Representations of Kronecker quivers and Steiner bundles on Grassmannians

Let \(\mathbb{k}\) be an algebraically closed field. Connections between representations of the generalized Kronecker quivers \(K_r\) and vector bundles on \(\mathbb{P}^{r-1}\) 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 \(\mathrm{Gr}_d(\mathbb{k}^r)\) and certain full subcategories \(\mathrm{rep}_{\mathrm{proj}}(K_r,d)\) of relative projective \(K_r\)-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 \(\mathrm{rep}_{\mathrm{proj}}(K_r,d)\), 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 \(r\!\ge\!3\), wild category of \(K_r\)-representations. These may be characterized as being right Hom-orthogonal to certain algebraic families of elementary test modules.