$(\pi,V) $ is the permutation representation of the symmetric group $S_5 $, $ V=C^5$ and the action of standard basis vectors of $ V$ is given by $\pi(\sigma)e_i=e_{\sigma(i)} $ for $\sigma\in S_5 $ $i=1,...,5$ Show that $(\pi , V) $ is not irreducible by finding a suitable G-invariant subspace.
Now I know that a subspace $W\in V $ is G-invariant if $\pi(g)W\in W $ for all $g \in G $, and V is irreducible if the only G-invariant subspaces are 0 and V.
How can I find G-invariant subspaces for this permutation representation, to prove reducibility? Thank you
The permutation representation of $S_n$ is $\mathbb C^n$ with elements of $S_n$ permuting the basis vectors $\{e_1, e_2, \ldots, e_n\}$. It has a trivial subrepresentation spanned by the vector $v = \sum_i e_i$. By Maschke's theorem there is a complement subrepresentation (given by the condition $\sum_i x_i = 0$, where $x_i$ is the i'th coordinate).