Exchange graphs for mutation-finite non-integer quivers of rank 3

TL;DR

This paper investigates the structure of exchange graphs for mutation-finite non-integer quivers of rank 3, revealing their geometric realizations and extending finite and affine classifications to non-integer cases.

Contribution

It introduces a geometric framework for rank 3 mutation-finite non-integer quivers and extends finite and affine type classifications to these quivers.

Findings

Exchange graphs of finite type are finite.

Exchange graphs of affine type are finite modulo a lattice.

Rank 3 mutation-finite non-integer quivers admit geometric realizations.

Abstract

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer weights of arrows. One can mutate such quivers according to usual rules of quiver mutation. Felikson and Tumarkin show that rank 3 mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows to define a notion of seeds (as Y-seeds), and hence, to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers in rank 3. The concept of finite and affine type generalises naturally to non-integer quivers. In particular, exchange graphs of finite type quivers are finite, while exchange graphs of affine quivers are finite modulo the action of a finite-dimensional lattice.