Representations of Kronecker quivers and Steiner bundles on Grassmannians

Abstract

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.