Hall polynomials and the Gabriel–Roiter submodules of simple homogeneous modules

sz´ant´o, csaba; sz¨oll˝osi, istv´an;

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