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LSpice
LSpice
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exterior Exterior square of $\operatorname{Sp}(4,\mathbb{C})$ is isomorphic to $\operatorname{SO}(5,\mathbb{C})$

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LSpice
LSpice
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exterior square of $Sp$\operatorname{Sp}(4,\mathbb{C})$ is isomorphic to $SO$\operatorname{SO}(5,\mathbb{C})$

I am studying exceptional isomorphisms recently, which arise due to the coincidence in Dynkin diagram.

I saw two forms of expressing the exceptional isomorphisms, one is isomorphisms between the spin group and the corresponding group, established by spin or half-spin representations; the other is isomorphism from symmetric squares, exterior squares or tensor squares of the relevant group to the special orthogonal group. But in this case, I don't know how to construct the induced symmetric bilinear form on the vector space. And in the case $\Lambda^2 Sp(4,\mathbb{C})\cong SO(5,\mathbb{C})$${\bigwedge}^2\operatorname{Sp}(4,\mathbb{C})\cong \operatorname{SO}(5,\mathbb{C})$, I think the dimension of exterior square of a 4-dimensional vector space is 6, and I am not clear how it gets mapped into $SO(5)$$\operatorname{SO}(5)$.

exterior square of $Sp(4,\mathbb{C})$ is isomorphic to $SO(5,\mathbb{C})$

I am studying exceptional isomorphisms recently, which arise due to the coincidence in Dynkin diagram.

I saw two forms of expressing the exceptional isomorphisms, one is isomorphisms between the spin group and the corresponding group, established by spin or half-spin representations; the other is isomorphism from symmetric squares, exterior squares or tensor squares of the relevant group to the special orthogonal group. But in this case, I don't know how to construct the induced symmetric bilinear form on the vector space. And in the case $\Lambda^2 Sp(4,\mathbb{C})\cong SO(5,\mathbb{C})$, I think the dimension of exterior square of a 4-dimensional vector space is 6, and I am not clear how it gets mapped into $SO(5)$.

exterior square of $\operatorname{Sp}(4,\mathbb{C})$ is isomorphic to $\operatorname{SO}(5,\mathbb{C})$

I am studying exceptional isomorphisms recently, which arise due to the coincidence in Dynkin diagram.

I saw two forms of expressing the exceptional isomorphisms, one is isomorphisms between the spin group and the corresponding group, established by spin or half-spin representations; the other is isomorphism from symmetric squares, exterior squares or tensor squares of the relevant group to the special orthogonal group. But in this case, I don't know how to construct the induced symmetric bilinear form on the vector space. And in the case ${\bigwedge}^2\operatorname{Sp}(4,\mathbb{C})\cong \operatorname{SO}(5,\mathbb{C})$, I think the dimension of exterior square of a 4-dimensional vector space is 6, and I am not clear how it gets mapped into $\operatorname{SO}(5)$.

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mhahthhh
mhahthhh
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exterior square of $Sp(4,\mathbb{C})$ is isomorphic to $SO(5,\mathbb{C})$

I am studying exceptional isomorphisms recently, which arise due to the coincidence in Dynkin diagram.

I saw two forms of expressing the exceptional isomorphisms, one is isomorphisms between the spin group and the corresponding group, established by spin or half-spin representations; the other is isomorphism from symmetric squares, exterior squares or tensor squares of the relevant group to the special orthogonal group. But in this case, I don't know how to construct the induced symmetric bilinear form on the vector space. And in the case $\Lambda^2 Sp(4,\mathbb{C})\cong SO(5,\mathbb{C})$, I think the dimension of exterior square of a 4-dimensional vector space is 6, and I am not clear how it gets mapped into $SO(5)$.