Dihedral Groups
The dihedral group Dn or Dih(2n) is of order 2n. It is the symmetry group of the regular n-gon.
(Some denote this group D2n because its order is 2n, but I prefer Dn - after all, one doesn't denote the symmetric group Sn by Sn! - nor J1 by J175560. On the other hand, Dih(2n) is fine as there's no conflict of notation.)
On this page:
D2
D3
D4
D5
D6
- The Schoenflies notation for Dn depends on which 3-dimensional representation is used: Cnv refers to the the 3-dimensional representation acting trivially in the 'vertical', while Dn denotes the representation in 3 dimensions where the reflections act by (-1) in the 'vertical'.
- A fixed generator of order n (rotation by \(2\pi/n\) is denoted ρ).
- For n even, there are two conjugacy classes of reflection, denoted κ and κ', with κ acting as reflections in a line joining two vertices, and κ' reflection in a line joining the mid-points of two opposite edges. There is an (outer) automorphism of Dn exchanging the two reflections.
- For n odd, all reflections are conjugate;
- Rotations through θ and -θ are conjugate for all n.
- This is close to the theory of Fourier series, and symmetric circulant matrices. See Notes for details.
D2 = Dih(4)
\(D_2 \simeq \mathbb{Z}_2\times\mathbb{Z}_2\) with generators κ and κ'. It is the symmetry group of the rectangle.
| D2
| e
| κ
| κ'
| ρ=κκ'
|
| A0
| 1
| 1
| 1
| 1
|
| A1
| 1
| 1
| -1
| -1
|
| A2
| 1
| -1
| 1
| -1
|
| A3
| 1
| -1
| -1
| 1
|
D3 = Dih(6)
| D3
| e
| ρ
| κ
|
| #
| 1
| 2
| 3
|
| A0
| 1
| 1
| 1
|
| A1
| 1
| 1
| -1
|
| E
| 2
| -1
| 0
|
D4 = Dih(8)
| D4
| e
| ρ
| ρ2
| κ
| κ'
|
| #
| 1
| 2
| 1
| 2
| 2
|
| A0
| 1
| 1
| 1
| 1
| 1
|
| A1
| 1
| 1
| 1
| -1
| -1
|
| B1
| 1
| -1
| 1
| 1
| -1
|
| B2
| 1
| -1
| 1
| -1
| 1
|
| E
| 2
| 0
| -2
| 0
| 0
|
- The permutation representation on the 4 vertices of the square is A0 + B1 + E
- The permutation representation on the 4 edges of the square is A0 + B2 + E
- The permutation representation on the 2 diagonals of the square is A0 + A1
- The orientation permutation representation on the 4 oriented edges of the square is A1 + B1 + E
- E is the standard representation for the action on the plane.
D5 = Dih(10)
| D5
| e
| \(\rho\)
| \(\rho^2\)
| \(\kappa\)
| notes
|
| #
| 1
| 2
| 2
| 5
| |D5|=10
|
| A0
| 1
| 1
| 1
| 1
| trivial rep
|
| A1
| 1
| 1
| 1
| -1
| "orientation" rep
|
| E1
| 2
| \(\gamma\)
| \(\bar\gamma\)
| 0
| symmetry of pentagon
|
| E2
| 2
| \(\bar\gamma\)
| \(\gamma\)
| 0
|
|
- \(\gamma = 2\cos(2\pi/5) = \frac12(\sqrt5-1)\) (= golden ratio)
- \(\bar\gamma = 2\cos(4\pi/5) = -\frac12(\sqrt5+1)\)
- There is an outer automorphism of D5, taking \(\rho\) to \(\rho^2\). This interchanges E1 and E2.
- The permutation representation on the 5 vertices of the pentagon is A0+ E1+ E2
D6 = Dih(12)
| D6
| e
| ρ
| ρ2
| ρ3
| κ
| κ'
| notes
|
| #
| 1
| 2
| 2
| 1
| 3
| 3
| |D6|=12
|
| A0
| 1
| 1
| 1
| 1
| 1
| 1
| trivial rep
|
| A1
| 1
| 1
| 1
| 1
| -1
| -1
| "orientation rep"
|
| B1
| 1
| -1
| 1
| -1
| 1
| -1
| alternating rep
|
| B2
| 1
| -1
| 1
| -1
| -1
| 1
|
|
| E1
| 2
| 1
| -1
| -2
| 0
| 0
| symmetry of hexagon
|
| E2
| 2
| -1
| -1
| 2
| 0
| 0
|
|
- The permutation representation on the 6 vertices of the hexagon is A0+ B1+ E1+ E2
- B1 is the alternating rep because it is the rep obtained if the vertices of the hexagon are weighted successively with +1,-1,+1,-1,+1,-1 (alternating signs)
- The permutation representation on the 3 diagonals joining opposite vertices of the hexagon is A0 + E2
- The 'oriented permutation' representation on the 3 oriented diagonals is B1+ E1
And so the pattern goes on ...
- n even: Dn has four 1-dimensional representations and ½(n-2) 2-dimensional representations.
- n odd: Dn has two 1-dimensional representation and ½(n-1) 2-dimensional representations.
- All the irreducible reps are absolutely irreducible.