Combinatorial aspects of extensions of Kronecker modules

TL;DR

This paper explores the combinatorial structure of extensions of Kronecker modules, showing they are field-independent up to Segre classes, and provides explicit descriptions and formulas for these extensions.

Contribution

It introduces a combinatorial framework for understanding Kronecker module extensions that is independent of the base field, extending known formulas to arbitrary fields.

Findings

Extensions are field independent up to Segre classes

Explicit descriptions of particular extensions are provided

A variant of Green's formula for Ringel-Hall numbers is established

Abstract

Let kK be the path algebra of the Kronecker quiver and consider the category 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 variant of the well known Green formula for Ringel-Hall numbers, valid over arbitrary fields. We end the paper with some results on extensions of preinjective Kronecker modules, involving the dominance ordering from partition combinatorics and its various generalizations.