Solution
= Solution
For $\lambda=(1^n)$, there is exactly one <standard Young tableau>, the column with entries $1,2,\ldots,n$ from top to bottom. Its <content vector of a standard Young tableau> is
$$
\boxed{C(T)=(0,-1,\ldots,-(n-1))}.
$$
Every adjacent distance is $-1$, so <Young orthogonal form> gives \b[$s_rw_T=-w_T$ for every $r$]. Therefore
$$
\boxed{S^{(1^n)}\cong\mathbb C_{\rm sgn}},
$$
the <sign representation>.