Essential dimension and genericity for quiver representations

TL;DR

This paper investigates the essential dimension of quiver representations, classifies quivers with the genericity property, and explores when essential and generic essential dimensions coincide, with detailed results for Kronecker quivers.

Contribution

It classifies quivers satisfying the genericity property for all dimension vectors and analyzes the property for wild quivers and Kronecker quivers.

Findings

Classified quivers with the genericity property for all dimension vectors.

Showed that wild quivers satisfy the property for infinitely many Schur roots.

Constructed examples where the genericity property fails.

Abstract

We study the essential dimension of representations of a fixed quiver with given dimension vector. We also consider the question of when the genericity property holds, i.e., when essential dimension and generic essential dimension agree. We classify the quivers satisfying the genericity property for every dimension vector and show that for every wild quiver the genericity property holds for infinitely many of its Schur roots. We also construct a large class of examples, where the genericity property fails. Our results are particularly detailed in the case of Kronecker quivers.