Group theory tables

Question

Warning: very MathJax heavy.

To avoid MathJax issues, please do not link to the individual answers on this post. Instead, link to the question where users will be shown this guidance.

If you are using this on a computer: Please let the MathJax load (~1 minute) BEFORE clicking any of the desktop links below. This avoids the issues associated with MathJax-heavy answers (namely, the page jumps to the top of the desired answer before the MathJax loads; when the MathJax loads, the desired answer is no longer on your screen). To get back to the table of contents, use the (back to top) links. These also avoid refreshing the page.

If you are using this on a mobile device: Please use the mobile links provided below. These are direct links to the answers and work just fine (tested on iOS 10, 22-11-2016). Hyperlinks to anchors don't work on mobile, hence the need for these.

You can sort by active to view the answers in their intended order.

1. Desktop | Mobile Low-symmetry groups: $C_1$, $C_\mathrm{s}$, $C_\mathrm{i}$

2. Desktop | Mobile $C_n$ $(2 \leq n \leq 8)$

3. Desktop | Mobile $D_n$ $(2 \leq n \leq 6)$

4. Desktop | Mobile $C_{n\mathrm{v}}$ $(2 \leq n \leq 6)$

5. Desktop | Mobile $C_{n\mathrm{h}}$ $(2 \leq n \leq 6)$

6. Desktop | Mobile $D_{n\mathrm{h}}$ $(2 \leq n \leq 6)$

7. Desktop | Mobile $D_{n\mathrm{d}}$ $(2 \leq n \leq 6)$

8. Desktop | Mobile $S_n$ $(n = 4, 6, 8)$

9. Desktop | Mobile Cubic groups: $T, T_\mathrm{d}, T_\mathrm{h}, O, O_\mathrm{h}$

10. Desktop | Mobile Icosahedral groups: $I, I_\mathrm{h}$

11. Desktop | Mobile $C_{\infty\mathrm{v}}$ and $D_{\infty\mathrm{h}}$

12. Desktop | Mobile The full rotation group $\mathrm{R}_3$

13. Desktop | Mobile Direct product tables

14. Desktop | Mobile Tables of descent in symmetry

These tables are taken from the OUP website. There are some typographical corrections which were distributed internally in my university, and I spotted some errors myself. I have corrected the tables where necessary.

The character tables have also been checked against Appendix IIA in F.A. Cotton's Chemical Applications of Group Theory, 3rd ed.

Schönflies symbols are used throughout.

Answer 1

(back to top) $\,\,\,$

Direct product tables

• The tables are symmetric about the diagonal.

• For point groups without subscripted irreps (i.e. the irreps are simply $\mathrm{A}$, $\mathrm{B}$, $\mathrm{E}$, $\mathrm{T}$), simply treat $\mathrm{A_1} \equiv \mathrm{A_2} \equiv \mathrm{A}$, etc.

• Square brackets $[\;]$ are used to indicate the representation spanned by the antisymmetrised product of a degenerate representation with itself.*

General

$$\begin{array}{ccc} \begin{array}{c|cc} \hline & \mathrm{g} & \mathrm{u} \\ \hline \mathrm{g} & \mathrm{g} & \mathrm{u} \\ \mathrm{u} & & \mathrm{g} \\ \hline \end{array} & \hspace{10pt} & \begin{array}{c|cc} \hline & \mathrm{'} & \mathrm{''} \\ \hline \mathrm{'} & \mathrm{'} & \mathrm{''} \\ \mathrm{''} & & \mathrm{'} \\ \hline \end{array} \end{array}$$

$\,$

$C_2, C_3, C_6, D_3, D_6, C_\mathrm{2v}, C_\mathrm{3v}, C_\mathrm{6v}, C_\mathrm{2h}, C_\mathrm{3h}, C_\mathrm{6h}, D_\mathrm{3h}, D_\mathrm{6h}, D_\mathrm{3d}, S_6$

$$\begin{array}{c|cccccc} \hline & \mathrm{A_1} & \mathrm{A_2} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{E_1} & \mathrm{E_2} \\ \hline \mathrm{A_1} & \mathrm{A_1} & \mathrm{A_2} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{E_1} & \mathrm{E_2} \\ \mathrm{A_2} & & \mathrm{A_1} & \mathrm{B_2} & \mathrm{B_1} & \mathrm{E_1} & \mathrm{E_2}\\ \mathrm{B_1} & & & \mathrm{A_1} & \mathrm{A_2} & \mathrm{E_2} & \mathrm{E_1} \\ \mathrm{B_2} & & & & \mathrm{A_1} & \mathrm{E_2} & \mathrm{E_1} \\ \mathrm{E_1} & & & & & \mathrm{A_1 + [A_2] + E_2} & \mathrm{B_1 + B_2 + E_1} \\ \mathrm{E_2} & & & & & & \mathrm{A_1 + [A_2] + E_2} \\ \hline \end{array}$$

$\,$

$D_2, D_\mathrm{2h}$

$$\begin{array}{c|cccc} \hline & \mathrm{A} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{B_3} \\ \hline \mathrm{A} & \mathrm{A} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{B_3} \\ \mathrm{B_1} & & \mathrm{A} & \mathrm{B_3} & \mathrm{B_2} \\ \mathrm{B_2} & & & \mathrm{A} & \mathrm{B_1} \\ \mathrm{B_3} & & & & \mathrm{A} \\ \hline \end{array}$$

$\,$

$C_4, D_4, C_\mathrm{4v}, C_\mathrm{4h}, D_\mathrm{4h}, D_\mathrm{2d}, S_4$

$$\begin{array}{c|ccccc} \hline & \mathrm{A_1} & \mathrm{A_2} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{E} \\ \hline \mathrm{A_1} & \mathrm{A_1} & \mathrm{A_2} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{E} \\ \mathrm{A_2} & & \mathrm{A_1} & \mathrm{B_2} & \mathrm{B_1} & \mathrm{E} \\ \mathrm{B_1} & & & \mathrm{A_1} & \mathrm{A_2} & \mathrm{E} \\ \mathrm{B_2} & & & & \mathrm{A_1} & \mathrm{E} \\ \mathrm{E} & & & & & \mathrm{A_1 + [A_2] + B_1 + B_2} \\ \hline \end{array}$$

$\,$

$C_5, D_5, C_\mathrm{5v}, C_\mathrm{5h}, D_\mathrm{5h}, D_\mathrm{5d}$

$$\begin{array}{c|cccc} \hline & \mathrm{A_1} & \mathrm{A_2} & \mathrm{E_1} & \mathrm{E_2} \\ \hline \mathrm{A_1} & \mathrm{A_1} & \mathrm{A_2} & \mathrm{E_1} & \mathrm{E_2} \\ \mathrm{A_2} & & \mathrm{A_1} & \mathrm{E_1} & \mathrm{E_2} \\ \mathrm{E_1} & & & \mathrm{A_1 + [A_2] + E_2} & \mathrm{E_1 + E_2} \\ \mathrm{E_2} & & & & \mathrm{A_1 + [A_2] + E_1} \\ \hline \end{array}$$

$\,$

$D_\mathrm{4d}, S_8$

$$\begin{array}{c|ccccccc}\hline & \mathrm{A_1} & \mathrm{A_2} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{E_1} & \mathrm{E_2} & \mathrm{E_3} \\ \hline \mathrm{A_1} & \mathrm{A_1} & \mathrm{A_2} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{E_1} & \mathrm{E_2} & \mathrm{E_3} \\ \mathrm{A_2} & & \mathrm{A_1} & \mathrm{B_2} & \mathrm{B_1} & \mathrm{E_1} & \mathrm{E_2} & \mathrm{E_3} \\ \mathrm{B_1} & & & \mathrm{A_1} & \mathrm{A_2} & \mathrm{E_3} & \mathrm{E_2} & \mathrm{E_1} \\ \mathrm{B_2} & & & & \mathrm{A_1} & \mathrm{E_3} & \mathrm{E_2} & \mathrm{E_1} \\ \mathrm{E_1} & & & & & \mathrm{A_1 + [A_2] + E_2} & \mathrm{E_1 + E_3} & \mathrm{B_1 + B_2 + E_2} \\ \mathrm{E_2} & & & & & & \mathrm{A_1 + [A_2] + B_1 + B_2} & \mathrm{E_1 + E_3} \\ \mathrm{E_3} & & & & & & & \mathrm{A_1 + [A_2] + E_2} \\ \hline \end{array}$$

$\,$

$T, T_\mathrm{h}, T_\mathrm{d}, O, O_\mathrm{h}$

$$\begin{array}{c|ccccc} \hline & \mathrm{A_1} & \mathrm{A_2} & \mathrm{E} & \mathrm{T_1} & \mathrm{T_2} \\ \hline \mathrm{A_1} & \mathrm{A_1} & \mathrm{A_2} & \mathrm{E} & \mathrm{T_1} & \mathrm{T_2} \\ \mathrm{A_2} & & \mathrm{A_1} & \mathrm{E} & \mathrm{T_2} & \mathrm{T_1} \\ \mathrm{E} & & & \mathrm{A_1 + [A_2] + E} & \mathrm{T_1 + T_2} & \mathrm{T_1 + T_2} \\ \mathrm{T_1} & & & & \mathrm{A_1 + E + [T_1] + T_2} & \mathrm{A_2 + E + T_1 + T_2} \\ \mathrm{T_2} & & & & & \mathrm{A_1 + E + [T_1] + T_2} \\ \hline \end{array}$$

$\,$

$D_\mathrm{6d}$

$$\small\begin{array}{c|ccccccccc} \hline & \mathrm{A_1} & \mathrm{A_2} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{E_1} & \mathrm{E_2} & \mathrm{E_3} & \mathrm{E_4} & \mathrm{E_5} \\ \hline \mathrm{A_1} & \mathrm{A_1} & \mathrm{A_2} & \mathrm{B_1} & \mathrm{B_2} & \mathrm{E_1} & \mathrm{E_2} & \mathrm{E_3} & \mathrm{E_4} & \mathrm{E_5} \\ \mathrm{A_2} & & \mathrm{A_1} & \mathrm{B_2} & \mathrm{B_1} & \mathrm{E_1} & \mathrm{E_2} & \mathrm{E_3} & \mathrm{E_4} & \mathrm{E_5} \\ \mathrm{B_1} & & & \mathrm{A_1} & \mathrm{A_2} & \mathrm{E_1} & \mathrm{E_2} & \mathrm{E_3} & \mathrm{E_4} & \mathrm{E_5} \\ \mathrm{B_2} & & & & \mathrm{A_1} & \mathrm{E_1} & \mathrm{E_2} & \mathrm{E_3} & \mathrm{E_4} & \mathrm{E_5} \\ \mathrm{E_1} & & & & & \begin{aligned}\mathrm{A_1 + [A_2]}\\ \mathrm{+ E_2}\quad\end{aligned} & \mathrm{E_1 + E_3} & \mathrm{E_2 + E_4} & \mathrm{E_3 + E_5} & \mathrm{B_1 + B_2 + E_4} \\ \mathrm{E_2} & & & & & & \begin{aligned}\mathrm{A_1 + [A_2]}\\ \mathrm{+ E_4}\quad\end{aligned} & \mathrm{E_1 + E_5} & \mathrm{B_1 + B_2 + E_2} & \mathrm{E_3 + E_5} \\ \mathrm{E_3} & & & & & & & \begin{aligned}\mathrm{A_1 + [A_2]}\\ \mathrm{+ B_1 + B_2}\end{aligned} & \mathrm{E_1 + E_5} & \mathrm{E_2 + E_4} \\ \mathrm{E_4} & & & & & & & & \mathrm{A_1 + [A_2] + E_4} & \mathrm{E_1 + E_3} \\ \mathrm{E_5} & & & & & & & & & \mathrm{A_1 + [A_2] + E_2} \\ \hline \end{array}$$

$\,$

$I, I_\mathrm{h}$

$$\begin{array}{c|ccccc} \hline & \mathrm{A} & \mathrm{T_1} & \mathrm{T_2} & \mathrm{G} & \mathrm{H} \\ \hline \mathrm{A} & \mathrm{A} & \mathrm{T_1} & \mathrm{T_2} & \mathrm{G} & \mathrm{H} \\ \mathrm{T_1} & & \mathrm{A + [T_1] + H} & \mathrm{G + H} & \mathrm{T_2 + G + H} & \mathrm{T_1 + T_2 + G + H} \\ \mathrm{T_2} & & & \mathrm{A + [T_2] + H} & \mathrm{T_1 + G + H} & \mathrm{T_1 + T_2 + G + H} \\ \mathrm{G} & & & & \begin{aligned}\mathrm{A + [T_1 + T_2]}\\\mathrm{+ G + H}\quad\,\,\end{aligned} & \mathrm{T_1 + T_2 + G + 2H} \\ \mathrm{H} & & & & & \mathrm{A + [T_1 + T_2 + G] + G + 2H} \\ \hline \end{array}$$

$\,$

$C_\mathrm{\infty v}, D_\mathrm{\infty h}$

In general: $$\begin{array}{c|cccccc} \hline \Lambda = & 0 & 1 & 2 & 3 & 4 & \cdots \\ \hline & \Sigma & \Pi & \Delta & \Phi & \Gamma & \cdots \\ \hline \end{array}$$

$$\begin{align} \Lambda_1 \times \Lambda_2 &= |\Lambda_1 - \Lambda_2| + (\Lambda_1 + \Lambda_2) \\ \Lambda \times \Lambda &= \Sigma^+ + [\Sigma^-] + 2\Lambda \end{align}$$

For the first few irreps:

$$\begin{array}{c|ccccc} \hline & \Sigma^+ & \Sigma^- & \Pi & \Delta & \cdots \\ \hline \Sigma^+ & \Sigma^+ & \Sigma^- & \Pi & \Delta & \cdots \\ \Sigma^- & & \Sigma^+ & \Pi & \Delta & \cdots \\ \Pi & & & \Sigma^+ + [\Sigma^-] + \Delta & \Pi + \Phi & \cdots \\ \Delta & & & & \Sigma^+ + [\Sigma^-] + \Gamma & \cdots \\ \vdots & & & & & \ddots \\ \hline \end{array}$$

$\,$

The full rotation group, $\mathrm{R_3}$

$$\begin{align} \Gamma^{(j)} \times \Gamma^{(j')} &= \Gamma^{(j+j')} + \Gamma^{(j+j'-1)} + \cdots + \Gamma^{(|j-j'|)} \\ \Gamma^{(j)} \times \Gamma^{(j)} &= \Gamma^{(2j)} + \Gamma^{(2j-2)} + \cdots + \Gamma^{(0)} + \left[\Gamma^{(2j-1)} + \Gamma^{(2j-3)} + \cdots + \Gamma^{(1)}\right] \end{align}$$

_{*The direct product of a degenerate species with itself may be resolved into a symmetric direct product, and an anti-symmetric direct product. In vibrational spectroscopy, the symmetry species of the overtones of a degenerate fundamental are obtained from the symmetric direct products. In the determination of electronic terms, the symmetric and anti- symmetric direct products for orbital angular momentum are taken with the appropriate spin functions to ensure that the total wave functions are anti-symmetric.}

(back to top)

Answer 2

(back to top)

Cubic groups: $T, T_\mathrm{d}, T_\mathrm{h}, O, O_\mathrm{h}$

$$\begin{array}{c|cccc|cc}\hline T & E & 4C_3 & 4C_3^2 & 3C_2 & & \varepsilon = \exp(2\pi\mathrm{i}/3) \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & & x^2+y^2+z^2 \\ \mathrm{E} & \left\{ \begin{aligned}1 \\ 1 \end{aligned} \right. & \begin{aligned}\varepsilon \\ \varepsilon^* \end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned} & \left.\begin{aligned}1 \\ 1 \end{aligned}\right\} & & (2z^2-x^2-y^2,x^2-y^2) \\ \mathrm{T} & 3 & 0 & 0 & -1 & (x,y,z),(R_x,R_y,R_z) & (xy,xz,yz) \\ \hline \end{array}$$

$\,$

$$\small\begin{array}{c|cccccccc|cc} \hline T_\mathrm{h} & E & 4C_3 & 4C_3^2 & 3C_2 & i & 4S_6 & 4S_6^5 & 3\sigma_\mathrm{d} & & \varepsilon = \exp(2\pi\mathrm{i}/3) \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2+z^2 \\ \mathrm{E_g} & \left\{ \begin{aligned}1 \\ 1\end{aligned} \right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned} 1\\ 1\end{aligned} & \begin{aligned}1 \\ 1\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \left.\begin{aligned}1 \\ 1\end{aligned}\right\} & & \begin{aligned}(2z^2-x^2-y^2,\\ x^2-y^2)\,\,\,\,\,\, \end{aligned} \\ \mathrm{T_g} & 3 & 0 & 0 & -1 & 3 & 0 & 0 & -1 & (R_x,R_y,R_z) & (xy,xz,yz) \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{E_u} & \left\{ \begin{aligned}1 \\ 1\end{aligned} \right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned} 1\\ 1\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned} & \left.\begin{aligned}-1 \\ -1\end{aligned}\right\} & & \\ \mathrm{T_u} & 3 & 0 & 0 & -1 & -3 & 0 & 0 & 1 & (x,y,z) & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccccc|cc} \hline T_\mathrm{d} & E & 8C_3 & 3C_2 & 6S_4 & 6\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & & x^2+y^2+z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & & \\ \mathrm{E} & 2 & -1 & 2 & 0 & 1 & & (2z^2-x^2-y^2,x^2-y^2) \\ \mathrm{T_1} & 3 & 0 & -1 & 1 & -1 & (R_x,R_y,R_z) & \\ \mathrm{T_2} & 3 & 0 & -1 & -1 & 1 & (x,y,z) & (xy,xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccccc|cc}\hline O & E & 8C_3 & 3C_2 = C_4^2 & 6C_4 & 6C_2' & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & & x^2+y^2+z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & & \\ \mathrm{E} & 2 & -1 & 2 & 0 & 0 & & (2z^2-x^2-y^2,x^2-y^2) \\ \mathrm{T_1} & 3 & 0 & -1 & 1 & -1 & (x,y,z),(R_x,R_y,R_z) & \\ \mathrm{T_2} & 3 & 0 & -1 & -1 & 1 & & (xy,xz,yz) \\ \hline \end{array}$$

$\,$

$$\small\begin{array}{c|cccccccccc|cc}\hline O_\mathrm{h} & E & 8C_3 & 6C_2 & 6C_4 & \begin{aligned}3C_2 \\ \scriptsize=C_4^2\end{aligned} & i & 6S_4 & 8S_6 & 3\sigma_\mathrm{h} & 6\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2+z^2 \\ \mathrm{A_{2g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_g} & 2 & -1 & 0 & 0 & 2 & 2 & 0 & -1 & 2 & 0 & & \begin{aligned}(2z^2-x^2-y^2,\\ x^2-y^2)\,\,\,\,\,\, \end{aligned} \\ \mathrm{T_{1g}} & 3 & 0 & -1 & 1 & -1 & 3 & 1 & 0 & -1 & -1 & (R_x,R_y,R_z) & \\ \mathrm{T_{2g}} & 3 & 0 & 1 & -1 & -1 & 3 & -1 & 0 & -1 & 1 & & (xy,xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_u} & 2 & -1 & 0 & 0 & 2 & -2 & 0 & 1 & -2 & 0 & & \\ \mathrm{T_{1u}} & 3 & 0 & -1 & 1 & -1 & -3 & -1 & 0 & 1 & 1 & (x,y,z) & \\ \mathrm{T_{2u}} & 3 & 0 & 1 & -1 & -1 & -3 & 1 & 0 & 1 & -1 & & \\ \hline \end{array}$$

(back to top)

Answer 3

(back to top)

$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccc|cc} \hline C_\mathrm{3v} & E & 2C_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & R_z & \\ \mathrm{E} & 2 & -1 & 0 & (x,y), (R_x,R_y) & (x^2-y^2,xy), (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{5v} & E & 2C_5 & 2C_5^2 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\\hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

(back to top)

Answer 4

(back to top)

Tables of descent in symmetry

These tables show the correlations between the irreps of an overgroup with those of some of its subgroups. $\newcommand{\mirror}[1]{\sigma_\mathrm{#1}} \newcommand{\rowthree}[3]{\mathrm{#1}&\mathrm{#2}&\mathrm{#3}\\} \newcommand{\rowfour}[4]{\mathrm{#1}&\mathrm{#2}&\mathrm{#3}&\mathrm{#4}\\} \newcommand{\rowfive}[5]{\mathrm{#1}&\mathrm{#2}&\mathrm{#3}&\mathrm{#4}&\mathrm{#5}\\} \newcommand{\rowsix}[6]{\mathrm{#1}&\mathrm{#2}&\mathrm{#3}&\mathrm{#4}&\mathrm{#5}&\mathrm{#6}\\} \newcommand{\rowseven}[7]{\mathrm{#1}&\mathrm{#2}&\mathrm{#3}&\mathrm{#4}&\mathrm{#5}&\mathrm{#6}&\mathrm{#7}\\} \newcommand{\roweight}[8]{\mathrm{#1}&\mathrm{#2}&\mathrm{#3}&\mathrm{#4}&\mathrm{#5}&\mathrm{#6}&\mathrm{#7}&\mathrm{#8}\\}$

• A sub-heading $A$ identifies which symmetry element(s) in the overgroup is/are preserved as symmetry element(s) in the subgroup.
• A sub-heading $A \to B$ indicates that symmetry element $A$ in the overgroup becomes symmetry element $B$ in the subgroup.
• In the subgroup $C_\mathrm{s}$ there is only one mirror plane, and therefore no ambiguity. However, in $C_\mathrm{2v}$ there are two mirror planes. Which mirror plane becomes $\mirror{v}$, and which becomes $\mirror{v}'$, cannot be uniquely defined and must be set by a convention. This convention is either indicated clearly, or omitted if it does not affect the resulting irreps (because they appear together as a sum $\mathrm{B_1 + B_2}$). In the latter case, this is indicated by a dagger symbol, $^\dagger$. Similar ambiguity is present in the three degenerate rotation axes of $D_2$ and $D_\mathrm{2h}$.

Tables for any omitted overgroups, as well as correlations for most omitted subgroups (listed below each table), can be found in Appendix X-8 of Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra by Wilson, Decius, and Cross. The tables presented here have been checked against this source. If I have time, I may add more tables from there.

$$\begin{array}{ccc} \begin{array}{c|ccc} \hline C_\mathrm{2v} & C_2 & C_\mathrm{s} & C_\mathrm{s} \\ & & \sigma(xz) & \sigma(yz) \\ \hline \rowfour{A_1}{A}{A' }{A' } \rowfour{A_2}{A}{A''}{A''} \rowfour{B_1}{B}{A' }{A''} \rowfour{B_2}{B}{A''}{A' } \hline \end{array} & \quad & \begin{array}{c|ccc} \hline C_\mathrm{3v} & C_3 & C_\mathrm{s} \\ \hline \rowthree{A_1}{A}{A' } \rowthree{A_2}{A}{A'' } \rowthree{E }{E}{A'+A''} \hline \end{array} \end{array}$$

$\,$

$$\begin{array}{c|cc} \hline C_\mathrm{4v} & C_\mathrm{2v} & C_\mathrm{2v} \\ & \mirror{v} & \mirror{d} \\ \hline \rowthree{A_1}{A_1 }{A_1 } \rowthree{A_2}{A_2 }{A_2 } \rowthree{B_1}{A_1 }{A_2 } \rowthree{B_2}{A_2 }{A_1 } \rowthree{E }{B_1+B_2}{B_1+B_2}\hline \end{array} \\ \text{also: }C_4, C_2, C_\mathrm{s}$$

$\,$

$$\begin{array}{c|cccccc} \hline D_\mathrm{2h} & D_2 & C_\mathrm{2v} & C_\mathrm{2v} & C_\mathrm{2v} & C_\mathrm{2h} & C_\mathrm{2h} & C_\mathrm{2h} \\ & & C_2(z) & C_2(y) & C_2(x) & C_2(z) & C_2(y) & C_2(x) \\ \hline \roweight{A_g }{A }{A_1}{A_1}{A_1}{A_g}{A_g}{A_g} \roweight{B_{1g}}{B_1}{A_2}{B_2}{B_1}{A_g}{B_g}{B_g} \roweight{B_{2g}}{B_2}{B_1}{A_2}{B_2}{B_g}{A_g}{B_g} \roweight{B_{3g}}{B_3}{B_2}{B_1}{A_2}{B_g}{B_g}{A_g} \roweight{A_u }{A }{A_2}{A_2}{A_2}{A_u}{A_u}{A_u} \roweight{B_{1u}}{B_1}{A_1}{B_1}{B_2}{A_u}{B_u}{B_u} \roweight{B_{2u}}{B_2}{B_2}{A_1}{B_1}{B_u}{A_u}{B_u} \roweight{B_{3u}}{B_3}{B_1}{B_2}{A_1}{B_u}{B_u}{A_u} \hline \end{array}$$

$$\begin{array}{c|cccccc} \hline D_\mathrm{2h} & C_2 & C_2 & C_2 & C_\mathrm{s} & C_\mathrm{s} & C_\mathrm{s} \\ \text{cont.} & C_2(z) & C_2(y) & C_2(x) & \mirror{xy} & \mirror{zx} & \mirror{yz} \\ \hline \rowseven{A_g }{A}{A}{A}{A' }{A' }{A' } \rowseven{B_{1g}}{A}{B}{B}{A' }{A''}{A''} \rowseven{B_{2g}}{B}{A}{B}{A''}{A' }{A''} \rowseven{B_{3g}}{B}{B}{A}{A''}{A''}{A' } \rowseven{A_u }{A}{A}{A}{A''}{A''}{A''} \rowseven{B_{1u}}{A}{B}{B}{A''}{A' }{A' } \rowseven{B_{2u}}{B}{A}{B}{A' }{A''}{A' } \rowseven{B_{3u}}{B}{B}{A}{A' }{A' }{A''} \hline \end{array}$$

$\,$

$$\begin{array}{c|ccccc} \hline D_\mathrm{3h} & C_\mathrm{3h} & C_\mathrm{3v} & C_\mathrm{2v} & C_\mathrm{s} & C_\mathrm{s} \\ & & & \mirror{h}\to\mirror{v}(yz) & \mirror{h} & \mirror{v} \\ \hline \rowsix{A_1' }{A' }{A_1}{A_1 }{A' }{A' } \rowsix{A_2' }{A' }{A_2}{B_2 }{A' }{A'' } \rowsix{E' }{E' }{E }{A_1+B_2}{2A' }{A'+A''} \rowsix{A_1''}{A''}{A_2}{A_2 }{A'' }{A'' } \rowsix{A_2''}{A''}{A_1}{B_1 }{A'' }{A' } \rowsix{E'' }{E''}{E }{A_2+B_1}{2A''}{A'+A''} \hline \end{array} \\ \text{also: }D_3, C_3, C_2$$

$\,$

$$\begin{array}{c|cccccc} \hline D_\mathrm{4h} & D_\mathrm{2d} & D_\mathrm{2d} & D_\mathrm{2h} & D_\mathrm{2h} & D_\mathrm{2} & D_\mathrm{2} \\ & C_2' \to C_2' & C_2'' \to C_2' & C_2' & C_2'' & C_2' & C_2'' \\ \hline \rowseven{A_{1g}}{A_1}{A_1}{A_g }{A_g }{A }{A } \rowseven{A_{2g}}{A_2}{A_2}{B_{1g} }{B_{1g} }{B_1 }{B_1 } \rowseven{B_{1g}}{B_1}{B_2}{A_g }{B_{1g} }{A }{B_1 } \rowseven{B_{2g}}{B_2}{B_1}{B_{1g} }{A_g }{B_1 }{A } \rowseven{E_g }{E }{E }{B_{2g}+B_{3g}}{B_{2g}+B_{3g}}{B_2+B_3}{B_2+B_3} \rowseven{A_{1u}}{B_1}{B_1}{A_u }{A_u }{A }{A } \rowseven{A_{2u}}{B_2}{B_2}{B_{1u} }{B_{1u} }{B_1 }{B_1 } \rowseven{B_{1u}}{A_1}{A_2}{A_u }{B_{1u} }{A }{B_1 } \rowseven{B_{2u}}{A_2}{A_1}{B_{1u} }{A_u }{B_1 }{A } \rowseven{E_u }{E }{E }{B_{2u}+B_{3u}}{B_{2u}+B_{3u}}{B_2+B_3}{B_2+B_3} \hline \end{array}$$

$$\begin{array}{c|cccc} \hline D_\mathrm{4h} & C_\mathrm{4h} & C_\mathrm{4v} & C_\mathrm{2v}{}^\dagger & C_\mathrm{2v}{}^\dagger \\ \text{(cont.)} & & & C_2,\mirror{v} & C_2,\mirror{d} \\ \hline \rowfive{A_{1g}}{A_g}{A_1}{A_1 }{A_1 } \rowfive{A_{2g}}{A_g}{A_2}{A_2 }{A_2 } \rowfive{B_{1g}}{B_g}{B_1}{A_1 }{A_2 } \rowfive{B_{2g}}{B_g}{B_2}{A_2 }{A_1 } \rowfive{E_g }{E_g}{E }{B_1+B_2}{B_1+B_2} \rowfive{A_{1u}}{A_u}{A_2}{A_2 }{A_2 } \rowfive{A_{2u}}{A_u}{A_1}{A_1 }{A_1 } \rowfive{B_{1u}}{B_u}{B_2}{A_2 }{A_1 } \rowfive{B_{2u}}{B_u}{B_1}{A_1 }{A_2 } \rowfive{E_u }{E_u}{E }{B_1+B_2}{B_1+B_2} \hline \end{array} \\ \text{also: }D_4, C_4, S_4, C_\mathrm{2v}(C_2'), C_\mathrm{2v}(C_2''), C_\mathrm{2h}(C_2), C_\mathrm{2h}(C_2'), C_\mathrm{2h}(C_2''), C_2(C_2), C_2(C_2'), C_2(C_2''), C_\mathrm{s}(\mirror{h}), C_\mathrm{s}(\mirror{v}), C_\mathrm{s}(\mirror{d}), C_\mathrm{i}$$

$\,$

$$\begin{array}{c|cccccc} \hline D_\mathrm{6h} & D_\mathrm{3d} & D_\mathrm{3d} & D_\mathrm{2h} & C_\mathrm{6v} & C_\mathrm{3v} & C_\mathrm{3v} \\ & C_2'' & C_2' & \mirror{h}\to\sigma(xy) & & \mirror{v} & \mirror{d} \\ & & & \mirror{v}\to\sigma(yz) & & & \\ \hline \rowseven{A_{1g}}{A_{1g}}{A_{1g}}{A_g }{A_1}{A_1}{A_1} \rowseven{A_{2g}}{A_{2g}}{A_{2g}}{B_{1g} }{A_2}{A_2}{A_2} \rowseven{B_{1g}}{A_{2g}}{A_{1g}}{B_{2g} }{B_2}{A_2}{A_1} \rowseven{B_{2g}}{A_{1g}}{A_{2g}}{B_{3g} }{B_1}{A_1}{A_2} \rowseven{E_{1g}}{E_{g} }{E_{g} }{B_{2g}+B_{3g}}{E_1}{E }{E } \rowseven{E_{2g}}{E_{g} }{E_{g} }{A_g+B_{1g} }{E_2}{E }{E } \rowseven{A_{1u}}{A_{1u}}{A_{1u}}{A_u }{A_2}{A_2}{A_2} \rowseven{A_{2u}}{A_{2u}}{A_{2u}}{B_{1u} }{A_1}{A_1}{A_1} \rowseven{B_{1u}}{A_{2u}}{A_{1u}}{B_{2u} }{B_1}{A_1}{A_2} \rowseven{B_{2u}}{A_{1u}}{A_{2u}}{B_{3u} }{B_2}{A_2}{A_1} \rowseven{E_{1u}}{E_{u} }{E_{u} }{B_{2u}+B_{3u}}{E_1}{E }{E } \rowseven{E_{2u}}{E_{u} }{E_{u} }{A_u+B_{1u} }{E_2}{E }{E } \hline \end{array}$$

$$\begin{array}{c|ccccc} \hline D_\mathrm{6h} & C_\mathrm{2v} & C_\mathrm{2v} & C_\mathrm{2h} & C_\mathrm{2h} & C_\mathrm{2h} \\ \text{(cont.)} & C_2' & C_2'' & C_2 & C_2' & C_2'' \\ & \mirror{h}\to\sigma(xz) & \mirror{h}\to\sigma(xz) & & & \\ \hline \rowsix{A_{1g}}{A_1 }{A_1 }{A_g }{A_g }{A_g } \rowsix{A_{2g}}{B_1 }{B_1 }{A_g }{B_g }{B_g } \rowsix{B_{1g}}{A_2 }{B_2 }{B_g }{A_g }{B_g } \rowsix{B_{2g}}{B_2 }{A_2 }{B_g }{B_g }{A_g } \rowsix{E_{1g}}{A_2+B_2}{A_2+B_2}{2B_g}{A_g+B_g}{A_g+B_g} \rowsix{E_{2g}}{A_1+B_1}{A_1+B_1}{2A_g}{A_g+B_g}{A_g+B_g} \rowsix{A_{1u}}{A_2 }{A_2 }{A_u }{A_u }{A_u } \rowsix{A_{2u}}{B_2 }{B_2 }{A_u }{B_u }{B_u } \rowsix{B_{1u}}{A_1 }{B_1 }{B_u }{A_u }{B_u } \rowsix{B_{2u}}{B_1 }{A_1 }{B_u }{B_u }{A_u } \rowsix{E_{1u}}{A_1+B_1}{A_1+B_1}{2B_u}{A_u+B_u}{A_u+B_u} \rowsix{E_{2u}}{A_2+B_2}{A_2+B_2}{2A_u}{A_u+B_u}{A_u+B_u} \hline \end{array} \\ \text{also: }D_6, D_\mathrm{3h}(C_2'), D_\mathrm{3h}(C_2''), C_\mathrm{6h}, C_6, C_\mathrm{3h}, D_3(C_2'), D_3(C_2''), S_6, D_2, C_3, C_2(C_2), C_2(C_2'), C_2(C_2''), C_\mathrm{s}(\mirror{h}), C_\mathrm{s}(\mirror{d}), C_\mathrm{s}(\mirror{v}), C_\mathrm{i} $$

$\,$

$$\begin{array}{c|cccc} \hline T_\mathrm{d} & T & D_\mathrm{2d} & C_\mathrm{3v} & C_\mathrm{2v}{}^\dagger \\ \hline \rowfive{A_1}{A}{A_1 }{A_1 }{A_1 } \rowfive{A_2}{A}{B_1 }{A_2 }{A_2 } \rowfive{E }{E}{A_1+B_1}{E }{A_1+A_2 } \rowfive{T_1}{T}{A_2+E }{A_2+E}{A_2+B_1+B_2} \rowfive{T_2}{T}{B_2+E }{A_1+E}{A_1+B_1+B_2} \hline \end{array} \\ \text{also: }S_4, D_2, C_3, C_2, C_\mathrm{s}$$

$\,$

$$\begin{array}{c|ccccccccc} \hline O_\mathrm{h} & O & T_\mathrm{d} & T_\mathrm{h} & D_\mathrm{4h} & D_\mathrm{2d} & C_\mathrm{4v} & C_\mathrm{2v}{}^\dagger \\ & & & & & & & C_2,\mirror{h},\mirror{d} \\ \hline \roweight{A_{1g}}{A_1}{A_1}{A_g}{A_{1g} }{A_1 }{A_1 }{A_1 } \roweight{A_{2g}}{A_2}{A_2}{A_g}{B_{1g} }{B_1 }{B_1 }{A_2 } \roweight{E_g }{E }{E }{E_g}{A_{1g}+B_{1g}}{A_1+B_1}{A_1+B_1}{A_1+A_2 } \roweight{T_{1g}}{T_1}{T_1}{T_g}{A_{2g}+E_g }{A_2+E }{A_2+E }{A_2+B_1+B_2} \roweight{T_{2g}}{T_2}{T_2}{T_g}{B_{2g}+E_g }{B_2+E }{B_2+E }{A_1+B_1+B_2} \roweight{A_{1u}}{A_1}{A_2}{A_u}{A_{1u} }{B_1 }{A_2 }{A_2 } \roweight{A_{2u}}{A_2}{A_1}{A_u}{B_{1u} }{A_1 }{B_2 }{A_1 } \roweight{E_u }{E }{E }{E_u}{A_{1u}+B_{1u}}{A_1+B_1}{A_2+B_2}{A_1+A_2 } \roweight{T_{1u}}{T_1}{T_2}{T_u}{A_{2u}+E_u }{B_2+E }{A_1+E }{A_1+B_1+B_2} \roweight{T_{2u}}{T_2}{T_1}{T_u}{B_{2u}+E_u }{A_2+E }{B_1+E }{A_2+B_1+B_2} \hline \end{array}$$

$$\begin{array}{c|ccc} \hline O_\mathrm{h} & D_\mathrm{3d} & D_3 & C_\mathrm{2h} \\ \text{(cont.)} & & & C_2 \\ \hline \rowfour{A_{1g}}{A_{1g} }{A_1 }{A_g } \rowfour{A_{2g}}{A_{2g} }{A_2 }{B_g } \rowfour{E_g }{E_g }{E }{A_g+B_g } \rowfour{T_{1g}}{A_{2g}+E_g}{A_2+E}{A_g+2B_g} \rowfour{T_{2g}}{A_{1g}+E_g}{A_1+E}{2A_g+B_g} \rowfour{A_{1u}}{A_{1u} }{A_1 }{A_u } \rowfour{A_{2u}}{A_{2u} }{A_2 }{B_u } \rowfour{E_u }{E_u }{E }{A_u+B_u } \rowfour{T_{1u}}{A_{2u}+E_u}{A_2+E}{A_u+2B_u} \rowfour{T_{2u}}{A_{1u}+E_u}{A_1+E}{2A_u+B_u} \hline \end{array} \\ \text{Note that }C_2' \equiv C_4^2\text{ and }C_2 \not\equiv C_4^2\text{.} \\ \text{also: }T, D_4, C_\mathrm{4h}, D_\mathrm{2h}(3C_2), D_\mathrm{2h}(3C_2'), C_\mathrm{3v}, S_6, C_4, S_4, C_\mathrm{2v}(C_2',2\mirror{h}), C_\mathrm{2v}(C_2',\mirror{h},\mirror{d}), D_2(3C_2), D_2(3C_2'), C_\mathrm{2h}(C_2'), C_3, C_2(C_2), C_2(C_2'), S_2, C_\mathrm{s}$$

$\,$

$$\begin{array}{c|ccc} \hline \mathrm{R_3} & O & D_4 & D_3 \\ \hline \rowfour{S}{A_1 }{A_1 }{A_1 } \rowfour{P}{T_1 }{A_2+E }{A_2+E } \rowfour{D}{E+T_2 }{A_1+B_1+B_2+E }{A_1+2E } \rowfour{F}{A_2+T_1+T_2 }{A_2+B_1+B_2+2E }{A_1+2A_1+2E} \rowfour{G}{A_1+E+T_1+T_2}{2A_1+A_2+B_1+B_2+2E}{2A_1+A_2+3E} \rowfour{H}{E+2T_1+2T_2 }{A_1+2A_2+B_1+B_2+3E}{A_1+2A_1+4E} \hline \end{array}$$

(back to top)

Answer 5

(back to top)

$C_{\infty\mathrm{v}}$ and $D_{\infty\mathrm{h}}$

$$\begin{array}{c|cccc|cc} \hline C_{\infty\mathrm{v}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1} \equiv \Sigma^+ & 1 & 1 & \cdots & 1 & z & x^2 + y^2, z^2 \\ \mathrm{A_2} \equiv \Sigma^- & 1 & 1 & \cdots & -1 & R_z & \\ \mathrm{E_1} \equiv \Pi & 2 & 2 \cos\phi & \cdots & 0 & (x,y), (R_x,R_y) & (xz,yz) \\ \mathrm{E_2} \equiv \Delta & 2 & 2 \cos 2\phi & \cdots & 0 & & (x^2-y^2,xy) \\ \mathrm{E_3} \equiv \Phi & 2 & 2 \cos 3\phi & \cdots & 0 & & \\ \vdots & \vdots & \vdots & \ddots & \vdots & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline D_{\infty\mathrm{h}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & i & 2S_\infty^\phi & \cdots & \infty C_2 & \\ \hline \mathrm{A_{1g}} \equiv \Sigma^+_{\mathrm{g}} & 1 & 1 & \cdots & 1 & 1 & 1 & \cdots & 1 & & x^2 + y^2, z^2 \\ \mathrm{A_{2g}} \equiv \Sigma^-_{\mathrm{g}} & 1 & 1 & \cdots & -1 & 1 & 1 & \cdots & -1 & R_z & \\ \mathrm{E_{1g}} \equiv \Pi_{\mathrm{g}} & 2 & 2\cos\phi & \cdots & 0 & 2 & -2\cos\phi & \cdots & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} \equiv \Delta_{\mathrm{g}} & 2 & 2\cos 2\phi & \cdots & 0 & 2 & 2\cos 2\phi & \cdots & 0 & & (x^2-y^2,xy) \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & & \\ \mathrm{A_{1u}} \equiv \Sigma^+_{\mathrm{u}} & 1 & 1 & \cdots & 1 & -1 & -1 & \cdots & -1 & z & \\ \mathrm{A_{2u}} \equiv \Sigma^-_{\mathrm{u}} & 1 & 1 & \cdots & -1 & -1 & -1 & \cdots & 1 & & \\ \mathrm{E_{1u}} \equiv \Pi_{\mathrm{u}} & 2 & 2\cos\phi & \cdots & 0 & -2 & 2\cos\phi & \cdots & 0 & (x,y) & \\ \mathrm{E_{2u}} \equiv \Delta_{\mathrm{u}} & 2 & 2\cos 2\phi & \cdots & 0 & -2 & -2\cos 2\phi & \cdots & 0 & & \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & & \\ \hline \end{array}$$

(back to top)

Answer 6

(back to top)

$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

(back to top)

Answer 7

(back to top)

$C_n$ $(2 \leq n \leq 8)$

$$\begin{array}{c|cc|cc}\hline C_2 & E & C_2 & & \\ \hline \mathrm{A} & 1 & 1 & z, R_z & x^2, y^2, z^2, xy \\ \mathrm{B} & 1 & -1 & x, y, R_x, R_y & xz, yz \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccc|cc} \hline C_3 & E & C_3 & C_3^2 & & \varepsilon = \exp(2\pi\mathrm{i}/3) \\ \hline \mathrm{A} & 1 & 1 & 1 & z, R_z & x^2+y^2, z^2 \\ \mathrm{E} & \left\{ \begin{aligned}1 \\ 1\end{aligned} \right. & \begin{aligned}\varepsilon \\ \varepsilon^* \end{aligned} & \left. \begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned}\right\} & (x, y), (R_x, R_y) & (x^2-y^2,xy), (xz, yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccc|cc} \hline C_4 & E & C_4 & C_2 & C_4^3 & & \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & z, R_z & x^2+y^2,z^2 \\ \mathrm{B} & 1 & -1 & 1 & -1 & & x^2-y^2,xy \\ \mathrm{E} & \left\{ \begin{aligned}1 \\ 1\end{aligned} \right. & \begin{aligned}\mathrm{i} \\ -\mathrm{i} \end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \left. \begin{aligned}-\mathrm{i} \\ \mathrm{i} \end{aligned}\right\} & (x, y), (R_x, R_y) & (xz, yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_5 & E & C_5 & C_5^2 & C_5^3 & C_5^4 & & \varepsilon = \exp(2\pi\mathrm{i}/5) \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & 1 & z, R_z & x^2 + y^2, z^2 \\ \mathrm{E_1} & \left\{ \begin{aligned}1 \\ 1\end{aligned} \right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned} & \left.\begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} \right\} & (x,y), (R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & \left\{ \begin{aligned}1 \\ 1\end{aligned} \right. & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \left.\begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned} \right\} & & (x^2-y^2,xy) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc}\hline C_6 & E & C_6 & C_3 & C_2 & C_3^2 & C_6^5 & & \varepsilon = \exp(2\pi\mathrm{i}/6) \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & 1 & 1 & z,R_z & x^2+y^2,z^2 \\ \mathrm{B} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_1} & \left\{ \begin{aligned}1 \\ 1\end{aligned} \right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned} & \begin{aligned}-1 \\ -1 \end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \left.\begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned}\right\} & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & \left\{ \begin{aligned}1 \\ 1\end{aligned} \right. & \begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \begin{aligned}1 \\ 1 \end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned} & \left.\begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned}\right\} & & (x^2-y^2,xy) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccccccc|cc}\hline C_7 & E & C_7 & C_7^2 & C_7^3 & C_7^4 & C_7^5 & C_7^6 & & \varepsilon = \exp(2\pi\mathrm{i}/7) \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & z, R_z & x^2 + y^2, z^2 \\ \mathrm{E_1} & \left\{\begin{aligned}1 \\ 1 \end{aligned} \right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^3 \\ \varepsilon^{*3}\end{aligned} & \begin{aligned}\varepsilon^{*3} \\ \varepsilon^3\end{aligned} & \begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned} & \left.\begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned}\right\} & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & \left\{\begin{aligned}1 \\ 1 \end{aligned} \right. & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^{*3} \\ \varepsilon^3\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}\varepsilon\\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^3 \\ \varepsilon^{*3}\end{aligned} & \left.\begin{aligned}\varepsilon^{*2} \\ \varepsilon^2 \end{aligned}\right\} & & (x^2-y^2,xy) \\ \mathrm{E_3} & \left\{\begin{aligned}1 \\ 1 \end{aligned} \right. & \begin{aligned}\varepsilon^3 \\ \varepsilon^{*3}\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \left.\begin{aligned}\varepsilon^{*3} \\ \varepsilon^3 \end{aligned}\right\} & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline C_8 & E & C_8 & C_4 & C_8^3 & C_2 & C_8^5 & C_4^3 & C_8^7 & & \varepsilon = \exp(2\pi\mathrm{i}/8) \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & z,R_z & x^2+y^2,z^2 \\ \mathrm{B} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_1} & \left\{\begin{aligned}1 \\ 1 \end{aligned} \right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \left.\begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned}\right\} & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & \left\{\begin{aligned}1 \\ 1 \end{aligned} \right. & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}1 \\ 1\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned} & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \left.\begin{aligned}-\mathrm{i} \\ \mathrm{i} \end{aligned}\right\} & & (x^2-y^2,xy) \\ \mathrm{E_3} & \left\{\begin{aligned}1 \\ 1 \end{aligned} \right. & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned} & \left.\begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned}\right\} & & \\ \hline \end{array}$$

(back to top)

Answer 8

(back to top)

The full rotation group, $\mathrm{R_3}$

$$ \chi^{(j)}(\phi) = \begin{cases} \Large\frac{\sin\left[\left(j+\frac{1}{2}\right)\phi\right]}{\sin\left(\frac{1}{2}\phi\right)} & \normalsize \phi \neq 0 \\ \\ 2j+1 & \phi = 0 \end{cases} $$

In $\mathrm{R_3}$, $j$ is confined to the natural numbers ($0,1,2, \ldots$), usually denoted with the symbol $l$ or $L$, and the irreducible representations $\Gamma^{(j)}$ are denoted with the letters

$$\begin{array}{c|ccccc} \hline j & 0 & 1 & 2 & 3 & 4 & 5 & \cdots \\ \hline & \mathrm{S} & \mathrm{P} & \mathrm{D} & \mathrm{F} & \mathrm{G} & \mathrm{H} & \cdots \\ \hline \end{array}$$

(back to top)

Answer 9

(back to top)

Icosahedral groups: $I, I_\mathrm{h}$

$$\begin{array}{c|ccccc|cc} \hline I & E & 12C_5 & 12C_5^2 & 20C_3 & 15C_2 & & \eta^{\pm} = \frac{1}{2}(1\pm\sqrt{5}) \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & 1 & & x^2+y^2+z^2 \\ \mathrm{T_1} & 3 & \eta^+ & \eta^- & 0 & -1 & (x,y,z),(R_x,R_y,R_z) & \\ \mathrm{T_2} & 3 & \eta^- & \eta^+ & 0 & -1 & & \\ \mathrm{G} & 4 & -1 & -1 & 1 & 0 & & \\ \mathrm{H} & 5 & 0 & 0 & -1 & 1 & & \begin{aligned}(2z^2-x^2-y^2,\,\,\,\,\, \\ x^2-y^2, xy, yz, xz)\end{aligned} \\ \hline \end{array}$$

$\,$

$$\small\begin{array}{c|cccccccccc|cc} \hline I_\mathrm{h} & E & 12C_5 & 12C_5^2 & 20C_3 & 15C_2 & i & 12S_{10} & 12S_{10}^3 & 20S_6 & 15\sigma & & \eta^{\pm} = \frac{1}{2}(1\pm\sqrt{5}) \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2+z^2 \\ \mathrm{T_{1g}} & 3 & \eta^+ & \eta^- & 0 & -1 & 3 & \eta^- & \eta^+ & 0 & -1 & \begin{aligned}(R_x,R_y,\\R_z)\,\,\,\,\end{aligned} & \\ \mathrm{T_{2g}} & 3 & \eta^- & \eta^+ & 0 & -1 & 3 & \eta^+ & \eta^- & 0 & -1 & & \\ \mathrm{G_g} & 4 & -1 & -1 & 1 & 0 & 4 & -1 & -1 & 1 & 0 & & \\ \mathrm{H_g} & 5 & 0 & 0 & -1 & 1 & 5 & 0 & 0 & -1 & 1 & & \begin{aligned}(2z^2-x^2-y^2, \\ x^2-y^2,\,\,\,\,\,\,\, \\ xy, yz, xz)\,\,\,\,\,\end{aligned} \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{T_{1u}} & 3 & \eta^+ & \eta^- & 0 & -1 & -3 & -\eta^- & -\eta^+ & 0 & 1 & (x,y,z) & \\ \mathrm{T_{2u}} & 3 & \eta^- & \eta^+ & 0 & -1 & -3 & -\eta^+ & -\eta^- & 0 & 1 & & \\ \mathrm{G_u} & 4 & -1 & -1 & 1 & 0 & -4 & 1 & 1 & -1 & 0 & & \\ \mathrm{H_u} & 5 & 0 & 0 & -1 & 1 & -5 & 0 & 0 & 1 & -1 & & \\ \hline \end{array}$$

(back to top)

Answer 10

(back to top)

$D_{n\mathrm{d}} (2 \leq n \leq 6)$

$$\begin{array}{c|ccccc|cc} \hline D_\mathrm{2d} & E & 2S_4 & C_2 & 2C_2' & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & z & xy & \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3d} & E & 2C_3 & 3C_2 & i & 2S_6 & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_g} & 2 & -1 & 0 & 2 & -1 & 0 & (R_x,R_y) & (x^2-y^2,xy),(xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_u} & 2 & -1 & 0 & -2 & 1 & 0 & (x,y) & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccccccc|cc} \hline D_\mathrm{4d} & E & 2S_8 & 2C_4 & 2S_8^3 & C_2 & 4C_2' & 4\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & z & \\ \mathrm{E_1} & 2 & \sqrt{2} & 0 & -\sqrt{2} & -2 & 0 & 0 & (x,y) & \\ \mathrm{E_2} & 2 & 0 & -2 & 0 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{E_3} & 2 & -\sqrt{2} & 0 & \sqrt{2} & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\small\begin{array}{c|cccccccc|cc} \hline D_\mathrm{5d} & E & 2C_5 & 2C_5^2 & 5C_2 & i & 2S_{10}^3 & 2S_{10} & 5\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_{1g}} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & \begin{aligned}(x^2-y^2,\\ xy)\,\,\,\,\,\end{aligned} \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_{1u}} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccccccccc|cc} \hline D_\mathrm{6d} & E & 2S_{12} & 2C_6 & 2S_4 & 2C_3 & 2S_{12}^5 & C_2 & 6C_2' & 6\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & z & \\ \mathrm{E_1} & 2 & \sqrt{3} & 1 & 0 & -1 & -\sqrt{3} & -2 & 0 & 0 & (x,y) & \\ \mathrm{E_2} & 2 & 1 & -1 & -2 & -1 & 1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{E_3} & 2 & 0 & -2 & 0 & 2 & 0 & -2 & 0 & 0 & & \\ \mathrm{E_4} & 2 & -1 & -1 & 2 & -1 & -1 & 2 & 0 & 0 & & \\ \mathrm{E_5} & 2 & -\sqrt{3} & 1 & 0 & -1 & \sqrt{3} & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

(back to top)

Answer 11

(back to top)

$C_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2h} & E & C_2 & i & \sigma_\mathrm{h} & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & R_z & x^2, y^2, z^2, xy \\ \mathrm{B_g} & 1 & -1 & 1 & -1 & R_x, R_y & xz, yz \\ \mathrm{A_u} & 1 & 1 & -1 & -1 & z & \\ \mathrm{B_u} & 1 & -1 & -1 & 1 & x, y & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{3h} & E & C_3 & C_3^2 & \sigma_\mathrm{h} & S_3 & S_3^5 & & \varepsilon = \exp(2\pi\mathrm{i}/3) \\ \hline \mathrm{A'} & 1 & 1 & 1 & 1 & 1 & 1 & R_z & x^2+y^2, z^2 \\ \mathrm{E'} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}1 \\ 1\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \left.\begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned}\right\} & (x,y) & (x^2-y^2, xy) \\ \mathrm{A''} & 1 & 1 & 1 & -1 & -1 & -1 & z & \\ \mathrm{E''} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \left.\begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned}\right\} & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccccc|cc} \hline C_\mathrm{4h} & E & C_4 & C_2 & C_4^3 & i & S_4^3 & \sigma_\mathrm{h} & S_4 & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & R_z & x^2+y^2, z^2 \\ \mathrm{B_g} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & x^2-y^2, xy \\ \mathrm{E_g} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned} & \begin{aligned}1 \\ 1\end{aligned} & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \left.\begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned}\right\} & (R_x,R_y) & (xz,yz) \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & z & \\ \mathrm{B_u} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & & \\ \mathrm{E_u} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned} & \begin{aligned}1 \\ 1\end{aligned} & \left.\begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned}\right\} & (x,y) & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccc|cc} \hline C_\mathrm{5h} & E & C_5 & C_5^2 & C_5^3 & C_5^4 & \sigma_\mathrm{h} & S_5 & S_5^7 & S_5^3 & S_5^9 & & \varepsilon = \exp(2\pi\mathrm{i}/5) \\ \hline \mathrm{A'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & R_z & x^2+y^2, z^2 \\ \mathrm{E_1'} & \left\{\begin{aligned}1\\ 1\end{aligned}\right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}1\\ 1\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned} & \left.\begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} \right\} & (x,y) & \\ \mathrm{E_2'} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned} & \begin{aligned}1 \\ 1\end{aligned} & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \left.\begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned}\right\} & & (x^2-y^2,xy) \\ \mathrm{A''} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & z & \\ \mathrm{E_1''} & \left\{\begin{aligned}1\\ 1\end{aligned}\right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}-1\\ -1\end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \begin{aligned}-\varepsilon^2 \\ -\varepsilon^{*2}\end{aligned} & \begin{aligned}-\varepsilon^{*2} \\ -\varepsilon^2\end{aligned} & \left.\begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned} \right\} & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\varepsilon^2 \\ \varepsilon^{*2}\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^{*2} \\ \varepsilon^2\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\varepsilon^2 \\ -\varepsilon^{*2}\end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \left.\begin{aligned}-\varepsilon^{*2} \\ -\varepsilon^2\end{aligned}\right\} & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline C_\mathrm{6h} & E & C_6 & C_3 & C_2 & C_3^2 & C_6^5 & i & S_3^5 & S_6^5 & \sigma_\mathrm{h} & S_6 & S_3 & & \varepsilon = \exp(2\pi\mathrm{i}/6) \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & R_z & x^2+y^2, z^2 \\ \mathrm{B_g} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_{1g}} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\varepsilon \\ \varepsilon^* \end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned} & \begin{aligned} -1 \\ -1 \end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned} & \begin{aligned}1 \\ 1\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^* \end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned} & \begin{aligned} -1 \\ -1 \end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned} & \left.\begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned}\right\} & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & \left\{\begin{aligned}1 \\ 1 \end{aligned}\right. & \begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned} & \begin{aligned} 1 \\ 1 \end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned} & \begin{aligned}1 \\ 1 \end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned} & \begin{aligned} 1 \\ 1 \end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned} & \left.\begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned}\right\} & & (x^2-y^2,xy) \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & z & \\ \mathrm{B_u} & 1 & -1 & 1 & -1 & 1 & -1 & -1 &1 & -1 & 1 & -1 & 1 & & \\ \mathrm{E_{1u}} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\varepsilon \\ \varepsilon^* \end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned} & \begin{aligned} -1 \\ -1 \end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned} & \begin{aligned} 1 \\ 1 \end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^* \end{aligned} & \left.\begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned}\right\} & (x,y) & \\ \mathrm{E_{2u}} & \left\{\begin{aligned}1 \\ 1 \end{aligned}\right. & \begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned} & \begin{aligned} 1 \\ 1 \end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon \end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^* \end{aligned} & \begin{aligned}-1 \\ -1 \end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^* \end{aligned} & \begin{aligned} -1 \\ -1 \end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon \end{aligned} & \left.\begin{aligned}\varepsilon \\ \varepsilon^* \end{aligned}\right\} & & \\ \hline \end{array}$$

(back to top)

Answer 12

(back to top)

$D_n (n \leq 2 \leq 6)$

$$\begin{array}{c|cccc|cc} \hline D_2 & E & C_2(z) & C_2(y) & C_2(x) & & \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & & x^2, y^2, z^2 \\ \mathrm{B_1} & 1 & 1 & -1 & -1 & z, R_z & xy \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & y, R_y & xz \\ \mathrm{B_3} & 1 & -1 & -1 & 1 & x, R_x & yz \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccc|cc} \hline D_3 & E & 2C_3 & 3C_2 & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & z, R_z & \\ \mathrm{E} & 2 & -1 & 0 & (x,y),(R_x,R_y) & (x^2-y^2,xy), (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|ccccc|cc} \hline D_4 & E & 2C_4 & C_2 = C_4^2 & 2C_2' & 2C_2'' & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & z, R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y), (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccc|cc} \hline D_5 & E & 2C_5 & 2C_5^2 & 5C_2 & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & z, R_z & \\ \mathrm{E_1} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline D_6 & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & z, R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

(back to top)

Answer 13

(back to top)

Low-symmetry groups: $C_1$, $C_\mathrm{s}$, $C_\mathrm{i}$

$$\begin{array}{c|c} \hline C_1 & E \\ \hline \mathrm{A} & 1 \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cc|cc} \hline C_\mathrm{s} (= C_\mathrm{h}) & E & \sigma_\mathrm{h} & & \\ \hline \mathrm{A'} & 1 & 1 & x,y, R_z & x^2, y^2, z^2, xy \\ \mathrm{A''} & 1 & -1 & z, R_x, R_y & xz, yz \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cc|cc} \hline C_\mathrm{i} (= S_2) & E & i & & \\ \hline \mathrm{A_g} & 1 & 1 & R_x, R_y, R_z & x^2, y^2, z^2, xy, xz, yz \\ \mathrm{A_u} & 1 & -1 & x,y,z & \\ \hline \end{array}$$

(back to top)

Answer 14

(back to top)

$S_n (n = 4,6,8)$

$$\begin{array}{c|cccc|cc}\hline S_4 & E & S_4 & C_2 & S_4^3 & & \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & R_z & x^2+y^2, z^2 \\ \mathrm{B} & 1 & -1 & 1 & -1 & z & x^2-y^2, xy \\ \mathrm{E} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \left.\begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned}\right\} & (x,y), (R_x, R_y) & (xz, yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline S_6 & E & C_3 & C_3^2 & i & S_6^5 & S_6 & & \varepsilon = \exp(2\pi\mathrm{i}/3) \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & R_z & x^2+y^2, z^2 \\ \mathrm{E_g} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}1 \\ 1\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \left.\begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned}\right\} & (R_x,R_y) & (x^2-y^2, xy), (xz,yz) \\ \mathrm{A_u} & 1 & 1 & 1 & -1 & -1 & -1 & z & \\ \mathrm{E_u} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \left.\begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned}\right\} & (x,y) & \\ \hline \end{array}$$

$\,$

$$\small\begin{array}{c|cccccccc|cc} \hline S_8 & E & S_8 & C_4 & S_8^3 & C_2 & S_8^5 & C_4^3 & S_8^7 & & \varepsilon = \exp(2\pi\mathrm{i}/8) \\ \hline \mathrm{A} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & R_z & x^2+y^2, z^2 \\ \mathrm{B} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & z & \\ \mathrm{E_1} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned} & \begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned} & \left.\begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned}\right\} & (x,y) & \\ \mathrm{E_2} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned} & \begin{aligned}1 \\ 1\end{aligned} & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \left.\begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned}\right\} & & (x^2-y^2, xy) \\ \mathrm{E_3} & \left\{\begin{aligned}1 \\ 1\end{aligned}\right. & \begin{aligned}-\varepsilon^* \\ -\varepsilon\end{aligned} & \begin{aligned}-\mathrm{i} \\ \mathrm{i}\end{aligned} & \begin{aligned}\varepsilon \\ \varepsilon^*\end{aligned} & \begin{aligned}-1 \\ -1\end{aligned} & \begin{aligned}\varepsilon^* \\ \varepsilon\end{aligned} & \begin{aligned}\mathrm{i} \\ -\mathrm{i}\end{aligned} & \left.\begin{aligned}-\varepsilon \\ -\varepsilon^*\end{aligned}\right\} & (R_x, R_y) & (xz,yz) \\ \hline \end{array}$$

In the point group $S_8$, $(R_x,R_y)$ actually transforms as $\mathrm{E_3}$, not $\mathrm{E_1}$. This runs contrary to many other sources (which are, unfortunately, all incorrect). Please see: J. Chem. Educ. 2007, 84 (11), 1882 for further clarification on this matter.

(back to top)