Reflection Functors for Hereditary Algebras

where P, Q are regular matrices of size n x n and mxm, respectively. We call the pair {n,m) the dimension type of this a + 6-tuple, and we are concerned with the determination of the dimension types of the indecomposable representations. For the case where 6 = 0, this question has been answered in [3], where it was shown that the dimension types of the indecomposable representations are precisely the positive roots of the corresponding root system. We will show that the same is true also when b ^ 1. However, whereas in the case when b = 0, for every positive root there exists an indecomposable representation with trivial endomorphism ring, this is no longer true in general. Note that for b ^ 1, we will show that there are always even imaginary roots without indecomposable representations with trivial endomorphism ring, in contrast to a recent conjecture by Kac [2]. To be more precise, recall the definition of the root system in our case. Consider on IR the quadratic form q{x,y) = x+y — (a + b)xy. The integral vectors (x, y) with q{x> y) ^ 0 are called imaginary roots; the integral vectors (x, y) with q(x, y) = 1 are called real roots.