Dihedral Group D4
Definition
The set of symmetries of an unoriented 4 sides regular polygon (a square).
We label the vertices of the square $\ket{ABCD}$. The transformations are thus
$$ \begin{array}{c|ccc|c} e & \ket{ABCD} & \quad & b & \ket{BADC} \\ a & \ket{BCDA} & \quad &ab & \ket{ADCB} \\ a^{2} & \ket{CDAB} & \quad &a^{2}b & \ket{DCBA} \\ a^3 & \ket{DABC} & \quad & a^3b & \ket{CBAD} \\ \end{array} $$The transformations are
- $\{a^n\}$ rotations of the square by $n \pi / 4$
- $\{b, a^2b\}$ reflections about the axes bisecting opposing edges
- $\{ab, a^3b\}$ reflections about the axes joining opposing vertices
Multiplication Table
$$ \begin{array}{c|cccccccc} D_{4} & e & a & a^2 & a^3 & b & ab & a^2b & a^3b \\[6pt] \hline e & e & a & a^2 & a^3 & b & ab & a^2b & a^3b \\[6pt] a & a & a^2 & a^3 & e & ab & a^2b & a^3b & b \\[6pt] a^2 & a^2 & a^3 & e & a & a^2b & a^3b & b & ab \\[6pt] a^3 & a^3 & e & a & a^2 & a^3b & b & ab & a^2b \\[6pt] b & b & a^3b & a^2b & ab & e & a^3 & a^2 & a \\[6pt] ab & ab & b & a^3b & a^2b & a & e & a^3 & a^2 \\[6pt] a^2b & a^2b & ab & b & a^3b & a^2 & a & e & a^3 \\[6pt] a^3b & a^3b & a^2b & ab & b & a^3 & a^2 & a & e \end{array} $$Group Properties
The multiplication table implicitely verifies closure, identity and inverse properties.
Associativity in inherited by the fact that each transformation is equivalent to a $2\times2$ matrix transformation of the coordinates at each vertex, and matrix multiplication is associative.
Equivalence Classes
The equivalence classes were determined intuitively. They can also be verified explicitly.
$$ \begin{array}{c|c|c} \mathcal{C}_{1} & \{ e \} & \text{trivial} \\[6pt] \hline \mathcal{C}_{2} & \{ a^{2} \} & \text{center point rotation} \\[6pt] \hline \mathcal{C}_{3} & \{ a, a^3 \} & \text{other rotations} \\[6pt] \hline \mathcal{C}_{4} & \{b, a^2b \} & \text{products belong to $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$} \\[6pt] \hline \mathcal{C}_{5} & \{ab, a^3b \} & \text{products belong to $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$} \\[6pt] \end{array} $$Subgroups
Since $\lvert D_{4} \rvert = 8$, the allowed orders of subgroups are $2$ or $4$.
Closure is verified for each of these sets, meaning they are subgroups. Invariant subgroups are identified if they are the union of equivalence classes.
$$ \begin{array}{c|c|c} H_{1} & \{ e, a^2 \} & \text{invariant} \\[6pt] \hline H_{2} & \{ e, a, a^2, a^3 \} & \text{invariant} \\[6pt] \hline H_{3} & \{ e, b \} & \\[6pt] \hline H_{4} & \{e, ab \} & \\[6pt] \hline H_{5} & \{e, a^2b \} & \\[6pt] \hline H_{6} & \{ e, a^3b \} & \\[6pt] \hline H_{7} & \{ e, a^2, b, a^2b \} & \text{invariant} \\[6pt] \hline H_{8} & \{ e, a^2, ab, a^3b \} & \text{invariant} \end{array} $$Representations
Number of Representations
Remember that the sum of the squares of the dimension of each representation is equal to the order of the group. So, if we have $N$ representations, each of dimension $n_{i}$,
$$ \sum_{i}^N n_{i}^{2} = 8 $$We know that the trivial representation must be present, meaning
$$ \sum_{i}^{N-1} = n_{i}^{2} = 7 $$Therefore, there are either
- One irrep of dimension $2$ and three of dimension $1$, or
- Seven irreps of dimension $1$.
But remember! The number of representations is equal to the number of equivalence classes. Therefore, there are 5 representations, 4 one-dimensional and one two-dimensional.