\(\def\C{\mathbf{C}}\)
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.
| 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 \(\exp(2\pi ir/n)\).
More on Permutation Reps