1. representation of symmetric group

I'm stuck at the following exercise of the Fulton-Harris "Representation theory" (p.47 ex. 4.4 (c)):

Show that the representation of a partition of the symmetric group is the tensor product of the representation of the conjugate partition and the alternating representation.

2. Originally Posted by bloob
I'm stuck at the following exercise of the Fulton-Harris "Representation theory" (p.47 ex. 4.4 (c)):

Representation theory: a first course - Google Books

Show that the representation of a partition of the symmetric group is the tensor product of the representation of the conjugate partition and the alternating representation.

$V_{\lambda^\prime} \cong Ab_{\lambda^\prime} a_{\lambda^\prime}$ by Exercise 4.4 (a), where $\lambda^\prime$ is the conjugate partition of $\lambda$.
Note that $c_\lambda=a_\lambda b_\lambda=\sum_{g \in P_\lambda, h \in Q_\lambda}{\text{sgn}(h)e_{gh}}$ (here). Note also that $P_{\lambda^\prime} = Q_\lambda$.
Try first a simple one like $\lambda=(2, 1)$ and generalize the relationship between $V_\lambda$ and $V_{\lambda^\prime}$.