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Math Help - Linear representations

  1. #1
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    Linear representations

    Prove there exists a left ideal I of C[ S_{3}] (so the ring in the symmetric group S3 with complex coeffs) which does not contain the element e_{1}:= \frac{1}{6} \sum [y] nor the element e_{2}:= \frac{1}{6} \sum sgn(y) [y]

    (Where both the sums are over all y in S_{3})

    (Forgive my Latex; I'm still trying to get the hang of it)
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  2. #2
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    Quote Originally Posted by brisbane View Post
    Prove there exists a left ideal I of C[ S_{3}] (so the ring in the symmetric group S3 with complex coeffs) which does not contain the element e_{1}:= \frac{1}{6} \sum [y] nor the element e_{2}:= \frac{1}{6} \sum sgn(y) [y]

    (Where both the sums are over all y in S_{3})

    (Forgive my Latex; I'm still trying to get the hang of it)
    Consider M=\mathbb{C}[S_3] as a \mathbb{C}[S_3]-module of itself.

    Then, M is decomposed as (Dummit & Foote p869-870)

    M = z_1M \oplus z_2M \oplus \cdots \oplus z_rM, where z_i = \frac{\chi_i(1)}{|S_3|}\sum_{g \in S_3}\chi_i(g^{-1})g.

    Each z_iM is a submodule of M (two-sided ideal of M) and your e_1, e_2 correspond to z_1, z_2, respectively.

    Since S_3 has three conjugacy classes, there are three irreducible characters of \mathbb{C}[S_3].
    The z_3M is the ideal you are looking for, where z_3=\frac{2}{6}(2e - (123) - (132)).

    Note that z_3M has a dimension 4, which is not irreducible.
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