The group $SL_2$ (say, $SL_2(\mathbb{C})$, but we could take $SL_2$ of anything else, or probably regard $SL_2$ as a group scheme) acts on binary cubic forms. (Or binary $n$-ic forms in general.) What is the correct form of the action?
Let me give two different answers to this, both of which I have seen in the literature.
(1) Regard binary cubic forms as homomorphisms $\mathbb{A}^2 \rightarrow \mathbb{A}$, where $\mathbb{A}$ is affine space (i.e. just $\mathbb{C}$). We can define an action of $SL_2$ on $\mathbb{A}^2$ as follows: Write $g \in SL_2$ as a matrix, $v \in \mathbb{A}^2$ as a column vector, and then the action is just given by $g v$ (usual matrix multiplication). The action on $\mathbb{A}^1$ is trivial. With these conventions, for a binary cubic form $f$, write $(gf)(v) = f(g^{-1} v)$. This seems to be the representation-theoretic point of view; at any rate it is consistent with p. 4 of Fulton-Harris and this definition is given on p. 14 of Olver's Classical Invariant Theory.
(2) Use the definition $(gf)(v) = f(v g)$, where this time we write $v$ as a row vector, so that $v g$ is well-defined. This definition is quite common, appearing for example in Bhargava, Shankar, and Tsimerman's paper here among many other places (including papers I've co-authored!)
These definitions are equivalent (neither is "wrong") but they're not the same. Definition (1) feels "right" to me, where it seems that the point of (2) is to identify binary cubic forms with $\mathbb{A}^4$ instead of its dual. We would like to write $(gf)(v) = f(g(v))$, but this doesn't work for reasons that are more or less explained in (1).
A couple questions: First of all, is my explanation above accurate?
If it is, is there a good highbrow explanation of (2)? Somehow it feels like a hacky workaround to me. I think the notation (2) is much more natural than (1) in the paper I linked to (I am not trying to argue with their choice of notation) but it feels like cheating a bit, and it is confusing that the same group action is defined in different ways in the literature.
Is there a better perspective than the one I have offered, or do I just need to grin and bear it?
Thank you!