# Representation type via Euler characteristics and singularities of quiver Grassmannians

## TL;DR

This paper characterizes the representation type of acyclic quivers by analyzing the properties of their associated quiver Grassmannians, linking geometric features like singularities and Euler characteristics to algebraic classification.

## Contribution

It extends known results by connecting singularities and cell decompositions of quiver Grassmannians to the representation type of the underlying quiver.

## Key findings

- Finite type quivers have smooth, cell-decomposable Grassmannians.
- Tame type quivers have cell decompositions but may be singular.
- Wild type quivers have Grassmannians with negative Euler characteristic.

## 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.