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 D_n 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 $b,ab,a^{2}b,\ldots ,a^{n-1}b$ .

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

Irreducible representations

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 D_n 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 < k < n, the k/n representation of D_n 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 D_n on the star polygon $\{n/k\}$ . The 1/n representation is just the standard representation of D_n.

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 D_n 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 D_n 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 D_n:

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

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

Character tables

Odd case

When n is odd, D_n 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:

Failed to parse (unknown function "\begin{array}"): {\displaystyle \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}}

Failed to parse (unknown function "\begin{array}"): {\displaystyle \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, D_n 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:

Failed to parse (unknown function "\begin{array}"): {\displaystyle \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} }

Failed to parse (unknown function "\begin{array}"): {\displaystyle \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}}

Failed to parse (unknown function "\begin{array}"): {\displaystyle \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.