Linear representation theory of finite cyclic groups

This article discusses the linear representation theory of a finite cyclic group of order $$n$$, which for concreteness we take as the group of integers modulo n, i.e., the group $$\mathbb{Z}/n\mathbb{Z}$$ or $$\mathbb{Z}_n$$.

For the linear representation theory of the infinite cyclic group, see linear representation theory of group of integers.

Summary
We denote by $$\varphi(n)$$ the Euler totient function of $$n$$, defined as the number of numbers in $$\{1,2,\dots,n \}$$ that are relatively prime to $$n$$, or equivalently as the order of the multiplicative group of the ring of integers modulo $$n$$.

Group ring interpretation
For any commutative unital ring $$R$$, the group ring $$R[\mathbb{Z}_n]$$ is isomorphic to the ring:

$$R[t]/(t^n - 1)$$

where the isomorphism sends an element $$a \pmod n$$ to the equivalence class of $$t^a$$.

Case of ring that contains primitive roots of unity and is uniquely $$n$$-divisible
If $$R$$ contains a primitive $$n^{th}$$ root of unity, say $$\zeta$$, then we have:

$$t^n - 1 = (t - 1)(t - \zeta)(t - \zeta^2) \dots (t - \zeta^{n-1})$$

If $$R$$ is uniquely $$n$$-divisible, then by the Chinese remainder theorem, we get the following direct sum/direct product decomposition as rings:

$$R[t]/(t^n - 1) \cong R[t]/(t - 1) \times R[t]/(t - \zeta) \times R[t]/(t - \zeta^2) \times \dots \times R[t]/(t - \zeta^{n-1}) \cong R \times R \times R \times \dots \times R$$

(We could also use $$\oplus$$ instead of $$\times$$ in the line above).

The choice of decomposition using the Chinese remainder theorem also corresponds to the use of characters, as follows. We have an isomorphism:

$$R[t]/(t^n - 1) \cong Re_1 \oplus Re_2 \oplus \dots Re_n$$

where the $$e_j$$ are idempotents. Each idempotent is given as follows: take a character, and consider the element of the group ring obtained by multiplying each group element by its character value, and dividing the sum by the order of the group. The idempotents look like:

$$e_1 = \frac{1 + t + t^2 + \dots + t^{n-1}}{n}, e_2 = \frac{1 + \zeta t + \zeta^2t^2 + \dots + \zeta^{n-1}t^{n-1}}{n}, e_3 = \frac{1 + \zeta^2 t + \dots + \zeta^{2(n-1)}t^{n-1}}{n}, \dots$$

Note that in the case of fields, unique $$n$$-divisibility is equivalent to saying that the characteristic of the field is either zero or is a prime not dividing $$n$$.

Also note that the decomposition does not work over $$\mathbb{Z}[\zeta]$$, because of the absence of $$n$$-divisibility. This is true even though the representations can be realized over $$\mathbb{Z}[\zeta]$$.