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Formulas for the decomposition of tensor products of any simple and 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. 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