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Thread: Linear representations

  1. #1
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    Linear representations

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

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

    (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[$\displaystyle S_{3}$] (so the ring in the symmetric group S3 with complex coeffs) which does not contain the element $\displaystyle e_{1}$:= $\displaystyle \frac{1}{6}$ $\displaystyle \sum$ [y] nor the element $\displaystyle e_{2}$:= $\displaystyle \frac{1}{6}$ $\displaystyle \sum$ sgn(y) [y]

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

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

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

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

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

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

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