MSG 227.133 \(A_{2g}\)

\(D^0\) 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},0,\frac{1}{4}\right\}\right\} & \left\{4^-{}_{100},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{4_{010},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{4^-{}_{010},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{4_{001},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{4^-{}_{001},\left\{\frac{1}{4},0,\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},0\right\}\right\} & \left\{2_{\text{1-10}}\right\} & \left\{2_{011},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{\text{01-1}}\right\} & \left\{2_{101},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{2_{\text{10-1}}\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},0,\frac{1}{4}\right\}\right\} & \left\{\bar{4}^-{}_{100},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{\bar{4}_{010},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{\bar{4}^-{}_{010},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{\bar{4}_{001},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{\bar{4}^-{}_{001},\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_{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},0\right\}\right\} & \left\{m_{\text{1-10}}\right\} & \left\{m_{011},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{\text{01-1}}\right\} & \left\{m_{101},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{\text{10-1}}\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} & 2 & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & -2 & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 \\ \text{A}_2 & \text{AI} & 2 & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 2 & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ \text{A}_3 & \text{AI} & 2 & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & -2 & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 \\ \text{A}_4 & \text{AI} & 2 & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 2 & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & \sqrt{2} & -\sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ \text{A}_5 & \text{CI} & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ \text{A}_6 & \text{CI} & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 \\ \end{array} \right)\)
B \(\left(0,1,0\right)\) \(\left( \begin{array}{cccccccccccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{4_{010},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{4^-{}_{010},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{2_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{2_{101},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{2_{\text{10-1}}\right\} & \left\{\bar{1}\right\} & \left\{\bar{4}_{010},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{\bar{4}^-{}_{010},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{m_{101},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{\text{10-1}}\right\} \\ \text{B}_1 & \text{CI} & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\)
C \(\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\) \(\left( \begin{array}{cccccccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{\text{1-10}}\right\} & \left\{2_{\text{01-1}}\right\} & \left\{2_{\text{10-1}}\right\} & \left\{3_{111}\right\} & \left\{3^-{}_{111}\right\} & \left\{\bar{1}\right\} & \left\{m_{\text{1-10}}\right\} & \left\{m_{\text{01-1}}\right\} & \left\{m_{\text{10-1}}\right\} & \left\{\bar{3}_{111}\right\} & \left\{\bar{3}^-{}_{111}\right\} \\ \text{C}_1 & \text{C} & 1 & -\mathrm{i} & -\mathrm{i} &\mathrm{i}& -1 & 1 & -1 &\mathrm{i}&\mathrm{i}& -\mathrm{i} & 1 & -1 \\ \text{C}_2 & \text{C} & 1 & -\mathrm{i} & -\mathrm{i} &\mathrm{i}& -1 & 1 & 1 & -\mathrm{i} & -\mathrm{i} &\mathrm{i}& -1 & 1 \\ \text{C}_3 & \text{C} & 1 &\mathrm{i}&\mathrm{i}& -\mathrm{i} & -1 & 1 & -1 & -\mathrm{i} & -\mathrm{i} &\mathrm{i}& 1 & -1 \\ \text{C}_4 & \text{C} & 1 &\mathrm{i}&\mathrm{i}& -\mathrm{i} & -1 & 1 & 1 &\mathrm{i}&\mathrm{i}& -\mathrm{i} & -1 & 1 \\ \text{C}_5 & \text{C} & 2 & 0 & 0 & 0 & 1 & -1 & -2 & 0 & 0 & 0 & -1 & 1 \\ \text{C}_6 & \text{C} & 2 & 0 & 0 & 0 & 1 & -1 & 2 & 0 & 0 & 0 & 1 & -1 \\ \end{array} \right)\)
D \(\left(\frac{1}{2},1,0\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{011},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{\text{01-1}}\right\} & \left\{\bar{4}_{100},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{\bar{4}^-{}_{100},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{m_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} \\ \text{D}_1 & \text{AIII} & 1 & -\mathrm{i} & -1 & -\mathrm{i} & (-1)^{3/4} & \sqrt[4]{-1} & \sqrt[4]{-1} & -(-1)^{3/4} \\ \text{D}_2 & \text{AIII} & 1 & -\mathrm{i} & -1 & -\mathrm{i} & -(-1)^{3/4} & -\sqrt[4]{-1} & -\sqrt[4]{-1} & (-1)^{3/4} \\ \text{D}_3 & \text{AIII} & 1 & -\mathrm{i} & 1 &\mathrm{i}& (-1)^{3/4} & \sqrt[4]{-1} & -\sqrt[4]{-1} & (-1)^{3/4} \\ \text{D}_4 & \text{AIII} & 1 & -\mathrm{i} & 1 &\mathrm{i}& -(-1)^{3/4} & -\sqrt[4]{-1} & \sqrt[4]{-1} & -(-1)^{3/4} \\ \text{D}_5 & \text{CII} & 2 & 2\mathrm{i}& 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\)
\(D^1\) EAZ & character of irrep
\(a\) \(\left(0,t,0\right)\) \(\left( \begin{array}{cccccccccc} \text{} & \text{EAZ} & \{1\} & \left\{4_{010},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{4^-{}_{010},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{2_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{m_{101},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{\text{10-1}}\right\} \\ a_1 & \text{AI} & 2 & -\sqrt{2} e^{-\frac{1}{2}\mathrm{i}\pi t} & \sqrt{2} e^{-\frac{1}{2}\mathrm{i}\pi t} & 0 & 0 & 0 & 0 & 0 \\ a_2 & \text{AI} & 2 & \sqrt{2} e^{-\frac{1}{2}\mathrm{i}\pi t} & -\sqrt{2} e^{-\frac{1}{2}\mathrm{i}\pi t} & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\)
\(b\) \(\left(t,t,0\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{110},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} & \left\{m_{\text{1-10}}\right\} \\ b_1 & \text{CI} & 2 & 0 & 0 & 0 \\ \end{array} \right)\)
\(c\) \(\left(\frac{t}{2},1,0\right)\) \(\left( \begin{array}{cccccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{100},\left\{0,\frac{1}{4},\frac{1}{4}\right\}\right\} & \left\{m_{010},\left\{\frac{1}{4},0,\frac{1}{4}\right\}\right\} & \left\{m_{001},\left\{\frac{1}{4},\frac{1}{4},0\right\}\right\} \\ c_1 & \text{AIII} & 1 &\mathrm{i}& -\mathrm{i} e^{-\frac{1}{4}\mathrm{i}\pi t} & e^{-\frac{1}{4}\mathrm{i}\pi t} \\ c_2 & \text{AIII} & 1 &\mathrm{i}&\mathrm{i}e^{-\frac{1}{4}\mathrm{i}\pi t} & -e^{-\frac{1}{4}\mathrm{i}\pi t} \\ c_3 & \text{AIII} & 1 & -\mathrm{i} & -\mathrm{i} e^{-\frac{1}{4}\mathrm{i}\pi t} & -e^{-\frac{1}{4}\mathrm{i}\pi t} \\ c_4 & \text{AIII} & 1 & -\mathrm{i} &\mathrm{i}e^{-\frac{1}{4}\mathrm{i}\pi t} & e^{-\frac{1}{4}\mathrm{i}\pi t} \\ \end{array} \right)\)
\(d\) \(\left(\frac{t}{2},\frac{t}{2},\frac{t}{2}\right)\) \(\left( \begin{array}{cccccccc} \text{} & \text{EAZ} & \{1\} & \left\{3_{111}\right\} & \left\{3^-{}_{111}\right\} & \left\{m_{\text{1-10}}\right\} & \left\{m_{\text{01-1}}\right\} & \left\{m_{\text{10-1}}\right\} \\ d_1 & \text{C} & 1 & -1 & 1 & -\mathrm{i} & -\mathrm{i} &\mathrm{i}\\ d_2 & \text{C} & 1 & -1 & 1 &\mathrm{i}&\mathrm{i}& -\mathrm{i} \\ d_3 & \text{CI} & 2 & 1 & -1 & 0 & 0 & 0 \\ \end{array} \right)\)
\(e\) \(\left(1-\frac{t}{2},1-\frac{t}{2},\frac{t}{2}\right)\) \(\left( \begin{array}{cccc} \text{} & \text{EAZ} & \{1\} & \left\{m_{\text{1-10}}\right\} \\ e_1 & \text{C} & 1 & -\mathrm{i} \\ e_2 & \text{C} & 1 &\mathrm{i}\\ \end{array} \right)\)
\(f\) \(\left(\frac{1}{2},1-\frac{t}{2},\frac{t}{2}\right)\) \(\left( \begin{array}{cccc} \text{} & \text{EAZ} & \{1\} & \left\{2_{\text{01-1}}\right\} \\ f_1 & \text{CII} & 1 & -\mathrm{i} \\ f_2 & \text{CII} & 1 &\mathrm{i}\\ \end{array} \right)\)