Complex and Real Representations, their differences by decomposition

1. May 12, 2012

aquila457

1. The problem statement, all variables and given/known data

Decompose \mathbb{C}^{5}, the 5 dimensional complex Euclidean space) into invariant subspaces irreducible with respect to the group C_{5} \cong \mathbb{Z}_{5} of cyclic permutations of the basis vectors e_{1} through e_{5}.

Hint: The group is Abelian, so all the irreps are one-dimensional. Therefore, you can use the simplified form of the projection operators, with characters.

Further, try to do the same for \mathbb{R}^{5}, insisting that the basis vectors can only be combined with real coefficients. What is the difference between real and complex reps?

2. Relevant equations

This may be the right projection operator, unsure:
P^{\alpha}=d_{\alpha}\|G| \sum_{g} \chi^{(\alpha)}(g)*O_{g}

3. The attempt at a solution

I am confused by the term decompose, so my attempts have been floundering. I tried to write out the character table for \mathbb{Z}_{5} and I think I succeeded in that, but am unsure if it is needed. The hint about the projection operators served to confuse me more, although I readily understand the part about 1D irreps and Abelian. Is this asking me to construct reps (matrices) using cyclic permutations of C_{5}? If so, how am I supposed to use projection operators in this case to get them; This seems right however.

Any help would be wonderful.