson, A.; Tumarkin, P.

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We present a geometric realization for all mutation classes of quivers of rank 3 with real weights. This realization is via linear reflection groups for acyclic mutation classes and via groups generated by π-rotations for the cyclic ones. The geometric behavior of the model turns out to be controlled by the Markov constant p2 + q2 + r 2 − pqr, where p, q,r are the weights of arrows in a quiver. We also classify skew-symmetric mutation-finite real 3×3 matrices and explore the structure of acyclic representatives in finite and infinite mutation classes.


Felikson, A., & Tumarkin, P. (2019). Geometry of mutation classes of rank 3 quivers. Arnold Mathematical Journal, 5(1), 37-55.

Journal Article Type Article
Acceptance Date Feb 18, 2019
Online Publication Date Mar 4, 2019
Publication Date Mar 31, 2019
Deposit Date Oct 27, 2016
Publicly Available Date May 24, 2019
Journal Arnold Mathematical Journal
Print ISSN 2199-6792
Electronic ISSN 2199-6806
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 5
Issue 1
Pages 37-55
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Published Journal Article (Advance online version) (514 Kb)

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Copyright Statement
Advance online version © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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