MSG 60.417 \(A_u\)

\(D^0\) EAZ & character of irrep
A \(\left(0,0,0\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{1}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ \text{A}_1 & \text{A} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\ \text{A}_2 & \text{A} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ \text{A}_3 & \text{A} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\ \text{A}_4 & \text{A} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ \text{A}_5 & \text{A} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ \text{A}_6 & \text{A} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ \text{A}_7 & \text{A} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ \text{A}_8 & \text{A} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array} \right)\)
B \(\left(\frac{1}{2},0,0\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{1}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ \text{B}_1 & \text{A} & 2 & 0 & 0 & 0 & 0 & 0 & -2 & 0 \\ \text{B}_2 & \text{A} & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ \end{array} \right)\)
C \(\left(\frac{1}{2},\frac{1}{2},0\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{1}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ \text{C}_1 & \text{A} & 2 & 0 & 0 & 0 & -2 & 0 & 0 & 0 \\ \text{C}_2 & \text{A} & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ \end{array} \right)\)
D \(\left(0,\frac{1}{2},0\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{1}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ \text{D}_1 & \text{C} & 2 & 0 & -2 & 0 & 0 & 0 & 0 & 0 \\ \text{D}_2 & \text{C} & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\)
E \(\left(0,0,\frac{1}{2}\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{1}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ \text{E}_1 & \text{A} & 2 & 0 & 0 & 0 & 0 & -2 & 0 & 0 \\ \text{E}_2 & \text{A} & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ \end{array} \right)\)
F \(\left(\frac{1}{2},0,\frac{1}{2}\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{1}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ \text{F}_1 & \text{A} & 2 & 0 & 0 & 2\mathrm{i}& 0 & 0 & 0 & 0 \\ \text{F}_2 & \text{A} & 2 & 0 & 0 & -2\mathrm{i}& 0 & 0 & 0 & 0 \\ \end{array} \right)\)
G \(\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{1}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ \text{G}_1 & \text{A} & 2 & 2\mathrm{i}& 0 & 0 & 0 & 0 & 0 & 0 \\ \text{G}_2 & \text{A} & 2 & -2\mathrm{i}& 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\)
H \(\left(0,\frac{1}{2},\frac{1}{2}\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{1}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ \text{H}_1 & \text{C} & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2\mathrm{i}\\ \text{H}_2 & \text{C} & 2 & 0 & 0 & 0 & 0 & 0 & 0 & -2\mathrm{i}\\ \end{array} \right)\)
\(D^1\) EAZ & character of irrep
\(a\) \(\left(\frac{t}{2},0,0\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ a_1 & \text{A} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -1 & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ a_2 & \text{A} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ a_3 & \text{A} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & -1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ a_4 & \text{A} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ \end{array} \right)\)
\(b\) \(\left(0,\frac{t}{2},0\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ b_1 & \text{A} & 1 & -1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ b_2 & \text{A} & 1 & -1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ b_3 & \text{A} & 1 & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ b_4 & \text{A} & 1 & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ \end{array} \right)\)
\(c\) \(\left(\frac{1}{2},\frac{t}{2},0\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ c_1 & \text{D} & 2 & 0 & 0 & 0 \\ \end{array} \right)\)
\(d\) \(\left(\frac{1-t}{2},\frac{1}{2},0\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ d_1 & \text{C} & 2 & 0 & 0 & 0 \\ \end{array} \right)\)
\(e\) \(\left(\frac{t}{2},0,\frac{1}{2}\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ e_1 & \text{D} & 2 & 0 & 0 & 0 \\ \end{array} \right)\)
\(f\) \(\left(0,\frac{t}{2},\frac{1}{2}\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ f_1 & \text{A} & 1 & -1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ f_2 & \text{A} & 1 & -1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ f_3 & \text{A} & 1 & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ f_4 & \text{A} & 1 & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ \end{array} \right)\)
\(g\) \(\left(\frac{1}{2},\frac{t}{2},\frac{1}{2}\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ g_1 & \text{D} & 2 & 0 & 0 & 0 \\ \end{array} \right)\)
\(h\) \(\left(\frac{1-t}{2},\frac{1}{2},\frac{1}{2}\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ h_1 & \text{A} & 1 &\mathrm{i}e^{\frac{i \pi t}{2}} &\mathrm{i}& e^{\frac{i \pi t}{2}} \\ h_2 & \text{A} & 1 &\mathrm{i}e^{\frac{i \pi t}{2}} & -\mathrm{i} & -e^{\frac{i \pi t}{2}} \\ h_3 & \text{A} & 1 & -\mathrm{i} e^{\frac{i \pi t}{2}} &\mathrm{i}& -e^{\frac{i \pi t}{2}} \\ h_4 & \text{A} & 1 & -\mathrm{i} e^{\frac{i \pi t}{2}} & -\mathrm{i} & e^{\frac{i \pi t}{2}} \\ \end{array} \right)\)
\(i\) \(\left(0,0,\frac{t}{2}\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} \\ i_1 & \text{A} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -1 & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ i_2 & \text{A} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ i_3 & \text{A} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & -1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ i_4 & \text{A} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ \end{array} \right)\)
\(j\) \(\left(\frac{1}{2},0,\frac{t}{2}\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} \\ j_1 & \text{A} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -1 & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ j_2 & \text{A} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ j_3 & \text{A} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & -1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} \\ j_4 & \text{A} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} \\ \end{array} \right)\)
\(k\) \(\left(\frac{1}{2},\frac{1}{2},\frac{t}{2}\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} \\ k_1 & \text{D} & 2 & 0 & 0 & 0 \\ \end{array} \right)\)
\(l\) \(\left(0,\frac{1}{2},\frac{t}{2}\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{001},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{100},\left\{\frac{1}{2},\frac{1}{2},0\right\}\right\} & \left\{m_{010},\left\{0,0,\frac{1}{2}\right\}\right\} \\ l_1 & \text{C} & 2 & 0 & 0 & 0 \\ \end{array} \right)\)