Full exceptional collections of vector bundles on rank-two linear GIT quotients

TL;DR

This paper constructs full strong exceptional collections of vector bundles on certain rank-two GIT quotients, using irreducible representations with weights in a bounded region, and introduces a method to generate more examples via decorated quiver varieties.

Contribution

It provides explicit constructions of full strong exceptional collections on rank-two GIT quotients and a new method to create additional examples using decorated quiver varieties.

Findings

Established full strong exceptional collections on specific GIT quotients.

Linked vector bundles to irreducible representations with bounded weights.

Proposed a construction method for more examples using decorated quiver varieties.

Abstract

We produce full strong exceptional collections consisting of vector bundles on the geometric invariant theory quotient of certain linear actions of a split reductive group $G$ of rank two. The vector bundles correspond to irreducible $G$-representations whose weights lie in an explicit bounded region in the weight space of $G$. We also describe a method for constructing more examples of linear GIT quotients with full strong exceptional collections of this kind as "decorated" quiver varieties.