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Let $\rho : S_n \to \mathrm{GL}(\mathbb{C}^n)$, where $\rho(\sigma)(x_1, \ldots, x_n) = (x_{\sigma^{-1}(1)}, \ldots, x_{\sigma^{-1}(n)})$. How can you prove that $W = \{ (x_1, \ldots, x_n) : x_1 + \cdots + x_n = 0 \} \subset \mathbb{C}^n$ is an irreducible representation?

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2 Answers 2

up vote 5 down vote accepted

Use the trace formula for $\rho$ to get that $\rho$ can be decomposed into two irreducible representations.

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The first time I was assigned this exercise I was explicitly told not to use character theory. There is a straightforward direct proof (showing directly that the orbit of any nonzero vector is all of $W$) and I implore you to try to find it.

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The orbit of any nonzero vector is a discrete set, and hence not W. I think you meant the span of the orbit :-) –  Jason DeVito Dec 1 '10 at 14:10
    
Yes, that's the "internal" notion of orbit here :P –  Qiaochu Yuan Dec 1 '10 at 15:56

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