$\newcommand\Alt{\bigwedge\nolimits}$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!



To describe an invariant trivector in dimension 8 geometrically
Here's another very nice (but still algebraic) interpretation that explains some of the geometry: Recall that $\operatorname{SL}(2,\mathbb{C})$ has a $2$-to-$1$ representation into $\operatorname{SL}(3,\mathbb{C})$ so that the Lie algebra splits as
$$
{\frak{sl}}(3,\mathbb{C}) = {\frak{sl}}(2,\mathbb{C})\oplus {\frak{m}}
$$
where ${\frak{m}}$ is the ($5$-dimensional) orthogonal complement of ${\frak{sl}}(2,\mathbb{C})$ using the Killing form of ${\frak{sl}}(3,\mathbb{C})$. Note that ${\frak{m}}$ is an irreducible ${\frak{sl}}(2,\mathbb{C})$-module, and that every element $x\in {\frak{sl}}(3,\mathbb{C})$ can be written uniquely as $x = x\_0 + x\_1$ with $x\_0\in {\frak{sl}}(2,\mathbb{C})$ and $x\_1\in{\frak{m}}$. Note also that $[{\frak{m}},{\frak{m}}]= {\frak{sl}}(2,\mathbb{C})$.


This defines the desired pairing ${\frak{sl}}(2,\mathbb{C})\times \bigwedge\nolimits^2({\frak{m}})\to\mathbb{C}$: Send $(x\_0,y\_1,z\_1)$ to $\operatorname{tr}(x\_0[y\_1,z\_1])$. Of course, this makes the $\operatorname{SL}(2,\mathbb{C})$-invariance of the pairing obvious.


