A remark on quiver varieties andweyl groups

TL;DR

This paper explores the action of Weyl groups on quiver varieties, providing new descriptions of their algebraic structure and proving key geometric properties such as connectedness and normality in specific cases.

Contribution

It introduces a Weyl group action on quiver varieties and describes generators of their projective rings, along with proving connectedness and normality under certain conditions.

Findings

Weyl group acts on quiver varieties with generic parameters.

Connectedness of smooth quiver varieties $M(d,v)$ is established.

Normality of $M_0(d,v)$ is proved for finite type quivers with regular weights.

Abstract

In this paper we define an action of the Weyl group on the quiver varieties $M_{m,\grl}(d,v)$ with generic $(m,\grl)$. To do it we describe a set of generators of the projective ring of a quiver variety. We also prove connectness for the smooth quiver variety $M(d,v)$ and normality for $M_0(d,v)$ in the case of a quiver of finite type and $d-v$ a regular weight.