Consider vector spaces $V$ and $W$ of dimensions $m$ and $n\geq m$, respectively. Both the special linear group $\operatorname{SL}(V)$ as well as the general linear group $\operatorname{GL}(V)$ act smoothly and freely from the right on the smooth manifold of monomorphism $\operatorname{Mono}(V;W)$ by precomposition. Denote the space of totally decomposable $m$-vectors in $W$ by $\bigwedge^m_sW$.
Fix a basis $e_1,\ldots,e_m$ of $V$ and define the surjective map $$\pi\colon \operatorname{Mono}(V;W)\to \bigwedge\nolimits^m_sW\setminus\{0\}, A\mapsto Ae_1 \wedge \cdots \wedge Ae_m.$$ I can show that the map $\Phi\colon \operatorname{Mono}(V;W)/\operatorname{SL}(V) \to \bigwedge^m_sW\setminus\{0\}$ defined by $\Phi([A]_{\operatorname{SL}}) = \pi(A)$ is well-defined and bijective. Analoguously, I can show that $\Psi\colon \operatorname{Mono}(V;W)/\operatorname{GL}(V) \to \bigwedge^m_sW\setminus\{0\}$ defined by $\Phi([A]_{\operatorname{GL}}) = \{t \pi(A)\colon t\in\mathbb{R}^{\times}\}$ is well-defined and bijective. Hence the following diagram commutes:
My main issue is whether $\Phi$ and $\Psi$ are diffeomorphisms. To even start to show this each space should be a smooth manifold. I know that by the Plücker embedding $\mathbb{P}(\bigwedge^m_sW\setminus\{0\})$ is the Grassmannian and thus a compact smooth manifold of dimension $m(n-m)$. The quotient on the bottom right also should have dimension $nm-m^2=m(n-m)$ but I'm unsure how to show that it's a smooth manifold. If $\operatorname{GL}$ acted properly then the quotient manifold theorem would be my friend. Is this the case?
The quotient on the middle right should have dimension $nm-(m^2-1)=m(n-m)+1$. The totally decomposable $m$-vectors are a projective variety of "non-projective" dimension $m(n-m)+1$. So these arguments at least indicate to me, thankfully, that my bijectivity proofs are not bogus. I'm unsure if the spaces on the middle level are actually manifolds. On this level, also, the map $\pi$ does depend on the chosen basis whereas (an analogous map) on the bottom level does not. Maybe another indication that the bottom-level compactification makes things "better"?
What "regularity" of $\Phi$ could I even expect? I want to show that the totally decomposable vectors are isomorphic to the middle right quotient in an appropriate category.
Let $U$ be the tautological (rank-$m$) vector bundle over the Grassmannian $\mathrm{Gr}(m,W)$ and let $$ P := \mathrm{Iso}(V,U) $$ be the corresponding principal $\mathrm{GL}(V)$-bundle over $\mathrm{Gr}(m,W)$. Then $$ P \cong \mathrm{Mono}(V;W). $$ Indeed, the map from left to right is given by taking an isomorphism $V \to U$ to the composition $V \to U \to W$, where the second arrow is the tautological embedding, and it is easy to see this map is an isomorphism and, moreover, it is $\mathrm{GL}(V)$-equivariant. Taking quotients by $\mathrm{GL}(V)$ and $\mathrm{SL}(V)$, you can obtain the two isomorphism you are interested in.