Combinatorial aspects of extensions of Kronecker modules

Let kK   be the path algebra of the Kronecker quiver and consider the category mod-kKmod-kK of finite dimensional right modules over kK (called Kronecker modules). We prove that extensions of Kronecker modules are field independent up to Segre classes, so they can be described purely combinatorially. We use in the proof explicit descriptions of particular extensions and a correspondence between exact “crosses” and “frames”, which (in case k is finite) is the main tool in the proof of the well known Green formula for Ringel–Hall numbers. We end the paper with some results on extensions of preinjective (or dually preprojective) Kronecker modules, involving the dominance ordering from partition combinatorics and its various generalizations.