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  Tensorial Approach vs Schur Functor Approach to Finite-Dimensional Representation of GL(n, $\mathbb{C}$) ?

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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 $\bar{V}$ and its dual of the conjugate representation $\bar{V}^*$. My questions:

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

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

asked Oct 30, 2015 in Mathematics by DarKnightS (125 points) [ revision history ]

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