[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that for the dihedral group [math]D_N\subset S_N[/math] we have

[[math]] law(\chi)=\begin{cases} \left(\frac{3}{4}-\frac{1}{2N}\right)\delta_0+\frac{1}{4}\delta_2+\frac{1}{2N}\delta_N&(N\ even)\\ &\\ \left(\frac{1}{2}-\frac{1}{2N}\right)\delta_0+\frac{1}{2}\delta_1+\frac{1}{2N}\delta_N&(N\ odd) \end{cases} [[/math]]

and look as well at truncated characters.