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Durham University

Department of Mathematical Sciences

Academic Staff

Publication details for Pavel Tumarkin

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

Author(s) from Durham


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.