Positive root of a quiver ($q_Q(\mathbf{n})=1$)

In the positive definite case this is a nonnegative integer vector with $q_Q(\mathbf{n})=1$, so it is nonzero. The positive definite Tits form indecomposable classification gives one indecomposable isomorphism class per positive root.

For the chain, with $n_0=n_{m+1}=0$, the quadratic form is $\frac{1}{2}{\sum }_{i=0}^m(n_{i+1}-n_i{)}^2$. Its value is one precisely for interval vectors: one $+1$ difference followed by one $-1$, all other differences zero. Thus there are $m(m+1)/2$ positive roots, independently of orientation.

The Ringel lemma on bricks makes every indecomposable a brick module; positive definiteness and the Ringel form then force $q_Q=1$ and rigidity. Conversely, a maximal-orbit representation at a positive root cannot split: nonzero cross extensions increase orbit dimension, while vanishing cross extensions make the quadratic form of a split sum at least two. Open dense orbits give uniqueness. A compact bound on the integer roots gives finite representation type.