User:Jim.belk/Representation Theory of the Dihedral Group Draft

In mathematics, the representation theory of dihedral groups is particular simple case of the representation theory of finite groups. It has important applications in group theory and chemistry.

The dihedral group Dn is generated by elements a and b with presentation:


 * $$\langle a,b \;|\; a^n = 1,\, b^2 = 1,\, bab = a^{-1} \rangle$$

It has order 2n, with elements $$1,a,a^2,\ldots,a^{n-1}$$ and $$b,ab,a^2b,\ldots,a^{n-1}b$$.

The main features of the representation theory depend on whether n is odd or even.

Standard representation
The dihedral group has a standard representation on the plane, defined as follows:
 * $$ a \mapsto \begin{pmatrix}

\cos 2\pi/n & -\sin 2\pi/n \\[0.5em] \sin 2\pi/n & \cos 2\pi/n\end{pmatrix},\;\;\;\; b \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ This representation represents the action of Dn on a regular polygon centered at the origin. The generator a acts as counterclockwise rotation by an angle of $$2\pi/n$$, and the generator b acts as reflection across the x-axis. This representation is faithful, and is irreducible for all $$n>2$$.

Other planar representations
For 0 &lt; k &lt; n, the k/n representation of Dn is defined as follows:


 * $$ a \mapsto \begin{pmatrix}

\cos 2\pi k/n & -\sin 2\pi k/n \\[0.5em] \sin 2\pi k/n & \cos 2\pi k/n\end{pmatrix},\;\;\;\; b \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$

In this representation, a acts as rotation by a multiple of $$2\pi/n$$, and b acts as a reflection. It can be thought of as the natural action of Dn on the star polygon $$\{n/k\}$$. The 1/n representation is just the standard representation of Dn.

The $$k/n$$ representation is faithful if and only if k and n are relatively prime (i.e. if and only if a acts as a rotation of order n). It is irreducible unless n is even and $$k=n/2$$, as can be seen by computing its character norm. The $$k/n$$ and $$(n-k)/n$$ representations are equivalent for each k, but the representations are otherwise non-isomorphic.

We conclude that the group Dn has $$(n-1)/2$$ irreducible planar representations when n is odd, and $$(n/2)-1$$ irreducible planar representations when n is even.

Linear representations
In addition to the trivial representation, the group Dn has the following one-dimensional representation:
 * $$a\mapsto 1,\;\;\;\;b\mapsto -1$$

This is the representation induced by the quotient map $$\mathrm{D}_n\rightarrow \mathbb{Z}_2$$.

When n is even, there are two more linear representations of Dn:


 * $$a\mapsto -1,\;\;\;\;b\mapsto 1$$


 * $$a\mapsto -1,\;\;\;\;b\mapsto -1$$

Odd case
When n is odd, Dn has the following $$\lfloor n/2\rfloor +2$$ conjugacy classes:


 * $$\begin{array}{l}

\{1\}                        \\[0.25em] \left\{ b, ab, a^2 b, \ldots, a^{n-1} b \right\} \\[0.25em] \left\{ a, a^{n-1} \right\},\, \left\{ a^2, a^{n-2} \right\},\, \ldots,\, \left\{ a^{\lfloor n/2 \rfloor}, a^{\lfloor n/2 \rfloor +1} \right\} \end{array}$$

Here are the character tables for the first few odd dihedral groups. The general pattern should be apparent:


 * $$\begin{array}{r|rrr}

& 1 & b & a \\ \mathrm{D}_3 & 1 & 3 & 2 \\ [0.25em] \hline \mathrm{trivial} & 1 & 1 & 1  \\[0.25em] \mathrm{det} & 1 & -1 & 1 \\[0.25em] 1/3 & 2 & 0 & -1 \\[0.25em] \end{array}\;\;\;\;\;\;\;\; \begin{array}{r|rrcc} & 1 & b & a & a^2 \\ \mathrm{D}_5 & 1 & 5 & 2 & 2  \\[0.25em] \hline \mathrm{trivial} & 1 & 1 & 1 & 1 \\[0.25em] \mathrm{det}    & 1 & -1 & 1 & 1 \\[0.25em] 1/5 & 2 & 0 & 2\cos \tfrac{2\pi}{5} & 2\cos \tfrac{4\pi}{5} \\[0.25em] 2/5 & 2 & 0 & 2\cos \tfrac{4\pi}{5} & 2\cos \tfrac{8\pi}{5} \end{array}$$


 * $$\begin{array}{r|rrccc}

& 1 & b & a & a^2 & a^3 \\ \mathrm{D}_7 & 1 & 7 & 2 & 2  &  2  \\[0.25em] \hline \mathrm{trivial} & 1 & 1 & 1 & 1 & 1 \\[0.25em] \mathrm{det}    & 1 & -1 & 1 & 1 & 1 \\[0.25em] 1/7 & 2 & 0 & \cos \tfrac{2\pi}{7} & \cos \tfrac{4\pi}{7} & \cos \tfrac{6\pi}{7} \\[0.25em] 2/7 & 2 & 0 & \cos \tfrac{4\pi}{7} & \cos \tfrac{8\pi}{7} & \cos \tfrac{12\pi}{7} \\[0.25em] 3/7 & 2 & 0 & \cos \tfrac{6\pi}{7} & \cos \tfrac{12\pi}{7} & \cos \tfrac{18\pi}{7} \end{array}$$

Even case
When n is even, Dn has the following $$(n/2)+3$$ conjugacy classes:


 * $$\begin{array}{l}

\{1\}                        \\[0.25em] \left\{ b, a^2 b, a^4 b, \ldots, \right\} \\[0.25em] \left\{ ab, a^3 b, a^5 b, \ldots \right\} \\[0.25em] \left\{ a, a^{n-1} \right\}, \left\{ a^2, a^{n-2} \right\}, \ldots, \left\{ a^{(n/2)-1}, a^{(n/2)+1} \right\} \\[0.25em] \left\{ a^{n/2} \right\} \end{array}$$

Here are the character tables for the first few even dihedral groups. The general pattern is similar to the pattern for D10:



\begin{array}{r|rrrrr} & 1 & b & ab & a & a^2 \\ \mathrm{D}_4 & 1 & 2 & 2 & 2 & 1  \\[0.25em] \hline \mathrm{trivial} & 1 & 1 &  1 &  1 & 1 \\[0.25em] \mathrm{det}    & 1 & -1 & -1 &  1 & 1 \\[0.25em] \mathrm{linear} & 1 &  1 & -1 & -1 & 1 \\[0.25em] \mathrm{linear} & 1 & -1 &  1 & -1 & 1 \\[0.25em] 1/4 & 2 & 0 & 0 & 0 & -2 \\[0.25em] \end{array} \;\;\;\;\;\;\;\; \begin{array}{r|rrrrrr} & 1 & b & ab & a & a^2 & a^3 \\ \mathrm{D}_6 & 1 & 3 & 3 & 2 & 2 & 1  \\[0.25em] \hline \mathrm{trivial} & 1 & 1 &  1 &  1 & 1 &  1 \\[0.25em] \mathrm{det}    & 1 & -1 & -1 &  1 & 1 &  1 \\[0.25em] \mathrm{linear} & 1 &  1 & -1 & -1 & 1 & -1 \\[0.25em] \mathrm{linear} & 1 & -1 &  1 & -1 & 1 & -1 \\[0.25em] 1/6 & 2 & 0 & 0 & 1 & -1 & -2 \\[0.25em] 2/6 & 2 & 0 & 0 & -1 & -1 & 2 \\[0.25em] \end{array} $$



\begin{array}{r|rrrrrrr} & 1 & b & ab & a & a^2 & a^3 & a^4 \\ \mathrm{D}_8 & 1 & 4 & 4 & 2 & 2 & 2 & 1  \\[0.25em] \hline \mathrm{trivial} & 1 & 1 &  1 &  1 & 1 &  1 & 1 \\[0.25em] \mathrm{det}    & 1 & -1 & -1 &  1 & 1 &  1 & 1 \\[0.25em] \mathrm{linear} & 1 &  1 & -1 & -1 & 1 & -1 & 1 \\[0.25em] \mathrm{linear} & 1 & -1 &  1 & -1 & 1 & -1 & 1 \\[0.25em] 1/8 & 2 & 0 & 0 & \sqrt{2} &  0 & -\sqrt{2} & -2 \\[0.25em] 2/8 & 2 & 0 & 0 &        0 & -2 &         0 &  2 \\[0.25em] 3/8 & 2 & 0 & 0 & -\sqrt{2} & 0 &  \sqrt{2} & -2 \end{array}$$



\begin{array}{r|rrrccccc} & 1 & b & ab & a & a^2 & a^3 & a^4 & a^5 \\ \mathrm{D}_{10} & 1 & 5 & 5 & 2 & 2 & 2 & 2 & 1  \\[0.25em] \hline \mathrm{trivial} & 1 & 1 &  1 &  1 & 1 &  1 & 1 &  1 \\[0.25em] \mathrm{det}    & 1 & -1 & -1 &  1 & 1 &  1 & 1 &  1 \\[0.25em] \mathrm{linear} & 1 &  1 & -1 & -1 & 1 & -1 & 1 & -1 \\[0.25em] \mathrm{linear} & 1 & -1 &  1 & -1 & 1 & -1 & 1 & -1 \\[0.25em] 1/10 & 2 & 0 & 0 & 2\cos\frac{\pi}{5} & 2\cos\frac{2\pi}{5} & 2\cos\frac{3\pi}{5} & 2\cos\frac{4\pi}{5} & 2\cos\frac{5\pi}{5} \\[0.25em] 2/10 & 2 & 0 & 0 & 2\cos\frac{2\pi}{5} & 2\cos\frac{4\pi}{5} & 2\cos\frac{6\pi}{5} & 2\cos\frac{8\pi}{5} &  2\cos\frac{10\pi}{5} \\[0.25em] 3/10 & 2 & 0 & 0 & 2\cos\frac{3\pi}{5} & 2\cos\frac{6\pi}{5} & 2\cos\frac{9\pi}{5} & 2\cos\frac{12\pi}{5} & 2\cos\frac{15\pi}{5} \\[0.25em] 4/10 & 2 & 0 & 0 & 2\cos\frac{4\pi}{5} & 2\cos\frac{8\pi}{5} & 2\cos\frac{12\pi}{5} & 2\cos\frac{16\pi}{5} &  2\cos\frac{20\pi}{5} \end{array} $$

Note that each of these tables is actually complete: the squares of the dimensions of the irreducible representations must sum to the order of the group, which is in this case 2n.