Mixed quantum skew Howe duality and link invariants of type A

TL;DR

This paper introduces a new ribbon category framework that unifies the construction of link invariants for superalgebras and the HOMFLY-PT polynomial, extending skew Howe duality to a mixed setting.

Contribution

It defines the category $ extsf{Sp}(eta)$, linking quantum superalgebra representations with link invariants and establishing a mixed skew Howe duality.

Findings

Unified framework for $ ext{gl}_{m|n}$ link invariants and HOMFLY-PT polynomial

Identification of $ extsf{Sp}(eta)$ with limits of quantum group quotients

Proof of a mixed skew Howe duality involving exterior powers and their duals

Abstract

We define a ribbon category $\mathsf{Sp}(\beta)$, depending on a parameter $\beta$, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta=m-n$ the monoidal category of representations of $U_q(\mathfrak{gl}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(\beta)$ with a direct limit of quotients of a dual idempotented quantum group $\dot{\mathsf{U}}_q(\mathfrak{gl}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(\beta)$ gives a unified natural setting for defining the colored $\mathfrak{gl}_{m|n}$ link invariant (for $\beta=m-n$) and the colored HOMFLY-PT polynomial (for $\beta$ generic).