Highest vectors of representations (total 29) ; the vectors are over the primal subalgebra. | \(g_{-5}\) | \(g_{-12}\) | \(g_{-19}\) | \(g_{-9}\) | \(g_{4}\) | \(g_{-16}\) | \(g_{-3}\) | \(g_{8}\) | \(g_{-13}\) | \(h_{4}\) | \(h_{5}\) | \(h_{3}\) | \(g_{13}\) | \(g_{-8}\) | \(g_{3}\) | \(g_{16}\) | \(g_{-4}\) | \(g_{9}\) | \(g_{19}\) | \(g_{12}\) | \(g_{5}\) | \(g_{23}+3/4g_{1}\) | \(g_{14}\) | \(g_{6}\) | \(g_{10}\) | \(g_{20}\) | \(g_{22}\) | \(g_{17}\) | \(g_{25}\) |
weight | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(2\omega_{1}\) | \(3\omega_{1}\) | \(3\omega_{1}\) | \(3\omega_{1}\) | \(3\omega_{1}\) | \(3\omega_{1}\) | \(3\omega_{1}\) | \(6\omega_{1}\) |
weights rel. to Cartan of (centralizer+semisimple s.a.). | \(2\psi_{2}-4\psi_{3}\) | \(-\psi_{1}+\psi_{2}-2\psi_{3}\) | \(-2\psi_{1}\) | \(\psi_{1}-2\psi_{3}\) | \(-\psi_{1}+2\psi_{2}-2\psi_{3}\) | \(-\psi_{2}\) | \(-2\psi_{1}+\psi_{2}\) | \(\psi_{1}+\psi_{2}-2\psi_{3}\) | \(2\psi_{1}-2\psi_{2}\) | \(0\) | \(0\) | \(0\) | \(-2\psi_{1}+2\psi_{2}\) | \(-\psi_{1}-\psi_{2}+2\psi_{3}\) | \(2\psi_{1}-\psi_{2}\) | \(\psi_{2}\) | \(\psi_{1}-2\psi_{2}+2\psi_{3}\) | \(-\psi_{1}+2\psi_{3}\) | \(2\psi_{1}\) | \(\psi_{1}-\psi_{2}+2\psi_{3}\) | \(-2\psi_{2}+4\psi_{3}\) | \(2\omega_{1}\) | \(3\omega_{1}+\psi_{2}-2\psi_{3}\) | \(3\omega_{1}-\psi_{1}\) | \(3\omega_{1}+\psi_{1}-\psi_{2}\) | \(3\omega_{1}-\psi_{1}+\psi_{2}\) | \(3\omega_{1}+\psi_{1}\) | \(3\omega_{1}-\psi_{2}+2\psi_{3}\) | \(6\omega_{1}\) |
Isotypical components + highest weight | \(\displaystyle V_{2\psi_{2}-4\psi_{3}} \) → (0, 0, 2, -4) | \(\displaystyle V_{-\psi_{1}+\psi_{2}-2\psi_{3}} \) → (0, -1, 1, -2) | \(\displaystyle V_{-2\psi_{1}} \) → (0, -2, 0, 0) | \(\displaystyle V_{\psi_{1}-2\psi_{3}} \) → (0, 1, 0, -2) | \(\displaystyle V_{-\psi_{1}+2\psi_{2}-2\psi_{3}} \) → (0, -1, 2, -2) | \(\displaystyle V_{-\psi_{2}} \) → (0, 0, -1, 0) | \(\displaystyle V_{-2\psi_{1}+\psi_{2}} \) → (0, -2, 1, 0) | \(\displaystyle V_{\psi_{1}+\psi_{2}-2\psi_{3}} \) → (0, 1, 1, -2) | \(\displaystyle V_{2\psi_{1}-2\psi_{2}} \) → (0, 2, -2, 0) | \(\displaystyle V_{0} \) → (0, 0, 0, 0) | \(\displaystyle V_{-2\psi_{1}+2\psi_{2}} \) → (0, -2, 2, 0) | \(\displaystyle V_{-\psi_{1}-\psi_{2}+2\psi_{3}} \) → (0, -1, -1, 2) | \(\displaystyle V_{2\psi_{1}-\psi_{2}} \) → (0, 2, -1, 0) | \(\displaystyle V_{\psi_{2}} \) → (0, 0, 1, 0) | \(\displaystyle V_{\psi_{1}-2\psi_{2}+2\psi_{3}} \) → (0, 1, -2, 2) | \(\displaystyle V_{-\psi_{1}+2\psi_{3}} \) → (0, -1, 0, 2) | \(\displaystyle V_{2\psi_{1}} \) → (0, 2, 0, 0) | \(\displaystyle V_{\psi_{1}-\psi_{2}+2\psi_{3}} \) → (0, 1, -1, 2) | \(\displaystyle V_{-2\psi_{2}+4\psi_{3}} \) → (0, 0, -2, 4) | \(\displaystyle V_{2\omega_{1}} \) → (2, 0, 0, 0) | \(\displaystyle V_{3\omega_{1}+\psi_{2}-2\psi_{3}} \) → (3, 0, 1, -2) | \(\displaystyle V_{3\omega_{1}-\psi_{1}} \) → (3, -1, 0, 0) | \(\displaystyle V_{3\omega_{1}+\psi_{1}-\psi_{2}} \) → (3, 1, -1, 0) | \(\displaystyle V_{3\omega_{1}-\psi_{1}+\psi_{2}} \) → (3, -1, 1, 0) | \(\displaystyle V_{3\omega_{1}+\psi_{1}} \) → (3, 1, 0, 0) | \(\displaystyle V_{3\omega_{1}-\psi_{2}+2\psi_{3}} \) → (3, 0, -1, 2) | \(\displaystyle V_{6\omega_{1}} \) → (6, 0, 0, 0) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Module label | \(W_{1}\) | \(W_{2}\) | \(W_{3}\) | \(W_{4}\) | \(W_{5}\) | \(W_{6}\) | \(W_{7}\) | \(W_{8}\) | \(W_{9}\) | \(W_{10}\) | \(W_{11}\) | \(W_{12}\) | \(W_{13}\) | \(W_{14}\) | \(W_{15}\) | \(W_{16}\) | \(W_{17}\) | \(W_{18}\) | \(W_{19}\) | \(W_{20}\) | \(W_{21}\) | \(W_{22}\) | \(W_{23}\) | \(W_{24}\) | \(W_{25}\) | \(W_{26}\) | \(W_{27}\) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. |
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| 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 | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) | \(3\omega_{1}\) \(\omega_{1}\) \(-\omega_{1}\) \(-3\omega_{1}\) | \(3\omega_{1}\) \(\omega_{1}\) \(-\omega_{1}\) \(-3\omega_{1}\) | \(3\omega_{1}\) \(\omega_{1}\) \(-\omega_{1}\) \(-3\omega_{1}\) | \(3\omega_{1}\) \(\omega_{1}\) \(-\omega_{1}\) \(-3\omega_{1}\) | \(3\omega_{1}\) \(\omega_{1}\) \(-\omega_{1}\) \(-3\omega_{1}\) | \(3\omega_{1}\) \(\omega_{1}\) \(-\omega_{1}\) \(-3\omega_{1}\) | \(6\omega_{1}\) \(4\omega_{1}\) \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) \(-4\omega_{1}\) \(-6\omega_{1}\) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer | \(2\psi_{2}-4\psi_{3}\) | \(-\psi_{1}+\psi_{2}-2\psi_{3}\) | \(-2\psi_{1}\) | \(\psi_{1}-2\psi_{3}\) | \(-\psi_{1}+2\psi_{2}-2\psi_{3}\) | \(-\psi_{2}\) | \(-2\psi_{1}+\psi_{2}\) | \(\psi_{1}+\psi_{2}-2\psi_{3}\) | \(2\psi_{1}-2\psi_{2}\) | \(0\) | \(-2\psi_{1}+2\psi_{2}\) | \(-\psi_{1}-\psi_{2}+2\psi_{3}\) | \(2\psi_{1}-\psi_{2}\) | \(\psi_{2}\) | \(\psi_{1}-2\psi_{2}+2\psi_{3}\) | \(-\psi_{1}+2\psi_{3}\) | \(2\psi_{1}\) | \(\psi_{1}-\psi_{2}+2\psi_{3}\) | \(-2\psi_{2}+4\psi_{3}\) | \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) | \(3\omega_{1}+\psi_{2}-2\psi_{3}\) \(\omega_{1}+\psi_{2}-2\psi_{3}\) \(-\omega_{1}+\psi_{2}-2\psi_{3}\) \(-3\omega_{1}+\psi_{2}-2\psi_{3}\) | \(3\omega_{1}-\psi_{1}\) \(\omega_{1}-\psi_{1}\) \(-\omega_{1}-\psi_{1}\) \(-3\omega_{1}-\psi_{1}\) | \(3\omega_{1}+\psi_{1}-\psi_{2}\) \(\omega_{1}+\psi_{1}-\psi_{2}\) \(-\omega_{1}+\psi_{1}-\psi_{2}\) \(-3\omega_{1}+\psi_{1}-\psi_{2}\) | \(3\omega_{1}-\psi_{1}+\psi_{2}\) \(\omega_{1}-\psi_{1}+\psi_{2}\) \(-\omega_{1}-\psi_{1}+\psi_{2}\) \(-3\omega_{1}-\psi_{1}+\psi_{2}\) | \(3\omega_{1}+\psi_{1}\) \(\omega_{1}+\psi_{1}\) \(-\omega_{1}+\psi_{1}\) \(-3\omega_{1}+\psi_{1}\) | \(3\omega_{1}-\psi_{2}+2\psi_{3}\) \(\omega_{1}-\psi_{2}+2\psi_{3}\) \(-\omega_{1}-\psi_{2}+2\psi_{3}\) \(-3\omega_{1}-\psi_{2}+2\psi_{3}\) | \(6\omega_{1}\) \(4\omega_{1}\) \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) \(-4\omega_{1}\) \(-6\omega_{1}\) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a. | \(\displaystyle M_{2\psi_{2}-4\psi_{3}}\) | \(\displaystyle M_{-\psi_{1}+\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{-2\psi_{1}}\) | \(\displaystyle M_{\psi_{1}-2\psi_{3}}\) | \(\displaystyle M_{-\psi_{1}+2\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{-\psi_{2}}\) | \(\displaystyle M_{-2\psi_{1}+\psi_{2}}\) | \(\displaystyle M_{\psi_{1}+\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{2\psi_{1}-2\psi_{2}}\) | \(\displaystyle M_{0}\) | \(\displaystyle M_{-2\psi_{1}+2\psi_{2}}\) | \(\displaystyle M_{-\psi_{1}-\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{2\psi_{1}-\psi_{2}}\) | \(\displaystyle M_{\psi_{2}}\) | \(\displaystyle M_{\psi_{1}-2\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{-\psi_{1}+2\psi_{3}}\) | \(\displaystyle M_{2\psi_{1}}\) | \(\displaystyle M_{\psi_{1}-\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{-2\psi_{2}+4\psi_{3}}\) | \(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\) | \(\displaystyle M_{3\omega_{1}+\psi_{2}-2\psi_{3}}\oplus M_{\omega_{1}+\psi_{2}-2\psi_{3}}\oplus M_{-\omega_{1}+\psi_{2}-2\psi_{3}} \oplus M_{-3\omega_{1}+\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{3\omega_{1}-\psi_{1}}\oplus M_{\omega_{1}-\psi_{1}}\oplus M_{-\omega_{1}-\psi_{1}}\oplus M_{-3\omega_{1}-\psi_{1}}\) | \(\displaystyle M_{3\omega_{1}+\psi_{1}-\psi_{2}}\oplus M_{\omega_{1}+\psi_{1}-\psi_{2}}\oplus M_{-\omega_{1}+\psi_{1}-\psi_{2}}\oplus M_{-3\omega_{1}+\psi_{1}-\psi_{2}}\) | \(\displaystyle M_{3\omega_{1}-\psi_{1}+\psi_{2}}\oplus M_{\omega_{1}-\psi_{1}+\psi_{2}}\oplus M_{-\omega_{1}-\psi_{1}+\psi_{2}}\oplus M_{-3\omega_{1}-\psi_{1}+\psi_{2}}\) | \(\displaystyle M_{3\omega_{1}+\psi_{1}}\oplus M_{\omega_{1}+\psi_{1}}\oplus M_{-\omega_{1}+\psi_{1}}\oplus M_{-3\omega_{1}+\psi_{1}}\) | \(\displaystyle M_{3\omega_{1}-\psi_{2}+2\psi_{3}}\oplus M_{\omega_{1}-\psi_{2}+2\psi_{3}}\oplus M_{-\omega_{1}-\psi_{2}+2\psi_{3}} \oplus M_{-3\omega_{1}-\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Isotypic character | \(\displaystyle M_{2\psi_{2}-4\psi_{3}}\) | \(\displaystyle M_{-\psi_{1}+\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{-2\psi_{1}}\) | \(\displaystyle M_{\psi_{1}-2\psi_{3}}\) | \(\displaystyle M_{-\psi_{1}+2\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{-\psi_{2}}\) | \(\displaystyle M_{-2\psi_{1}+\psi_{2}}\) | \(\displaystyle M_{\psi_{1}+\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{2\psi_{1}-2\psi_{2}}\) | \(\displaystyle 3M_{0}\) | \(\displaystyle M_{-2\psi_{1}+2\psi_{2}}\) | \(\displaystyle M_{-\psi_{1}-\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{2\psi_{1}-\psi_{2}}\) | \(\displaystyle M_{\psi_{2}}\) | \(\displaystyle M_{\psi_{1}-2\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{-\psi_{1}+2\psi_{3}}\) | \(\displaystyle M_{2\psi_{1}}\) | \(\displaystyle M_{\psi_{1}-\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{-2\psi_{2}+4\psi_{3}}\) | \(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\) | \(\displaystyle M_{3\omega_{1}+\psi_{2}-2\psi_{3}}\oplus M_{\omega_{1}+\psi_{2}-2\psi_{3}}\oplus M_{-\omega_{1}+\psi_{2}-2\psi_{3}} \oplus M_{-3\omega_{1}+\psi_{2}-2\psi_{3}}\) | \(\displaystyle M_{3\omega_{1}-\psi_{1}}\oplus M_{\omega_{1}-\psi_{1}}\oplus M_{-\omega_{1}-\psi_{1}}\oplus M_{-3\omega_{1}-\psi_{1}}\) | \(\displaystyle M_{3\omega_{1}+\psi_{1}-\psi_{2}}\oplus M_{\omega_{1}+\psi_{1}-\psi_{2}}\oplus M_{-\omega_{1}+\psi_{1}-\psi_{2}}\oplus M_{-3\omega_{1}+\psi_{1}-\psi_{2}}\) | \(\displaystyle M_{3\omega_{1}-\psi_{1}+\psi_{2}}\oplus M_{\omega_{1}-\psi_{1}+\psi_{2}}\oplus M_{-\omega_{1}-\psi_{1}+\psi_{2}}\oplus M_{-3\omega_{1}-\psi_{1}+\psi_{2}}\) | \(\displaystyle M_{3\omega_{1}+\psi_{1}}\oplus M_{\omega_{1}+\psi_{1}}\oplus M_{-\omega_{1}+\psi_{1}}\oplus M_{-3\omega_{1}+\psi_{1}}\) | \(\displaystyle M_{3\omega_{1}-\psi_{2}+2\psi_{3}}\oplus M_{\omega_{1}-\psi_{2}+2\psi_{3}}\oplus M_{-\omega_{1}-\psi_{2}+2\psi_{3}} \oplus M_{-3\omega_{1}-\psi_{2}+2\psi_{3}}\) | \(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\) |