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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?

  • 1
    The dual representation is already a representation; what do you mean by obtaining a new representation from it? The corresponding partition is the transpose of $\mu$.2012-01-10
  • 2
    No, it is not the transpose. See below.2012-01-11

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