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

TL;DR

This paper provides a unified geometric formula for denominators of normalized R-matrices between fundamental modules over quantum loop algebras of type ADE, linking representation theory with graded quiver varieties.

Contribution

It introduces a simple unified formula for R-matrix denominators and offers a geometric interpretation via graded quiver varieties, connecting to quantum affine Schur-Weyl duality.

Findings

Unified formula for R-matrix denominators in type ADE

Geometric interpretation using graded quiver varieties

Identification of cases where graded quiver varieties match nilpotent orbits

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 the Dynkin quivers and can be proved in a unified way using the geometry of 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 fundamental modules. We also study several cases when the graded quiver varieties are isomorphic to the graded nilpotent orbits of type A.