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Mathematics > Representation Theory
Title: Representations of Kronecker quivers and Steiner bundles on Grassmannians
(Submitted on 29 Feb 2024 (v1), last revised 9 Apr 2024 (this version, v2))
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.
Submission history
From: Daniel Bissinger [view email][v1] Thu, 29 Feb 2024 19:11:58 GMT (70kb)
[v2] Tue, 9 Apr 2024 14:55:12 GMT (71kb)
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