On some Ringel-Hall numbers in tame cases

Let k be a finite field and consider the finite dimensional path algebra kQ, where Q is a quiver of tame type i.e. of type Ã_n, D̃_n,Ẽ₆,Ẽ₇,Ẽ₈. Let H(kQ) be the corresponding Ringel-Hall algebra. We are going to determine the Ringel-Hall numbers of the form F^P`<sub>XP</sub> with P, P` preprojective indecomposables of defect -1 and F^I`<sub>IX</sub> with I, I` preinjective indecomposables of defect 1. It turns out that these numbers are either 1 or 0.