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Wu-Ki Tung discusses the "tensorial approach" to deriving all the finite dimensional irreducible representations of GL(n, $\mathbb{C}$) in chapter 13 of his book Group Theory in Physics claiming that all those representations can be constructed from the standard/defining representation over $V$, its dual representation $V^*$, its complex conjugate representation $\overline{\ V}$ and its dual of the conjugate representation $\overline{\ V}^*$. My questions:

I. Are $\overline{\ V}$ and $\overline{\ V}^*$ really necessary?

II. How to reconcile this "tensorial approach" with the construction using the Schur functor (See, e.g., Representation Theory: A First Course by Fulton and Harris) applied to $V$ and/or $V^*$?

I know all finite dimensional irreducible linear representations can be realized as a subspace of the tensor product. In physics, people usually care about $\text{SO}(n)$ or $\text{SU}(n)$, so it really makes no difference.

I guess that I can find the answer if I have time to read Fulton and Harris' thoroughly, but unfortunately that is not feasible for the moment. I want to get the general picture, so I appreciate it if anyone can help.

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  • $\begingroup$ Both results are correct. When Fulton and Harris classify $\text{SL}_n(\mathbb{C})$, they treat $\text{SL}_n(\mathbb{C})$ as a complex Lie group and representations have to be holomorphic, so complex conjugate representations are not allowed. In Tung's, however, he treats $\text{SL}_n(\mathbb{C})$ as a real Lie group, so representations of $\text{SL}_n(\mathbb{C})$ just need to be smooth and complex conjugate representations are permissible. $\endgroup$ – DarKnightS Jun 25 '17 at 17:15

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