Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl construction there corresponds to each partition of $d$ a representation of $GL(V)$. Taking $V=\mathbb{C}^n$ this in turn determines a representation of lie algebra $sl_n \mathbb{C}$. This is the approach adopted in Fulton and Harris.

My question is: What if we have an $sl_n\mathbb{C}$ structure on some space other than $\mathbb{C}^n$, do we have the same construction? To be precise, If we have such $sl_n$ representation $V$ then how can we decompose the tensor product $V^{\otimes d}$ into $sl_n$ irreps provided we have a $S_d$ irreducible decomposition.