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 P^r-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 rep₏ₑ₎₉ (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 rep₏ₑ₎₉ (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.