Distinguished Bases of Exceptional Modules

Abstract

An indecomposable representation M of a quiver Q=(Q ₀,Q ₁) is said to be exceptional provided $\operatorname{Ext}^{1}(M,M) = 0$. And it is called a tree module provided one can choose a set of bases of the vector spaces M _x (x∈Q ₀) such that the coefficient quiver is a tree quiver; we call a tree basis of M. It is known that 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 indecomposable modules which have a distinguished tree basis, the “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 $\mathbb{E}_{8}$. Also, the exceptional representations of the generalized Kronecker quivers are given (via the universal cover) 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.