# representation of symmetric group

• Jan 16th 2011, 05:53 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, 06:47 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.

$\displaystyle V_{\lambda^\prime} \cong Ab_{\lambda^\prime} a_{\lambda^\prime}$ 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 $\displaystyle V_{\lambda^\prime}$.