Group Representations of $C_5$

Question

If I have the cyclic group $C_5=\langle g\mid g^5=e\rangle$ and the left regular representation $V=\mathbb{C}C_5$. Would the matrices of this representation (in the standard basis) be defined by $\rho_g$, the $5\times 5$ matrix with $e^{\frac{2\pi i}{5}}$ on the diagonal and zeros elsewhere? If not, why?

Answer 1

No, they are not. I suppose that what you call the “standard basis” is $\{v_e,v_g,v_{g^2},v_{g^3},v_{g^4}\}$. Then the action of, say, $g$ on the vectors of this basis is

• $gv_e=v_g$;
• $gv_g=v_{g^2}$;
• $gv_{g^2}=v_{g^3}$;
• $gv_{g^3}=v_{g^4}$;
• $gv_{g^4}=v_e$.

It is therefore a permutation matrix, not a diagonal one. The same thing holds for the actions of the other elements of $C_5$, except, of course, for the action of $e$ (which is the identity map).