\(\def\C{\mathbf{C}}\)

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 \(\exp(2\pi ir/n)\).

More on Permutation Reps