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The product of simple modules over KLR algebras and quiver Grassmannians

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arXiv:2107.08244v5 Announce Type: replace
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.

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