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Let $G=S_n$ and let $V$ be the permutation module of $G$ with basis $\{x_1,\ldots,x_n\}.$

Let $\lambda, \mu \in \mathbb{C}$ to allow one to define a $\mathbb{C}G$-homomorphism $\rho:V \to V$ by $$\rho(x_j):=\lambda x_j+\mu\sum_{i \neq j}x_i.$$

By using the above fact or otherwise, how can we prove that $V$ is the direct sum of two non-isomorphic irreducible $\mathbb{C}G$ -submodules?

I tried to prove this by construction. A familiar irreducible submodule in this case is the $1$-dimensional space $U:=\operatorname{span}\{x_1+\cdots+x_n\}$. I intend to find another $(n-1)$-dimensional submodule $W$ which makes $V=U\oplus W$ hold, but it's hard to do so. Is there a way to use the fact instead of a random construction?

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    Find an inner product in your space which is preserved by the action of the group. Then the orthogonal complement of every submodule is a submodule. (Of course, you need to prove this at some point!)2012-10-13
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    Thanks! But probably that is not the expected way as I have not been introduced to the inner product and orthogonal complement yet.2012-10-13
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    Now that is very, very weird! One more dent off my faith in the way we teach math nowadays :-/2012-10-13
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    I am curious. How does one come across representation theory without a basic knowledge of linear algebra? I thought the later was virtually everywhere a prerequisite for the former.2012-12-06

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