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We study a mixed tensor product $\mathbf{3}^{\otimes m} \otimes\mathbf{\overline{3}}^{\otimes n}$ of the three-dimensional fundamentalrepresentations of the Hopf algebra $U_{q} s\ell(2|1)$, whenever $q$ is not aroot of unity. Formulas for the decomposition of tensor products of any simpleand projective $U_{q} s\ell(2|1)$-module with the generating modules$\mathbf{3}$ and $\mathbf{\overline{3}}$ are obtained. The centralizer of$U_{q} s\ell(2|1)$ on the chain is calculated. It is shown to be the quotient$\mathscr{X}_{m,n}$ of the quantum walled Brauer algebra. The structure ofprojective modules over $\mathscr{X}_{m,n}$ is written down explicitly. It isknown that the walled Brauer algebras form an infinite tower. We havecalculated the corresponding restriction functors on simple and projectivemodules over $\mathscr{X}_{m,n}$. This result forms a crucial step indecomposition of the mixed tensor product as a bimodule over$\mathscr{X}_{m,n}\boxtimes U_{q} s\ell(2|1)$. We give an explicit bimodulestructure for all $m,n$.

Bimodule structure of the mixed tensor product over $U_{q} s\ell(2|1)$ and quantum walled Brauer algebra