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