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$\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$. For an integer $p\ge 0$, write $R_p=S^p R$; then $R_1=R$ and $\dim R_p=p+1$.

Using Table 5 in the book of Onishchik and Vinberg, I computed that the representation $$ R_2\otimes\Alt^2 R_4 $$ contains the trivial representation with multiplicity one. I used the table as a black box.

Question. Let $V\subset R_2\otimes\Alt^2 R_4$ denote the corresponding one-dimensional subspace. How can one describe $V$ as a subspace geometrically?

Motivation: I want to consider a $\operatorname{PGL}(2,k)$-fixed trivector $$v\in V\subset R_2\otimes\Alt^2 R_4\subset \Alt^3(R_2\oplus R_4)$$ of the 8-dimensional vector space $W=R_2\oplus R_4$ over a field $k$, and then to twist all this using a Galois-cocycle of $\operatorname{PGL}(2,k)$. For this end I need a geometric description of $V$.

Feel free to add/edit tags!

$\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$. For an integer $p\ge 0$, write $R_p=S^p R$; then $R_1=R$ and $\dim R_p=p+1$.

Using Table 5 in the book of Onishchik and Vinberg, I computed that the representation $$ R_2\otimes\Alt^2 R_4 $$ contains the trivial representation with multiplicity one. I used the table as a black box.

Question. Let $V\subset R_2\otimes\Alt^2 R_4$ denote the corresponding one-dimensional subspace. How can one describe $V$ as a subspace geometrically?

Motivation: I want to consider a $\operatorname{PGL}(2,k)$-fixed trivector $$v\in V\subset R_2\otimes\Alt^2 R_4\subset \Alt^3(R_2\oplus R_4)$$ of the 8-dimensional vector space $W=R_2\oplus R_4$ over a field $k$ of characteristic 0, and then to twist all this using a Galois-cocycle of $\operatorname{PGL}(2,k)$. For this end I need a geometric description of $V$.

Feel free to add/edit tags!

Let $G={\rm SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$. For an integer $p\ge 0$, write $R_p=S^p R$; then $R_1=R$ and ${\rm dim}\, R_p=p+1$.

Using Table 5 in the book of Onishchik and Vinberg, I computed that the representation $$ R_2\otimes\Lambda^2 R_4 $$ contains the trivial representation with multiplicity one. I used the table as a black box.

Question. Let $V\subset R_2\otimes\Lambda^2 R_4$ denote the corresponding one-dimensional subspace. How can one describe $V$ as a subspace geometrically?

Motivation: I want to consider a ${\rm PGL}(2,k)$-fixed trivector $$v\in V\subset R_2\otimes\Lambda^2 R_4\subset \Lambda^3(R_2\oplus R_4)$$ of the 8-dimensional vector space $W=R_2\oplus R_4$ over a field $k$, and then to twist all this using a Galois-cocycle of ${\rm PGL}(2,k)$. For this end I need a geometric description of $V$.

Feel free to add/edit tags!

$\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$. For an integer $p\ge 0$, write $R_p=S^p R$; then $R_1=R$ and $\dim R_p=p+1$.

Using Table 5 in the book of Onishchik and Vinberg, I computed that the representation $$ R_2\otimes\Alt^2 R_4 $$ contains the trivial representation with multiplicity one. I used the table as a black box.

Question. Let $V\subset R_2\otimes\Alt^2 R_4$ denote the corresponding one-dimensional subspace. How can one describe $V$ as a subspace geometrically?

Motivation: I want to consider a $\operatorname{PGL}(2,k)$-fixed trivector $$v\in V\subset R_2\otimes\Alt^2 R_4\subset \Alt^3(R_2\oplus R_4)$$ of the 8-dimensional vector space $W=R_2\oplus R_4$ over a field $k$, and then to twist all this using a Galois-cocycle of $\operatorname{PGL}(2,k)$. For this end I need a geometric description of $V$.

Feel free to add/edit tags!