Distinguished bases of exceptional modules

Claus Michael Ringel

Published 2012-10-28, updated 2013-02-26Version 3

Exceptional modules are tree modules. A tree module usually has many tree bases and the corresponding coefficient quivers may look quite differently. The aim of this note is to introduce a class of exceptional modules which have a distinguished tree basis, we call them radiation modules (generalizing an inductive construction considered already by Kinser). For a Dynkin quiver, nearly all indecomposable representations turn out to be radiation modules, the only exception is the maximal indecomposable module in case E_8. Also, the exceptional representation of the generalized Kronecker quivers are given by radiation modules. Consequently, with the help of Schofield induction one can display all the exceptional modules of an arbitrary quiver in a nice way.