Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

TL;DR

This paper develops a boundary analogue of Schur-Weyl duality for quantum affine symmetric pairs, connecting modules over quantum symmetric pairs with orientifold KLR algebras, enriching the combinatorial and categorical framework.

Contribution

It constructs a functor linking modules over quantum symmetric pairs to orientifold KLR algebras, extending Schur-Weyl duality to boundary cases with new combinatorial structures.

Findings

Constructed a functor between quantum symmetric pair modules and orientifold KLR algebras.

Enriched the combinatorial model with poles of a trigonometric K-matrix.

Proved the functor recovers known dualities in quasi-split type AIII.

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_qL\mathfrak{g}$ the corresponding quantum affine algebra. We construct a functor ${}^{\theta}{\sf F}$ between finite-dimensional modules over a quantum symmetric pair of affine type $U_q\mathfrak{k}\subset U_qL{\mathfrak{g}}$ and an orientifold KLR algebra arising from a framed quiver with a contravariant involution, providing a boundary analogue of Kang-Kashiwara-Kim-Oh generalized Schur-Weyl duality. With respect to their construction, our combinatorial model is further enriched with the poles of a trigonometric K-matrix intertwining the action of $U_q\mathfrak{k}$ on finite-dimensional $U_qL{\mathfrak{g}}$-modules. By construction, ${}^{\theta}{\sf F}$ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, ${}^{\theta}{\sf F}$ is a functor of module categories. Relying on a suitable isomorphism \`a la Brundan-Kleshchev-Rouquier, we prove that ${}^{\theta}{\sf F}$ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type $\sf AIII$.