Non-Schurian indecomposables via intersection theory

TL;DR

This paper uses intersection theory and Schubert calculus to construct and analyze non-Schurian indecomposable representations of acyclic quivers with three vertices, revealing new geometric insights into their structure.

Contribution

It introduces a novel geometric approach to study non-Schurian indecomposables via intersection theory and Schubert calculus, linking quiver representations to Grassmannian subvarieties.

Findings

Subvarieties intersect in a way that preserves indecomposability

The intersection contains an open subset of Schurian representations

Dimension matches predictions by Kac's Theorem

Abstract

For an acyclic quiver with three vertices, we consider the canonical decomposition of a non-Schurian root and associate certain representations of a generalized Kronecker quiver. These representations correspond to points contained in the intersection of two subvarieties of a Grassmannian and give rise to representations of the original quiver, preserving indecomposability. We show that these subvarieties intersect using Schubert calculus. Provided that the intersection contains a Schurian representation, it already contains an open subset of Schurian representations whose dimension is what we expect by Kac's Theorem.