representation of symmetric group

**bloob** · January 16th 2011, 05:53 AM

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

http://books.google.com/books?id=6GU...page&q&f=false

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?

**TheArtofSymmetry** · January 17th 2011, 06:47 PM

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?

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

**bloob** · January 21st 2011, 03:32 PM

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

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