Solution
= Solution
For the $(n+1)$-dimensional irreducible,
$$
\chi_{V_n}(z)=z^n+z^{n-2}+\cdots+z^{-n}.
$$
The Clebsch--Gordan rule gives
$$
V_4\otimes V_4\cong V_8\oplus V_6\oplus V_4\oplus V_2\oplus V_0.
$$
The even total-spin summands are symmetric and the odd ones alternating, hence
$$
\boxed{S^2V_4\cong V_8\oplus V_4\oplus V_0,
\qquad
\Lambda^2V_4\cong V_6\oplus V_2.}
$$