MSG 228.135 \(A_{2g}\)

\(D^0\) EAZ & character of irrep
A \(\left(0,0,0\right)\) \(\left( \begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{4_{100},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{4^-{}_{100},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{4_{010},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{4^-{}_{010},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{4_{001},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{4^-{}_{001},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{2_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{2_{110},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{2_{\text{1-10}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{2_{011},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{\text{01-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{2_{101},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{2_{\text{10-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{3_{111}\right\} & \left\{3^-{}_{111}\right\} & \left\{3^-{}_{\text{1-1-1}},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{3_{\text{1-1-1}},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{3^-{}_{\text{-11-1}},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{3_{\text{-11-1}},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{3^-{}_{\text{-1-11}},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{3_{\text{-1-11}},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{\bar{1}\right\} & \left\{\bar{4}_{100},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{\bar{4}^-{}_{100},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{\bar{4}_{010},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{\bar{4}^-{}_{010},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{\bar{4}_{001},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{\bar{4}^-{}_{001},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{m_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{m_{110},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{m_{\text{1-10}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{011},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{\text{01-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{101},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{m_{\text{10-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{3}_{111}\right\} & \left\{\bar{3}^-{}_{111}\right\} & \left\{\bar{3}^-{}_{\text{1-1-1}},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{\bar{3}_{\text{1-1-1}},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{\bar{3}^-{}_{\text{-11-1}},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{\bar{3}_{\text{-11-1}},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{\bar{3}^-{}_{\text{-1-11}},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{\bar{3}_{\text{-1-11}},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} \\ \text{A}_1 & \text{AI} & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\ \text{A}_2 & \text{AI} & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \text{A}_3 & \text{AI} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\ \text{A}_4 & \text{AI} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \text{A}_5 & \text{CI} & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & -2 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \text{A}_6 & \text{CI} & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\ \text{A}_7 & \text{AI} & 3 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \text{A}_8 & \text{AI} & 3 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \text{A}_9 & \text{AI} & 3 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \text{A}_{10} & \text{AI} & 3 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\)
B \(\left(0,1,0\right)\) \(\left( \begin{array}{cccccccccccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{4_{010},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{4^-{}_{010},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{2_{101},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{2_{\text{10-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{1}\right\} & \left\{\bar{4}_{010},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{\bar{4}^-{}_{010},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{m_{101},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{m_{\text{10-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ \text{B}_1 & \text{AI} & 2 & 0 & 0 & 0 & -2 & 0 & 2 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \text{B}_2 & \text{AI} & 2 & 0 & 0 & 0 & -2 & 0 & -2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \text{B}_3 & \text{AI} & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 \\ \text{B}_4 & \text{AI} & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & -2 \\ \end{array} \right)\)
C \(\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\) \(\left( \begin{array}{cccccccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{\text{1-10}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{2_{\text{01-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{2_{\text{10-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{3_{111}\right\} & \left\{3^-{}_{111}\right\} & \left\{\bar{1}\right\} & \left\{m_{\text{1-10}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{\text{01-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{\text{10-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{3}_{111}\right\} & \left\{\bar{3}^-{}_{111}\right\} \\ \text{C}_1 & \text{AIII} & 2 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & -\mathrm{i} \sqrt{3} &\mathrm{i}\sqrt{3} \\ \text{C}_2 & \text{AIII} & 2 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 &\mathrm{i}\sqrt{3} & -\mathrm{i} \sqrt{3} \\ \text{C}_3 & \text{BDI} & 2 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\)
D \(\left(\frac{1}{2},1,0\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{011},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{\text{01-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{\bar{4}_{100},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{\bar{4}^-{}_{100},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{m_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} \\ \text{D}_1 & \text{D} & 2 & 0 & 0 & 0 & 1+\mathrm{i} & 1-\mathrm{i} & 0 & 0 \\ \text{D}_2 & \text{D} & 2 & 0 & 0 & 0 & -1-\mathrm{i} & -1+\mathrm{i} & 0 & 0 \\ \end{array} \right)\)
\(D^1\) EAZ & character of irrep
\(a\) \(\left(0,t,0\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{4_{010},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{4^-{}_{010},\left\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{m_{101},\left\{\frac{1}{4},\frac{1}{2},\frac{1}{4}\right\}\right\} & \left\{m_{\text{10-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ a_1 & \text{AI} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\mathrm{i} \pi t} & e^{-\mathrm{i} \pi t} \\ a_2 & \text{AI} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\mathrm{i} \pi t} & -e^{-\mathrm{i} \pi t} \\ a_3 & \text{AI} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\frac{1}{2}\mathrm{i}\pi t} & -e^{-\mathrm{i} \pi t} & -e^{-\mathrm{i} \pi t} \\ a_4 & \text{AI} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\frac{1}{2}\mathrm{i}\pi t} & 1 & e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\frac{1}{2}\mathrm{i}\pi t} & e^{-\mathrm{i} \pi t} & e^{-\mathrm{i} \pi t} \\ a_5 & \text{BDI} & 2 & 0 & 0 & -2 & 0 & 0 & 0 & 0 \\ \end{array} \right)\)
\(b\) \(\left(t,t,0\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{110},\left\{\frac{1}{4},\frac{1}{4},\frac{1}{2}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{m_{\text{1-10}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ b_1 & \text{AI} & 1 & -e^{-\mathrm{i} \pi t} & -e^{-\mathrm{i} \pi t} & e^{-2\mathrm{i}\pi t} \\ b_2 & \text{AI} & 1 & -e^{-\mathrm{i} \pi t} & e^{-\mathrm{i} \pi t} & -e^{-2\mathrm{i}\pi t} \\ b_3 & \text{AI} & 1 & e^{-\mathrm{i} \pi t} & -e^{-\mathrm{i} \pi t} & -e^{-2\mathrm{i}\pi t} \\ b_4 & \text{AI} & 1 & e^{-\mathrm{i} \pi t} & e^{-\mathrm{i} \pi t} & e^{-2\mathrm{i}\pi t} \\ \end{array} \right)\)
\(c\) \(\left(\frac{t}{2},1,0\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} \\ c_1 & \text{BDI} & 2 & 0 & 0 & 0 \\ \end{array} \right)\)
\(d\) \(\left(\frac{t}{2},\frac{t}{2},\frac{t}{2}\right)\) \(\left( \begin{array}{cccccccc} \text{} & \text{EAZ} & \{1\} & \left\{3_{111}\right\} & \left\{3^-{}_{111}\right\} & \left\{m_{\text{1-10}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{\text{01-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} & \left\{m_{\text{10-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ d_1 & \text{AI} & 1 & 1 & 1 & -e^{-\frac{3}{2}\mathrm{i}\pi t} & -e^{-\frac{3}{2}\mathrm{i}\pi t} & -e^{-\frac{3}{2}\mathrm{i}\pi t} \\ d_2 & \text{AI} & 1 & 1 & 1 & e^{-\frac{3}{2}\mathrm{i}\pi t} & e^{-\frac{3}{2}\mathrm{i}\pi t} & e^{-\frac{3}{2}\mathrm{i}\pi t} \\ d_3 & \text{CI} & 2 & -1 & -1 & 0 & 0 & 0 \\ \end{array} \right)\)
\(e\) \(\left(1-\frac{t}{2},1-\frac{t}{2},\frac{t}{2}\right)\) \(\left( \begin{array}{cccc} \text{} & \text{EAZ} & \{1\} & \left\{m_{\text{1-10}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ e_1 & \text{AI} & 1 & -e^{\frac{i \pi t}{2}} \\ e_2 & \text{AI} & 1 & e^{\frac{i \pi t}{2}} \\ \end{array} \right)\)
\(f\) \(\left(\frac{1}{2},1-\frac{t}{2},\frac{t}{2}\right)\) \(\left( \begin{array}{cccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{\text{01-1}},\left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\}\right\} \\ f_1 & \text{D} & 1 & 1 \\ f_2 & \text{D} & 1 & -1 \\ \end{array} \right)\)