Recall that $S_5$ acts on $\mathbb C^5$ by permuting its coordinates and that $\mathbb C^5$ decomposes as $V\oplus W$ where $W$ is the trivial representation and $V$ has dimension $4$. Show that the alternating square of $V$, which has dimension $6$, is irreducible. What is the tableau which corresponds to this representation?
Of course, $V$ is the natural representation and $W$ is the trivial representation. I could show that the alternating square of $V$ is irreducible. But, I don't have any idea how to find the tableau which corresponds to this representation.
As David A Craven points out, the degrees of the characters of $S_5$ are $1,1,4,4,5,5,6$. But, how does that connect with the shape of the tableau? In general, I don't know any relation between degree of a character and tableau corresponding to that character. It would help if anyone can provide any information about it.
I know the Hook Length formula in the sense that the number of standard Young tableaux of a given shape is given by $\frac{n!}{\prod h_{ij}}$ but I really don't know how to connect that to degree of representation.
