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Math Help - representation of symmetric group

  1. #1
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    representation of symmetric group

    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?
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  2. #2
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    Quote Originally Posted by bloob View Post
    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}.
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  3. #3
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    Thanks a lot for your help. I think I've figured out the solution now.
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