The product of simple modules over KLR algebras and quiver Grassmannians

TL;DR

This paper links the multiplication of simple modules over KLR algebras with quiver Grassmannians for Dynkin quivers, providing new criteria for simplicity and proving a recent conjecture in specific cases.

Contribution

It establishes a novel connection between KLR module products and quiver Grassmannian components, advancing understanding of module simplicity criteria.

Findings

Derived a necessary condition for simple module products over KLR algebras.

Linked induction functor with irreducible components of quiver Grassmannians.

Proved a conjecture by Lapid and Minguez in certain cases.

Abstract

In this paper, we study the product of two simple modules over KLR algebras using the quiver Grassmannians for Dynkin quivers. More precisely, we establish a bridge between the Induction functor on the category of modules of KLR algebras and the irreducible components of quiver Grassmannians for Dynkin quivers via a sort of extension varieties, which is an analogue of the extension group in Hall algebras. As a result, we give a necessary condition when the product of two simple modules over a KLR algebra is simple using the set of irreducible components of quiver Grassmannians. In particular, in some special cases, we provide a proof for the conjecture recently proposed by Lapid and Minguez.