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Given a finite-volume hyperbolic 3-manifold, we compose a lift of the holonomy in $SL(2,\mathbf{C})$ with the $n$-dimensional irreducible representation of $SL(2,\mathbf{C})$ in $SL(n,\mathbf{C})$. In this paper we give local coordinates of the $SL(n,\mathbf{C})$-character variety around the character of this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.