Distinguished bases of exceptional modules

TL;DR

This paper introduces radiation modules, a special class of exceptional modules with distinguished bases, enabling a unified and explicit description of all exceptional modules for various quivers.

Contribution

It defines radiation modules with distinguished bases, generalizing previous constructions, and shows nearly all indecomposables in Dynkin quivers are radiation modules.

Findings

Nearly all indecomposable representations in Dynkin quivers are radiation modules.

Radiation modules include exceptional representations of generalized Kronecker quivers.

Schofield induction can be used to explicitly display all exceptional modules.

Abstract

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.