Decompose the permutation representation into irreducible representations.Construct three non-isomorphic irreducible representations from $S_3$

Question

$S_3$ works on $\mathbb{C^3}$ with the permutation representation. I have to decompose this into irreducible representations and construct three non isomorphic irreducible representations from $S_3$ and verify $\sum_{i=1}^h m_i = |G|$ where m_i is the degree of a irreducible representations. This is my first encounter with representation theory and I honestly have no idea where to start. I mean, I know what I have to do, I just don't know how. Thanks :)

Answer 1

Let $V=\mathbb{C}^3$. If $e_1,e_2,e_3$ are the standard basis vectors of $V$, then notice that the subspace $W$ spanned by $e_1+e_2+e_3$ is invariant. Since $W$ is one-dimensional, it must be irreducible. It is a copy of the trivial representation of $S_3$ sitting inside of $V$.

Next, the $2$-dimensional subspace $U$ spanned by $e_1-e_2$ and $e_2-e_3$ is also invariant under the action of $S_3$ on $V$. In fact, $U$ is the subspace $\{(a,b,c)\in V\mid a+b+c=0\}$. Try to show that $U$ is also irreducible (Hint: If $v=(a,b,c)$ is any nonzero vector in $U$, then $(b,a,c)$ and $(a,c,b)$ are nonzero vectors in the subrepresentation generated by $v$. Now subtract each of these new vectors from $v$).

This decomposes $V$ into the direct sum of $W$ and $U$. The third irreducible representation is the alternating representation, which is $1$-dimensional, and an element $\sigma\in S_3$ acts as $\mathrm{sgn}(\sigma)$.

The three irreducible representations of degrees $2$, $1$, and $1$ are a complete list, as $2^2+1^2+1^2=6=|S_3|$.