$$\def\C{\mathbf{C}}$$

# Cyclic Groups

• The cyclic group $$\C_n$$ is of order $$n$$. We use additive notation, so the identity element is 0.
• As the group is Abelian, no two distinct elements are conjugate, so there are $$n$$ conjugacy classes each containing 1 element
• This is close to the theory of Fourier series.
 C2 0 1 A0 1 1 A1 1 -1
 C3 0 1 2 A0 1 1 1 E 2 -1 -1
• The representation E is irreducible but not absolutely irredicible. Indeed, over $$\mathbb{C}$$ it splits as $$E = E_+ \oplus E_-$$, where $$E_+$$ has character $$(1,\,\omega,\,\bar\omega)$$ and $$E_-$$ has character $$(1,\,\bar\omega,\,\omega)$$, where $$\omega$$ is a cube root of unity.
 C4 0 1 2 3 A0 1 1 1 1 A1 1 -1 1 -1 E 2 0 -2 0
 C5 0 1 2 3 4 A0 1 1 1 1 1 E1 2 $$\gamma$$ $$\bar\gamma$$ $$\bar\gamma$$ $$\gamma$$ E2 2 $$\bar\gamma$$ $$\gamma$$ $$\gamma$$ $$\bar\gamma$$
• $$\gamma = \textstyle\frac12(\sqrt5-1)$$ and $$\bar\gamma = -\textstyle\frac12(\sqrt5+1)$$
 C6 0 1 2 3 4 5 A0 1 1 1 1 1 1 A1 1 -1 1 -1 1 -1 E1 2 1 -1 -2 -1 1 E2 2 -1 -1 2 -1 -1
##### And so the pattern goes on ...
• n even: $$\C_n$$ has two 1-dimensional representations and ½(n-2) 2-dimensional representations.
• n odd: $$\C_n$$ has one 1-dimensional representation and ½(n-1) 2-dimensional representations.
• Denote by $$E_r$$ the 2-d rep where the generator of $$\C_n$$ has character $$2\cos(2\pi r/n)$$, so acts on the plane by rotation through an angle of $$2\pi r/n$$ (for $$1 \leq r \leq n/2$$). These 2-dimensional reps are irreducible but not absolutely irreducible, and their complexification splits as a sum of two 1-d reps.
• Indeed, the complexification $$E_r^{\mathbb{C}} = U_r \oplus U_{n-r}$$, where $$U_r$$ is the 1-d complex rep with the generator of $$\C_n$$ acting as multiplication by $$\exp(2\pi ir/n)$$.

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