# representation of symmetric group

• Jan 16th 2011, 06:53 AM
bloob
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.

• Jan 17th 2011, 07:47 PM
TheArtofSymmetry
Quote:

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 $V_{\lambda^\prime}$.