Representation type via Euler characteristics and singularities of quiver Grassmannians

Abstract

In this paper, 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 type are smooth and admit cell decompositions, it turns out that all quiver Grassmannians for indecomposable representations of tame quivers 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.

Frieze varieties: A characterization of the finite-tame-wild trichotomy for acyclic quivers

“…In [30], Skowroński and Weyman characterized tameness in terms of semi-invariants. Recently, Lorscheid and Weist characterized tameness using quiver Grassmannians [22]. To our knowledge, our characterization is the first one in terms of numerical invariants that are integers.…”

Cell decompositions for rank two quiver Grassmannians

“…For instance, quiver Grassmannians attached to exceptional representations are smooth [11]. For Dynkin quivers and tame quivers of typesà orD, it is known that every quiver Grassmannian attached to an indecomposable representation admits a cell decomposition, see [9,14] and references therein. It has been conjectured that this is also true for exceptional representations of any quiver, in particular for preprojective and preinjective representations.…”

Plücker Relations for Quiver Grassmannians

“…Example 2 (Family with jumping Euler characteristic). The following example is taken from [5]. Let Q be a quiver of extended Dynkin quiver type A 2 with arrows a, b and c as illustrated below.…”