Character table of $S_4$.

Question

I know there are potentially quite a few posts related to this question, but I guess I had a specific one that wasn't found in the ones that came up.

I was reading the notes pertaining to this particular character table and the way the standard representation characters were justified were as follows:

Let $X$ be the set ${1, 2, 3, 4}$. Then $\rho_{\Bbb C[X]}$ contains $\rho_{\text{triv}}$. In reality, $\rho_{\Bbb C[X]} = \rho_{\text{stand}}\oplus \rho_{\text{triv}}.$ So $\chi_{\text{stand}} = \chi_{\Bbb C[X]} − \chi_{\text{triv}}$. So we should take the difference $(4, 2, 0, 1, 0) − (1, 1, 1, 1, 1)=(3, 1, -1, 0, -1)$.

In this sum, where did the first vector come from? Was a bit confused regarding this.

Thanks in advance.

Answer 1

Having thought about this a bit more, I think that the numbers from the first vector come from the various characters of the regular representation.

So for example, the one related to the permutation $(1\ 2)$, would be the matrix $$\begin{pmatrix} 0 &1 &0 &0 \\ 1 &0 &0 &0 \\ 0 &0 &1 &0 \\ 0 &0 &0 &1 \end{pmatrix},$$ which has character $2$.