Suppose we have a partition $\mu$ of $n$. There is an associated polynomial irreducible representation $\phi_{\mu}$ of $GL_n(\mathbb C)$.
How do I obtain a new representation of $GL_n(\mathbb C)$ from the dual representation $\phi_{\mu}^{*}$? What is the relation between $\mu$ and the partition associated to this new representation?
I tried to think about Young tableaux, but how can I find an isomorphism between a dual space representation and something to which I can associate a partition?