= Solution
The additive <cyclic group> $\mathbb Z_n$ is generated by $1$, with $n\cdot1=0$. Every complex <irreducible representation> is one-dimensional by <1/a>. If $R(1)=z$, the defining relation requires $z^n=1$. Conversely each such <root of unity> defines a <group homomorphism>. Hence the complete list is
$$
\boxed{\rho_k(g)=\exp\!\left(\frac{2\pi i kg}{n}\right),\qquad k=0,\ldots,n-1,\quad g\in\mathbb Z_n.}
$$
Changing the representative of $g$ changes the exponent by an integer multiple of $2\pi i$, so the formula is well-defined. These <representations> are pairwise inequivalent because their values on the generator differ. The case $k=0$ is the <trivial representation>.
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