Categorical geometric skew Howe duality

TL;DR

This paper develops a categorification of the R-matrix isomorphism for minuscule representations of U_q(sl(n)) using derived categories of coherent sheaves, advancing the geometric understanding of quantum group representations.

Contribution

It constructs an equivalence between derived categories related to affine Grassmannian convolution products, providing a geometric categorification of quantum skew Howe duality.

Findings

Categorification of U_q(sl(2)) representations related to U_q(sl(n))

Equivalence between derived categories of coherent sheaves

Progress towards geometric categorification of Reshitikhin-Turaev invariants

Abstract

We categorify the R-matrix isomorphism between tensor products of minuscule representations of U_q(sl(n)) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of U_q(sl(2)) which are related to representations of U_q(sl(n)) by quantum skew Howe duality. The resulting equivalence is part of the program of algebro-geometric categorification of Reshitikhin-Turaev tangle invariants developed by the first two authors.