How is the standard representation of $S_n$ the direct sum of trivial representation and permutation representation?

Question

I need to prove that the character of standard representation is equal to the number of elements fixed by a permutation-1. For that I need to prove that permutation representation is equal to direct sum of trivial representation and standard representation. Please explain. Thanks.

Answer 1

Firstly, I think you've gotten the standard and permutation representations confused, but also I'm not sure you need to go down the route that you are suggesting. Let's see:

Let $\rho_n : S_n \to V$ be the permutation representation of $S_n$ over a field $\mathsf{k}$, by which I mean that $V$ is an $n$-dimensional vector space over $\mathsf{k}$ with a basis $\mathcal{B} = \left\{ e_{\sigma} \mid \sigma \in S_n \right\}$ consisting of elements indexed by the elements $\sigma$ of $S_n$ (in any order, doesn't really matter), and $\rho_n$ acts by

$$ \rho_n(\tau)(e_{\sigma}) = e_{\tau\sigma}. $$

Now since given any $\tau \in S_n$, $\rho_n(\tau)$ preserves $\mathcal{B}$ as a set, and in particular acts as a permutation on $\mathcal{B}$, the matrix of $\rho_n(\tau)$ with respect to $\mathcal{B}$ is a permutation matrix, which is to say that its entries are taken from $\{0,1\}$ an every row and column has precisely one $1$. Then the character $\chi_n$ of $\rho_n$ is defined to be the trace of $\rho_n$. That is $\chi_n(\tau) = \operatorname{trace}(\rho_n(\tau))$ for any $\tau \in S_n$. Now think what it means for the $k^{\text{th}}$ diagonal entry of $\rho_n(\tau)$ to be a $1$ rather than a $0$. It means exactly that the $k^{\text{th}}$ element of $\mathcal{B}$, $e_{\sigma_k}$ say is fixed by $\rho_n(\tau)$, i.e $\rho_n(\tau)(e_{\sigma_k}) = e_{\sigma_k}$. And then when $e_{\sigma_k}$ is fixed by $\rho_n(\tau)$, $e_{\sigma_k}$ contributes exactly "+1" to the trace of $\rho_n(\tau)$, and this is the only situation where an element of $\mathcal{B}$ can affect the trace. Hence the trace of $\rho_n(\tau)$ is precisely the number of fixed points of $\rho_n(\tau)$.