The theorem of Bo Chen and Hall polynomials

Abstract

Let $\Lambda$ be the path algebra of a Dynkin quiver. A recent result of Bo Chen asserts that $\operatorname{Hom}(X, Y/X) = 0$ for any Gabriel-Roiter inclusion $X \subseteq Y$. The aim of the present note is to give an interpretation of this result in terms of Hall polynomials, and to extend it in this way to representation-directed split algebras. We further show its relevance when dealing with arbitrary representation-finite split algebras.