Homological approach to the Hernandez-Leclerc construction and quiver varieties

TL;DR

This paper establishes an isomorphism between two algebras associated with Dynkin quivers, enabling explicit descriptions of orbit closures of quiver representations as affine quotients, bridging categorical and combinatorial approaches.

Contribution

It proves the isomorphism between the categorical algebra and Hernandez-Leclerc's combinatorial algebra, linking quiver Grassmannians and Nakajima quiver varieties.

Findings

Isomorphism between the two algebras is established.

Explicit realization of orbit closures as affine quotients.

Bridging categorical and combinatorial frameworks for quiver varieties.

Abstract

In a previous paper the authors have attached to each Dynkin quiver an associative algebra. The definition is categorical and the algebra is used to construct desingularizations of arbitrary quiver Grassmannians. In the present paper we prove that this algebra is isomorphic to an algebra constructed by Hernandez-Leclerc defined combinatorially and used to describe certain graded Nakajima quiver varieties. This approach is used to get an explicit realization of the orbit closures of representations of Dynkin quivers as affine quotients.