Representations of Kronecker quivers and Steiner bundles on Grassmannians

Let k be an algebraically closed field. Connections between representations of the generalized Kronecker quivers Kᵣ and vector bundles on Pʳ⁻¹ have been known for quite some time. This article is concerned with a particular aspect of this correspondence, involving more generally Steiner bundles on Grassmannians Grd(kʳ) and certain full subcategories repₚᵣₒⱼ(Kᵣ,d) of relative projective Kᵣ-representations. Building on a categorical equivalence first explicitly established by Jardim and Prata, we employ representation-theoretic techniques provided by Auslander-Reiten theory and reflection functors to organize indecomposable Steiner bundles in a manner that facilitates the study of bundles enjoying certain properties such as uniformity and homogeneity. Conversely, computational results on Steiner bundles motivate investigations in repₚᵣₒⱼ(Kᵣ,d), which elicit the conceptual sources of some recent work on the subject. From a purely representation-theoretic vantage point, our paper initiates the investigation of certain full subcategories of the, for r\!≥\!3, wild category of Kᵣ-representations. These may be characterized as being right Hom-orthogonal to certain algebraic families of elementary test modules.