Highest vectors of representations (total 23) ; the vectors are over the primal subalgebra. | \(g_{-5}\) | \(h_{1}\) | \(h_{3}\) | \(h_{5}\) | \(g_{5}\) | \(g_{11}\) | \(g_{14}\) | \(g_{15}\) | \(g_{17}\) | \(g_{4}\) | \(g_{8}\) | \(g_{9}\) | \(g_{12}\) | \(g_{23}\) | \(g_{24}\) | \(g_{25}\) | \(g_{18}\) | \(g_{20}\) | \(g_{21}\) | \(g_{22}\) | \(g_{13}\) | \(g_{16}\) | \(g_{19}\) |
weight | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(\omega_{1}\) | \(\omega_{1}\) | \(\omega_{1}\) | \(\omega_{1}\) | \(\omega_{2}\) | \(\omega_{2}\) | \(\omega_{2}\) | \(\omega_{2}\) | \(2\omega_{1}\) | \(2\omega_{1}\) | \(2\omega_{1}\) | \(\omega_{1}+\omega_{2}\) | \(\omega_{1}+\omega_{2}\) | \(\omega_{1}+\omega_{2}\) | \(\omega_{1}+\omega_{2}\) | \(2\omega_{2}\) | \(2\omega_{2}\) | \(2\omega_{2}\) |
weights rel. to Cartan of (centralizer+semisimple s.a.). | \(-4\psi_{3}\) | \(0\) | \(0\) | \(0\) | \(4\psi_{3}\) | \(\omega_{1}-\psi_{1}-2\psi_{3}\) | \(\omega_{1}+\psi_{1}-2\psi_{3}\) | \(\omega_{1}-\psi_{1}+2\psi_{3}\) | \(\omega_{1}+\psi_{1}+2\psi_{3}\) | \(\omega_{2}-\psi_{2}-2\psi_{3}\) | \(\omega_{2}+\psi_{2}-2\psi_{3}\) | \(\omega_{2}-\psi_{2}+2\psi_{3}\) | \(\omega_{2}+\psi_{2}+2\psi_{3}\) | \(2\omega_{1}-2\psi_{1}\) | \(2\omega_{1}\) | \(2\omega_{1}+2\psi_{1}\) | \(\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}\) | \(\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}\) | \(\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}\) | \(\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}\) | \(2\omega_{2}-2\psi_{2}\) | \(2\omega_{2}\) | \(2\omega_{2}+2\psi_{2}\) |
Isotypical components + highest weight | \(\displaystyle V_{-4\psi_{3}} \) → (0, 0, 0, 0, -4) | \(\displaystyle V_{0} \) → (0, 0, 0, 0, 0) | \(\displaystyle V_{4\psi_{3}} \) → (0, 0, 0, 0, 4) | \(\displaystyle V_{\omega_{1}-\psi_{1}-2\psi_{3}} \) → (1, 0, -1, 0, -2) | \(\displaystyle V_{\omega_{1}+\psi_{1}-2\psi_{3}} \) → (1, 0, 1, 0, -2) | \(\displaystyle V_{\omega_{1}-\psi_{1}+2\psi_{3}} \) → (1, 0, -1, 0, 2) | \(\displaystyle V_{\omega_{1}+\psi_{1}+2\psi_{3}} \) → (1, 0, 1, 0, 2) | \(\displaystyle V_{\omega_{2}-\psi_{2}-2\psi_{3}} \) → (0, 1, 0, -1, -2) | \(\displaystyle V_{\omega_{2}+\psi_{2}-2\psi_{3}} \) → (0, 1, 0, 1, -2) | \(\displaystyle V_{\omega_{2}-\psi_{2}+2\psi_{3}} \) → (0, 1, 0, -1, 2) | \(\displaystyle V_{\omega_{2}+\psi_{2}+2\psi_{3}} \) → (0, 1, 0, 1, 2) | \(\displaystyle V_{2\omega_{1}-2\psi_{1}} \) → (2, 0, -2, 0, 0) | \(\displaystyle V_{2\omega_{1}} \) → (2, 0, 0, 0, 0) | \(\displaystyle V_{2\omega_{1}+2\psi_{1}} \) → (2, 0, 2, 0, 0) | \(\displaystyle V_{\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}} \) → (1, 1, -1, -1, 0) | \(\displaystyle V_{\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}} \) → (1, 1, 1, -1, 0) | \(\displaystyle V_{\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}} \) → (1, 1, -1, 1, 0) | \(\displaystyle V_{\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}} \) → (1, 1, 1, 1, 0) | \(\displaystyle V_{2\omega_{2}-2\psi_{2}} \) → (0, 2, 0, -2, 0) | \(\displaystyle V_{2\omega_{2}} \) → (0, 2, 0, 0, 0) | \(\displaystyle V_{2\omega_{2}+2\psi_{2}} \) → (0, 2, 0, 2, 0) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Module label | \(W_{1}\) | \(W_{2}\) | \(W_{3}\) | \(W_{4}\) | \(W_{5}\) | \(W_{6}\) | \(W_{7}\) | \(W_{8}\) | \(W_{9}\) | \(W_{10}\) | \(W_{11}\) | \(W_{12}\) | \(W_{13}\) | \(W_{14}\) | \(W_{15}\) | \(W_{16}\) | \(W_{17}\) | \(W_{18}\) | \(W_{19}\) | \(W_{20}\) | \(W_{21}\) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. |
| Cartan of centralizer component.
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Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above | \(0\) | \(0\) | \(0\) | \(\omega_{1}\) \(-\omega_{1}\) | \(\omega_{1}\) \(-\omega_{1}\) | \(\omega_{1}\) \(-\omega_{1}\) | \(\omega_{1}\) \(-\omega_{1}\) | \(\omega_{2}\) \(-\omega_{2}\) | \(\omega_{2}\) \(-\omega_{2}\) | \(\omega_{2}\) \(-\omega_{2}\) | \(\omega_{2}\) \(-\omega_{2}\) | \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) | \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) | \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) | \(\omega_{1}+\omega_{2}\) \(-\omega_{1}+\omega_{2}\) \(\omega_{1}-\omega_{2}\) \(-\omega_{1}-\omega_{2}\) | \(\omega_{1}+\omega_{2}\) \(-\omega_{1}+\omega_{2}\) \(\omega_{1}-\omega_{2}\) \(-\omega_{1}-\omega_{2}\) | \(\omega_{1}+\omega_{2}\) \(-\omega_{1}+\omega_{2}\) \(\omega_{1}-\omega_{2}\) \(-\omega_{1}-\omega_{2}\) | \(\omega_{1}+\omega_{2}\) \(-\omega_{1}+\omega_{2}\) \(\omega_{1}-\omega_{2}\) \(-\omega_{1}-\omega_{2}\) | \(2\omega_{2}\) \(0\) \(-2\omega_{2}\) | \(2\omega_{2}\) \(0\) \(-2\omega_{2}\) | \(2\omega_{2}\) \(0\) \(-2\omega_{2}\) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer | \(-4\psi_{3}\) | \(0\) | \(4\psi_{3}\) | \(\omega_{1}-\psi_{1}-2\psi_{3}\) \(-\omega_{1}-\psi_{1}-2\psi_{3}\) | \(\omega_{1}+\psi_{1}-2\psi_{3}\) \(-\omega_{1}+\psi_{1}-2\psi_{3}\) | \(\omega_{1}-\psi_{1}+2\psi_{3}\) \(-\omega_{1}-\psi_{1}+2\psi_{3}\) | \(\omega_{1}+\psi_{1}+2\psi_{3}\) \(-\omega_{1}+\psi_{1}+2\psi_{3}\) | \(\omega_{2}-\psi_{2}-2\psi_{3}\) \(-\omega_{2}-\psi_{2}-2\psi_{3}\) | \(\omega_{2}+\psi_{2}-2\psi_{3}\) \(-\omega_{2}+\psi_{2}-2\psi_{3}\) | \(\omega_{2}-\psi_{2}+2\psi_{3}\) \(-\omega_{2}-\psi_{2}+2\psi_{3}\) | \(\omega_{2}+\psi_{2}+2\psi_{3}\) \(-\omega_{2}+\psi_{2}+2\psi_{3}\) | \(2\omega_{1}-2\psi_{1}\) \(-2\psi_{1}\) \(-2\omega_{1}-2\psi_{1}\) | \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) | \(2\omega_{1}+2\psi_{1}\) \(2\psi_{1}\) \(-2\omega_{1}+2\psi_{1}\) | \(\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}\) \(-\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}\) \(\omega_{1}-\omega_{2}-\psi_{1}-\psi_{2}\) \(-\omega_{1}-\omega_{2}-\psi_{1}-\psi_{2}\) | \(\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}\) \(-\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}\) \(\omega_{1}-\omega_{2}+\psi_{1}-\psi_{2}\) \(-\omega_{1}-\omega_{2}+\psi_{1}-\psi_{2}\) | \(\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}\) \(-\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}\) \(\omega_{1}-\omega_{2}-\psi_{1}+\psi_{2}\) \(-\omega_{1}-\omega_{2}-\psi_{1}+\psi_{2}\) | \(\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}\) \(-\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}\) \(\omega_{1}-\omega_{2}+\psi_{1}+\psi_{2}\) \(-\omega_{1}-\omega_{2}+\psi_{1}+\psi_{2}\) | \(2\omega_{2}-2\psi_{2}\) \(-2\psi_{2}\) \(-2\omega_{2}-2\psi_{2}\) | \(2\omega_{2}\) \(0\) \(-2\omega_{2}\) | \(2\omega_{2}+2\psi_{2}\) \(2\psi_{2}\) \(-2\omega_{2}+2\psi_{2}\) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a. | \(\displaystyle M_{-4\psi_{3}}\) | \(\displaystyle M_{0}\) | \(\displaystyle M_{4\psi_{3}}\) | \(\displaystyle M_{\omega_{1}-\psi_{1}-2\psi_{3}}\oplus M_{-\omega_{1}-\psi_{1}-2\psi_{3}}\) | \(\displaystyle M_{\omega_{1}+\psi_{1}-2\psi_{3}}\oplus M_{-\omega_{1}+\psi_{1}-2\psi_{3}}\) | \(\displaystyle M_{\omega_{1}-\psi_{1}+2\psi_{3}}\oplus M_{-\omega_{1}-\psi_{1}+2\psi_{3}}\) | \(\displaystyle M_{\omega_{1}+\psi_{1}+2\psi_{3}}\oplus M_{-\omega_{1}+\psi_{1}+2\psi_{3}}\) | \(\displaystyle M_{\omega_{2}-\psi_{2}-2\psi_{3}}\oplus M_{-\omega_{2}-\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{\omega_{2}+\psi_{2}-2\psi_{3}}\oplus M_{-\omega_{2}+\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{\omega_{2}-\psi_{2}+2\psi_{3}}\oplus M_{-\omega_{2}-\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{\omega_{2}+\psi_{2}+2\psi_{3}}\oplus M_{-\omega_{2}+\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{2\omega_{1}-2\psi_{1}}\oplus M_{-2\psi_{1}}\oplus M_{-2\omega_{1}-2\psi_{1}}\) | \(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\) | \(\displaystyle M_{2\omega_{1}+2\psi_{1}}\oplus M_{2\psi_{1}}\oplus M_{-2\omega_{1}+2\psi_{1}}\) | \(\displaystyle M_{\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}}\oplus M_{\omega_{1}-\omega_{2}-\psi_{1}-\psi_{2}} \oplus M_{-\omega_{1}-\omega_{2}-\psi_{1}-\psi_{2}}\) | \(\displaystyle M_{\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}}\oplus M_{-\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}}\oplus M_{\omega_{1}-\omega_{2}+\psi_{1}-\psi_{2}} \oplus M_{-\omega_{1}-\omega_{2}+\psi_{1}-\psi_{2}}\) | \(\displaystyle M_{\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}}\oplus M_{\omega_{1}-\omega_{2}-\psi_{1}+\psi_{2}} \oplus M_{-\omega_{1}-\omega_{2}-\psi_{1}+\psi_{2}}\) | \(\displaystyle M_{\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}}\oplus M_{-\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}}\oplus M_{\omega_{1}-\omega_{2}+\psi_{1}+\psi_{2}} \oplus M_{-\omega_{1}-\omega_{2}+\psi_{1}+\psi_{2}}\) | \(\displaystyle M_{2\omega_{2}-2\psi_{2}}\oplus M_{-2\psi_{2}}\oplus M_{-2\omega_{2}-2\psi_{2}}\) | \(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\) | \(\displaystyle M_{2\omega_{2}+2\psi_{2}}\oplus M_{2\psi_{2}}\oplus M_{-2\omega_{2}+2\psi_{2}}\) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Isotypic character | \(\displaystyle M_{-4\psi_{3}}\) | \(\displaystyle 3M_{0}\) | \(\displaystyle M_{4\psi_{3}}\) | \(\displaystyle M_{\omega_{1}-\psi_{1}-2\psi_{3}}\oplus M_{-\omega_{1}-\psi_{1}-2\psi_{3}}\) | \(\displaystyle M_{\omega_{1}+\psi_{1}-2\psi_{3}}\oplus M_{-\omega_{1}+\psi_{1}-2\psi_{3}}\) | \(\displaystyle M_{\omega_{1}-\psi_{1}+2\psi_{3}}\oplus M_{-\omega_{1}-\psi_{1}+2\psi_{3}}\) | \(\displaystyle M_{\omega_{1}+\psi_{1}+2\psi_{3}}\oplus M_{-\omega_{1}+\psi_{1}+2\psi_{3}}\) | \(\displaystyle M_{\omega_{2}-\psi_{2}-2\psi_{3}}\oplus M_{-\omega_{2}-\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{\omega_{2}+\psi_{2}-2\psi_{3}}\oplus M_{-\omega_{2}+\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{\omega_{2}-\psi_{2}+2\psi_{3}}\oplus M_{-\omega_{2}-\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{\omega_{2}+\psi_{2}+2\psi_{3}}\oplus M_{-\omega_{2}+\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{2\omega_{1}-2\psi_{1}}\oplus M_{-2\psi_{1}}\oplus M_{-2\omega_{1}-2\psi_{1}}\) | \(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\) | \(\displaystyle M_{2\omega_{1}+2\psi_{1}}\oplus M_{2\psi_{1}}\oplus M_{-2\omega_{1}+2\psi_{1}}\) | \(\displaystyle M_{\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\psi_{1}-\psi_{2}}\oplus M_{\omega_{1}-\omega_{2}-\psi_{1}-\psi_{2}} \oplus M_{-\omega_{1}-\omega_{2}-\psi_{1}-\psi_{2}}\) | \(\displaystyle M_{\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}}\oplus M_{-\omega_{1}+\omega_{2}+\psi_{1}-\psi_{2}}\oplus M_{\omega_{1}-\omega_{2}+\psi_{1}-\psi_{2}} \oplus M_{-\omega_{1}-\omega_{2}+\psi_{1}-\psi_{2}}\) | \(\displaystyle M_{\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}}\oplus M_{-\omega_{1}+\omega_{2}-\psi_{1}+\psi_{2}}\oplus M_{\omega_{1}-\omega_{2}-\psi_{1}+\psi_{2}} \oplus M_{-\omega_{1}-\omega_{2}-\psi_{1}+\psi_{2}}\) | \(\displaystyle M_{\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}}\oplus M_{-\omega_{1}+\omega_{2}+\psi_{1}+\psi_{2}}\oplus M_{\omega_{1}-\omega_{2}+\psi_{1}+\psi_{2}} \oplus M_{-\omega_{1}-\omega_{2}+\psi_{1}+\psi_{2}}\) | \(\displaystyle M_{2\omega_{2}-2\psi_{2}}\oplus M_{-2\psi_{2}}\oplus M_{-2\omega_{2}-2\psi_{2}}\) | \(\displaystyle M_{2\omega_{2}}\oplus M_{0}\oplus M_{-2\omega_{2}}\) | \(\displaystyle M_{2\omega_{2}+2\psi_{2}}\oplus M_{2\psi_{2}}\oplus M_{-2\omega_{2}+2\psi_{2}}\) |
2 & | 0\\ |
0 & | 2\\ |