Exceptional modules are tree modules

Claus Michael Ringel

Abstract. We consider representations of a quiver over an arbitrary field. 
Recall that an indecomposable representation M without self-extensions is said
to be exceptional. We are going to show that exceptional representations can be
exhibited using matrices involving as coefficients just 0 and 1. Actually, if
d is the dimension of M, there exists such a matrix presentation with precisely
d-1 non-zero coefficients: the corresponding "coefficient quiver" is a tree.