Categorification of tensor powers of the vector representation of $U_q(\mathfrak{gl}(1|1))$

We consider the monoidal subcategory of finite dimensional representations of $U_q(\mathfrak{ gl}(1|1))$ generated by the vector representation, and we provide a diagram calculus for the intertwining operators, which allows to compute explicitly the canonical basis. We construct then a categorification of these representations and of the action of both $U_q(\mathfrak{ gl}(1|1))$ and the intertwining operators using subquotient categories of the BGG category $\mathcal{ O}(\mathfrak{ gl}_n)$.

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