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In this paper, we study the product of two simple modules over KLR algebrasusing the quiver Grassmannians for Dynkin quivers. More precisely, we establisha bridge between the Induction functor on the category of modules of KLRalgebras and the irreducible components of quiver Grassmannians for Dynkinquivers via a sort of extension varieties, which is an analogue of theextension group in Hall algebras. As a result, we give a necessary conditionwhen the product of two simple modules over a KLR algebra is simple using theset of irreducible components of quiver Grassmannians. In particular, in somespecial cases, we provide a proof for the conjecture recently proposed by Lapidand Minguez.

The product of simple modules over KLR algebras and quiver Grassmannians