General Representations of Quivers

Abstract

We investigate general representations of a quiver Q. There are two main questions that we shall address. The first, which was raised by Kac, is to find the dimension vectors of the summands of a general representation of dimension vector α; the second is to find the dimension vectors of the subrepresentations of a general representation of dimension vector α. These two questions are closely related. One consequence of our investigation is to find an algorithm to compute the canonical decomposition of an arbitrary dimension vector in terms of the Euler form of the quiver. We may also compute the dimension vectors of subrepresentations of a general representation and the minimal dimension of Ext(R, S) as R and S run through representations of dimension vectors α and β respectively.