- Describe the irreducible complex representations and character table of the dihedral group \(D_4\) (order 8).
- Explicitly construct the 2-dimensional irreducible representation.
- Provide its geometric interpretation as symmetries of the square.
Solution
Write \[ D_4=\langle r,s\mid r^4=s^2=1,\ srs=r^{-1}\rangle. \] Its conjugacy classes are \[ \{1\},\quad \{r^2\},\quad \{r,r^3\},\quad \{s,sr^2\},\quad \{sr,sr^3\}. \] Hence there are five irreducible complex characters. Since \[ [D_4,D_4]=\langle r^2\rangle, \qquad D_4^{\mathrm{ab}}\cong C_2\times C_2, \] there are four one-dimensional characters, obtained by choosing independently \(r\mapsto\pm1\) and \(s\mapsto\pm1\).
The remaining irreducible representation is the geometric representation \[ \rho(r)=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \qquad \rho(s)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. \] These matrices satisfy \(r^4=s^2=1\) and \(srs=r^{-1}\), so they define a representation. Its character is \[ \chi_5=(2,-2,0,0,0) \] on the five classes above, and \[ \langle\chi_5,\chi_5\rangle =\frac18(4+4)=1, \] so it is irreducible.
Thus the character table is \[ \begin{array}{c|ccccc} &1&r^2&\{r,r^3\}&\{s,sr^2\}&\{sr,sr^3\}\\\hline \chi_1&1&1&1&1&1\\ \chi_2&1&1&-1&1&-1\\ \chi_3&1&1&1&-1&-1\\ \chi_4&1&1&-1&-1&1\\ \chi_5&2&-2&0&0&0 \end{array} \] with class sizes \(1,1,2,2,2\).
Geometrically, \(\rho(r)\) is rotation of the square by \(\pi/2\), and \(\rho(s)\) is reflection across the \(x\)-axis.