= Complex irreducible representations of a finite cyclic group
{title2=$\chi_j(g^r)=e^{2\pi ijr/n},\quad0\leq j<n$}
For $C_n=\langle g:g^n=e\rangle$, the complex <irreducible representations> are precisely the $n$ one-dimensional maps $g\mapsto e^{2\pi ij/n}$, with $0\leq j<n$. Their <characters> have the displayed values, forming the discrete Fourier character table. Indeed a representing matrix $A$ satisfies $A^n=I$. Applying the <fundamental theorem of algebra> to its <characteristic polynomial> gives an <eigenvalue> and an <eigenvector> over the <complex numbers>. Its line is an <invariant subspace> under every power of $A$; irreducibility therefore forces the entire space to be that line. Its <eigenvalue> is an $n$th <root of unity>. Different roots give inequivalent one-dimensional representations, and there are exactly $n$ possibilities.
All <conjugacy classes> are singletons because the <group> is <Abelian>. The number of the displayed irreducibles equals the group order, and their squared dimensions sum to $n$, in agreement with the <sum of squares of irreducible representation dimensions>. Over the real numbers, conjugate nonreal characters instead combine into the planes described by the <irreducible real representations of a finite cyclic group>.
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