Free resolutions of orbit closures of Dynkin quivers

TL;DR

This paper applies geometric techniques to compute minimal free resolutions of orbit closures in Dynkin quivers, revealing normality, rational singularities, and explicit generators, especially for type A quivers.

Contribution

It introduces a method to determine free resolutions of orbit closures in Dynkin quivers and provides explicit generators for type A cases.

Findings

Orbit closures of 1-step representations are normal with rational singularities.

Explicit minimal generators of defining ideals are described for type A quivers.

An algorithm for generating defining ideals in type A quivers is developed.

Abstract

We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type $A$, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type $A$ quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation.