Geometry of mutation classes of rank $3$ quivers
Abstract
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 $\pi$rotations for the cyclic ones. The geometric behavior of the model turns out to be controlled by the Markov constant $p^2+q^2+r^2pqr$, where $p,q,r$ are the elements of exchange matrix. We also classify skewsymmetric mutationfinite real $3\times 3$ matrices and explore the structure of acyclic representatives in finite and infinite mutation classes.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.08828
 Bibcode:
 2016arXiv160908828F
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Metric Geometry;
 Mathematics  Rings and Algebras;
 13F60;
 20H15;
 51F15
 EPrint:
 27 pages, 11 figures