Table 1 Character table of the group \(C_{3h}\) with \(\xi ^3=1\). Two common notations are used for the IRs of the single and double group. The reduction of symmetry from \(D_{3h}\) to \(C_{3h}\) is accompanied by the compatibility relations \(A_1^{\prime },A_2^{\prime }\rightarrow A^{\prime }\), \(E^{\prime }\rightarrow E_1^{\prime }\bigoplus E_2^{\prime }\), \(A_1^{\prime \prime },A_2^{\prime \prime }\rightarrow A^{\prime \prime }\), \(E^{\prime \prime }\rightarrow E_1^{\prime \prime }\bigoplus E_2^{\prime \prime }\), \(E_{1/2}\rightarrow {^1}E_{1/2}\bigoplus {^2}E_{1/2}\), \(E_{3/2}\rightarrow {^1}E_{3/2}\bigoplus {^2}E_{3/2}\), and \(E_{5/2}\rightarrow {^1}E_{5/2}\bigoplus {^2}E_{5/2}\).
\(C_{3h}\) | E | \(C_{3}\) | \(C_{3}^2\) | \(\sigma _{h}\) | \(S_{3}\) | \(S_{3}^5\) | linear | quadratic | |
|---|---|---|---|---|---|---|---|---|---|
\(A^{\prime }\) | \(\Gamma _1\) | 1 | 1 | 1 | 1 | 1 | 1 | \(R_z\) | \(x^2+y^2\), \(z^2\) |
\(A^{\prime \prime }\) | \(\Gamma _4\) | 1 | 1 | 1 | −1 | −1 | −1 | z |  |
\(E^{\prime }_{1}\) | \(\Gamma _2\) | 1 | \(\xi\) | \(\xi ^2\) | 1 | \(\xi\) | \(\xi ^2\) | \(x+iy\) | \((x^2-y^2,xy)\) |
\(E^{\prime }_{2}\) | \(\Gamma _3\) | 1 | \(\xi ^2\) | \(\xi\) | 1 | \(\xi ^2\) | \(\xi\) | \(x-iy\) | |
\(E^{\prime \prime }_{1}\) | \(\Gamma _5\) | 1 | \(\xi\) | \(\xi ^2\) | −1 | \(-\xi\) | \(-\xi ^2\) | \(R_x+iR_y\) | (xz, yz) |
\(E^{\prime \prime }_{2}\) | \(\Gamma _6\) | 1 | \(\xi ^2\) | \(\xi\) | −1 | \(-\xi ^2\) | \(-\xi\) | \(R_x-iR_y\) | |
\(^1E_{1/2}\) | \(\Gamma _7\) | 1 | \(-\xi ^2\) | \(-\xi\) | i | \(i\xi ^2\) | \(-i\xi\) | \(\left| \frac{1}{2},\frac{1}{2}\right\rangle ,\left| \frac{3}{2},\frac{1}{2}\right\rangle\) | |
\(^2E_{1/2}\) | \(\Gamma _8\) | 1 | \(-\xi\) | \(-\xi ^2\) | \(-i\) | \(-i\xi\) | \(i\xi ^2\) | \(\left| \frac{1}{2},-\frac{1}{2}\right\rangle ,\left| \frac{3}{2},-\frac{1}{2}\right\rangle\) | |
\(^1E_{3/2}\) | \(\Gamma _{11}\) | 1 | \(-1\) | \(-1\) | i | i | \(-i\) | \(\left| \frac{3}{2},\frac{3}{2}\right\rangle ,\left| \frac{5}{2},\frac{3}{2}\right\rangle\) | |
\(^2E_{3/2}\) | \(\Gamma _{12}\) | 1 | \(-1\) | \(-1\) | \(-i\) | \(-i\) | i | \(\left| \frac{3}{2},-\frac{3}{2}\right\rangle ,\left| \frac{5}{2},-\frac{3}{2}\right\rangle\) | |
\(^1E_{5/2}\) | \(\Gamma _9\) | 1 | \(-\xi\) | \(-\xi ^2\) | i | \(i\xi\) | \(-i\xi ^2\) | \(\left| \frac{5}{2},\frac{5}{2}\right\rangle\) | |
\(^2E_{5/2}\) | \(\Gamma _{10}\) | 1 | \(-\xi ^2\) | \(-\xi\) | \(-i\) | \(-i\xi ^2\) | \(i\xi\) | \(\left| \frac{5}{2},-\frac{5}{2}\right\rangle\) | |
