In the book Group Theory in a Nutshell for Physicist of A.Zee, the reader has been asked to compute the characters of a given $4$ dimensional representation of $S_4$. Taking the 4 elements to permute in a column, we can use this $4$dim representation like this:
$$\begin{bmatrix}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}a \\ b \\ c \\ d\end{bmatrix} = \begin{bmatrix}c \\ a \\ b \\ d\end{bmatrix}$$
Where the 4x4 Matrix corresponds to the cycle $(1 2 3)$.
Let's call the representation $D(g)$, where $g\in S_4$ and its character as $C$.
So, taking one representative for each $S_4$ conjugacy class, we can have:
- $C(D((e))) = 4$
- $C(D((1 2 3))) = 1$
- $C(D((1 2))) = 2$
- $C(D((1 2)(3 4))) = 0$
- $C(D((1 2 3 4))) = 0$
I've checked them carefully and they seem to me correct.
Now, as usual, I start wondering if this representation is reducible. Currently we know from the $S_4$ character table we have 5 irreps with dimension $1, 1, 2, 3, 3$.
So this representation is not reducible but could be gotten from the above ones.
I was wondering if it could be gotten as a direct product of the matrices of the $2$dim representation.
May be you give me some hints if this could be right or eventually point me to the right solution, please?
Thanks in advance.
EDIT:
I've tried to build the Canonical Jordan Form of the given 4x4 matrix, getting:
$$\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & -\frac{1}{2}(1 + i\sqrt 3) & 0 \\ 0 & 0 & 0 & -\frac{1}{2}(1 - i\sqrt 3)\end{bmatrix}$$
Now, I should be able to recognize some blocks of dimension 1 and 3 in there, related to the 2 irreps. My bet is the first block is the top left most 1, i.e. the $S_4$ trivial representation, and the other 3 entries are forming the other 3dim irrep. Inspecting the character of this latter for the group element $(1 2 3)$ gives $0$, which is exactly the character of the 3dim standard representation. Of course I should check the other $S_4$ class characters to have a complete match, but my bet is my representation is the direct sum of the $S_4$ trivial representation and the standard one. Is there a precise way to check claim this without checking all characters and match the character table for the standard representation?
Tx