Let k be an arbitrary field and Q be an acyclic quiver of tame type (that is, of typeud˜ An, ˜Dn, ˜E6, ˜E7, ˜E8). Consider the path algebra kQ, the category of finite-dimensional rightudmodules mod-kQ, and the minimal positive imaginary root of Q, denoted by δ. In the first part ofudthe paper, we deduce that the Gabriel–Roiter (GR) inclusions in preprojective indecomposablesudand homogeneous modules of dimension δ, as well as their GR measures are field independent (audsimilar result due to Ringel being true in general over Dynkin quivers). Using this result, we canudprove in a more general setting a theorem by Bo Chen which states that the GR submodule P ofuda homogeneous module R of dimension δ is preprojective of defect −1 and so the pair (R/P, P)udis a Kronecker pair. The generalization consists in considering the originally missing case ˜E8 andudusing arbitrary fields (instead of algebraically closed ones). Our proof is based on the idea ofudRingel (used in the Dynkin quiver context) of comparing all possible Hall polynomials with theudspecial form they take in case of a GR inclusion. For this purpose, we determine (with the helpudof a program written in GAP) a list of tame Hall polynomials which may have further interestingudapplications.
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