# Thread: representation of symmetric group

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.

Can anybody help me please?

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.

Can anybody help me please?

$\displaystyle V_{\lambda^\prime} \cong Ab_{\lambda^\prime} a_{\lambda^\prime}$ by Exercise 4.4 (a), where $\displaystyle \lambda^\prime$ is the conjugate partition of $\displaystyle \lambda$.

Note that $\displaystyle c_\lambda=a_\lambda b_\lambda=\sum_{g \in P_\lambda, h \in Q_\lambda}{\text{sgn}(h)e_{gh}}$ (here). Note also that $\displaystyle P_{\lambda^\prime} = Q_\lambda$.

Try first a simple one like $\displaystyle \lambda=(2, 1)$ and generalize the relationship between $\displaystyle V_\lambda$ and $\displaystyle V_{\lambda^\prime}$.

3. Thanks a lot for your help. I think I've figured out the solution now.