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Irreducible Modules/Submodules & Group Algebras (G = D6)

  1. Jul 3, 2011 #1
    1. The problem statement, all variables and given/known data
    [tex]G = D_6 = \left\{a,b: a^{3} = b^{2} = 1, b^{-1}ab = a^{-1}\right\}[/tex]

    Must show that the group algebra: [tex]\mathbb{C}G = U_{1} \oplus U_{2} \oplus U_{3} \oplus U_{4}[/tex]

    where U_{i} are irreducible CG-submodules.

    2. Relevant equations
    We have [tex]w = e^{2 \pi i/3}:[/tex]

    [tex]v_0 = 1 + a + a^{2}; v_1 = 1 + w^{2}a + wa^{2}; v_2 = 1 + wa + w^{2}a^{2}.[/tex]

    [tex] w_0 = bv_0; w_1 = bv_1; w_2 = bv_2 [/tex]


    3. The attempt at a solution
    I've shown that for i = 0,1,2: [tex]v_{i}a = w^{i}v_{i}[/tex]

    And so clearly, [tex]sp(v_{i})[/tex] is a C<a>-Module, as va in sp(v_{i}) for all v_{i} in sp(v_{i}) and r in C<a> etc.

    And I've also shown that [tex]w_{0}b = v_{0}, w_{1}b = v_{2}, w_{2}b = v_{1}[/tex]

    To show that sp(v_0, bv_0), sp(v_1, bv_2), sp(v_2, bv_1) are C<b>-Modules.

    And so all of these are CG-submodules of CG, as a and b are generators of the group G.

    So now I need to show that there are 4 irreducible CG-submodules, presumably from the above, or if some of them aren't irreducible they reduce to the required submodules.

    However, I'm a little stuck at this stage.
     
  2. jcsd
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