Graded quiver varieties and singularities of normalized R-matrices for fundamental modules

Abstract

We present a simple unified formula expressing the denominators of the normalized R-matrices between the fundamental modules over the quantum loop algebras of type ADE. It has an interpretation in terms of representations of Dynkin quivers and can be proved in a unified way using geometry of the graded quiver varieties. As a by-product, we obtain a geometric interpretation of Kang–Kashiwara–Kim’s generalized quantum affine Schur–Weyl duality functor when it arises from a family of the fundamental modules. We also study several cases when the graded quiver varieties are isomorphic to unions of the graded nilpotent orbits of type A.