Highest vectors of representations (total 2) ; the vectors are over the primal subalgebra. | \(g_{2}+4/3g_{1}\) | \(g_{4}\) |
weight | \(2\omega_{1}\) | \(6\omega_{1}\) |
Isotypical components + highest weight | \(\displaystyle V_{2\omega_{1}} \) → (2) | \(\displaystyle V_{6\omega_{1}} \) → (6) | ||||||||||||
Module label | \(W_{1}\) | \(W_{2}\) | ||||||||||||
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. | Semisimple subalgebra component.
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Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above | \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) | \(6\omega_{1}\) \(4\omega_{1}\) \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) \(-4\omega_{1}\) \(-6\omega_{1}\) | ||||||||||||
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer | \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) | \(6\omega_{1}\) \(4\omega_{1}\) \(2\omega_{1}\) \(0\) \(-2\omega_{1}\) \(-4\omega_{1}\) \(-6\omega_{1}\) | ||||||||||||
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a. | \(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\) | \(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\) | ||||||||||||
Isotypic character | \(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\) | \(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\) |