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

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 Added by Ryo Fujita
 Publication date 2019
  fields
and research's language is English
 Authors Ryo Fujita




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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-Kims 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.



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Let $U_q(mathfrak{g})$ be a twisted affine quantum group of type $A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $mathfrak{g}_{0}$ be the finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin quiver of type $mathfrak{g}_{0}$, we define a full subcategory ${mathcal C}_{Q}^{(2)}$ of the category of finite-dimensional integrable $U_q(mathfrak{g})$-modules, a twisted version of the category ${mathcal C}_{Q}$ introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur-Weyl duality, we construct an exact faithful KLR-type duality functor ${mathcal F}_{Q}^{(2)}: Rep(R) rightarrow {mathcal C}_{Q}^{(2)}$, where $Rep(R)$ is the category of finite-dimensional modules over the quiver Hecke algebra $R$ of type $mathfrak{g}_{0}$ with nilpotent actions of the generators $x_k$. We show that ${mathcal F}_{Q}^{(2)}$ sends any simple object to a simple object and induces a ring isomorphism $K(Rep(R)) simeq K({mathcal C}_{Q}^{(2)})$.
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