Representation type via Euler characteristics and singularities of quiver Grassmannians

Abstract

In this text, we characterize the representation type of an acyclic quiver by the properties of its associated quiver Grassmannians. This characterization utilizes and extends known results about singular quiver Grassmannians and cell decompositions into affine spaces. While all quiver Grassmannians for indecomposable representations of quivers of finite representation types $A$ and $D$ are smooth and admit cell decompositions, it turns out that all quiver Grassmannians for indecomposable representations of quivers of tame types $A$ and $D$ admit cell decompositions, but some of these quiver Grassmannians are singular (even as varieties). A quiver is wild if and only if there exists a quiver Grassmannian with negative Euler characteristic.

17 pages

References in corpus (4)

• Moduli of Representations, Quiver Grassmannians, and Hilbert Schemes
• Quiver Grassmannians of extended Dynkin type D - Part 2: Schubert decompositions and F-polynomials
• Quiver Grassmannians of type extended Dynkin type D - Part 1: Schubert systems and decompositions into affine spaces
• Plücker relations for quiver Grassmannians

Cited by in corpus (2)

• Coefficient Quivers, ${\mathbb{F}}_1$-Representations, and Euler Characteristics of Quiver Grassmannians
• Plücker relations for quiver Grassmannians