Specific question on Sn modules

Question

Let $L_{-1}$ denote the 1-dimensional sign-representation of the symmetric group $S_n$ and V the standard $(n - 1)$-dimensional module for $S_n$. How to prove that V and $V \otimes L_{-1}$ are not isomorphic if n > 3.

Answer 1

When $n > 3$, the element $(12)$ has trace $n - 3 \neq 0$ on $V$. Thus, $V$ and $L_{-1} \otimes V$ has different characters.