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

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.