Period 2 quivers and their T- and Y-systems

TL;DR

This paper classifies period 2 quivers with up to 6 vertices, explores their associated T- and Y-systems, and identifies cases leading to known recurrences like Somos-4 and Somos-5.

Contribution

It provides a comprehensive classification of period 2 quivers for small N and analyzes their T- and Y-systems, revealing new periodicities and recurrences.

Findings

Classified all solutions for N=3,4,5 vertices

Identified solutions for N=6 vertices

Found cases leading to Somos-4 and Somos-5 recurrences

Abstract

We give a general definition for period $2$ quivers before making some prudent assumptions to reduce the number of possibilities. Finding these quivers requires solving a complicated system of equations between the number of arrows in the quiver. We find all solutions of these systems for quivers with $N=3,4,5$ vertices and give some solutions for $N=6$. We then consider the $T$- and $Y$-systems associated with these quivers, some of which exhibit periodic quantities which lead to reductions; in a few cases we obtain the Somos-4 and Somos-5 recurrences.