Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category $\mathsf{Sp}(β)$, depending on a parameter $β$, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $β=m-n$ 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 $\mathsf{Sp}(β)$ 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 $\mathsf{Sp}(β)$ gives a unified natural setting for defining the colored $\mathfrak{gl}_{m|n}$ link invariant (for $β=m-n$) and the colored HOMFLY-PT polynomial (for $β$ generic).

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