Let $\lambda:n=n_1+\cdots + n_k$ be a partition of $n$ ($n_1\ge n_2\ge\cdots$).
Let $D_{\lambda}$ be a Young diagram associated with partition $\lambda$.
Let $P_{\lambda}$ and $Q_{\lambda}$ be respectively the row, resp. column stabilizer of $D_{\lambda}$.
Define $a_{\lambda}=\sum_{x\in P_{\lambda}} x$, and $b_{\lambda}=\sum_{y\in Q_{\lambda}} sign(y)y$ in $A=\mathbb{C}[S_n]$. Then
a) The minimal left ideal $Aa_{\lambda}b_{\lambda}$ and $Ab_{\lambda}a_{\lambda}$ give (isomorphic) simple $\mathbb{C}[S_n]$-modules.
b) Some scalar (rational) multiple of $a_{\lambda}b_{\lambda}$ is an idempotent.
It is an exercise in the book of Fulton-Harris to show, using a) and b), that
If $\lambda'$ is conjugate partition of $\lambda$ then $A a_{\lambda'}b_{\lambda'}=A a_{\lambda}b_{\lambda}\otimes U'$ where $U'$ is alternating representation of $S_n$.
I didn't get any direction to use a) and b) for proving this. How should I proceed?
I tried through a small example, whose computation is below (I don't know to type Young diagram in this site, so I typed in tex and posted here picture.)
How these two modules (left ideals) differ by alternating representation, I didn't get idea. The only thing clear in computation is that there is change of sign in the generating elements of the left ideals; but I didn't get direction to write precise justification.