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

[OMM!]

Homework Statement

$$G = D_6 = \left\{a,b: a^{3} = b^{2} = 1, b^{-1}ab = a^{-1}\right\}$$

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

where U_{i} are irreducible CG-submodules.

Homework Equations

We have $$w = e^{2 \pi i/3}:$$

$$v_0 = 1 + a + a^{2}; v_1 = 1 + w^{2}a + wa^{2}; v_2 = 1 + wa + w^{2}a^{2}.$$

$$w_0 = bv_0; w_1 = bv_1; w_2 = bv_2$$

The Attempt at a Solution

I've shown that for i = 0,1,2: $$v_{i}a = w^{i}v_{i}$$

And so clearly, $$sp(v_{i})$$ 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 $$w_{0}b = v_{0}, w_{1}b = v_{2}, w_{2}b = v_{1}$$

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.

 

[lippyka]

I know that an irreducible module is one where the only submodule is the zero submodule. I think I need to show that none of the above submodules reduce to the zero submodule, but I'm not sure how to do this.