Is the determinant bundle the pullback of the $\mathcal O(1)$ on $\mathbb P^n$ under the Plücker embedding? Let $V$ be a $n$-dimensional complex vector space and consider the Grassmannian of complex $k$-planes $Gr(k,V)$. The Plücker embedding is an embedding $p:Gr(k,V) \to \mathbb P^M$ where $M = \left(\begin{array}{c} n \\ k \end{array}\right)$, given by
$$ \text{span}\{ u_1,\ldots,u_k\} \mapsto [u_1\wedge u_2 \wedge \cdots \wedge u_k].$$
The Grassmannian comes equipped with a tautological vector bundle $T \to Gr(k,V)$ of rank $k$, where
$$ T = \{ (W,w): W \in Gr(k,V), w \in W\}.$$
To this bundle we thus have the associated determinant bundle $\det T\to Gr(k,V)$ whose fibre over over $W$ is $\wedge^k W$.
Projective space $\mathbb P^M$ comes with its own bundle $\mathcal O(1) \to \mathbb P^M$.

Is it the case that $p^*\mathcal O(1) = \det T$?

I suspect that this is indeed true. My evidence is that $\det T$ is a prequantum line bundle and hence is very ample, admitting a embedding of $Gr(k,V)$ into $\mathbb P^n$ for some $n$. Moreoever, the Chern connection on $T$ is locally $d+\partial\bar\partial \log \det(Z^t Z)$ for a global section $Z$, while the Chern/prequantum connection on $\mathbb P^n$  is $d + \partial \bar\partial \log( |z_i|^2)$, being far too similar for mere coincidence (of course, most Chern connections look like this).
Alternatively, the Plücker embedding acting at $W$ arises by taking determinants of $k\times k$ subminors of the matrix whose columns span $W$, and in some sense these determinants feel as though they are algebraic sections of the determinant bundle, and hence would precisely give the very ample embedding, but I'm not sure how to show this more explicitly. 
 A: Let $\mathbb{V}$ be a $\mathbb{C}$-vector space of dimension $n$ and $G(k,n)$ the Grassmannian of $k$-planes of $\mathbb{V}$; considering the canonical projection:
\begin{equation}
\pi:\bigwedge^k\mathbb{V}-\{\underline0\}\to\mathbb{P}\left(\bigwedge^k\mathbb{V}\right)\cong\mathbb{P}^M_{\mathbb{C}}\,\text{where}\,M=\binom{n}{k}-1.
\end{equation}
Let $\{e_1,\dots,e_n\}$ be a basis of $\mathbb{V}$, then:
\begin{gather}
\forall i\in\{1,\dots,k\},\,v^i=a_i^je_j\in\mathbb{V};\\
v^1\wedge\dots\wedge v^k=a_1^je_j\wedge\dots\wedge a_k^je_j=\sum_{1\leq i_1<\dots<i_k\leq n}\sum_{\sigma\in Sym(k)}sign(\sigma)a^{i_1,\dots,i_k}_{\sigma(i_1),\dots,\sigma(i_1)}e_{i_1}\wedge\dots\wedge e_{i_k}=\\
=\sum_{1\leq i_1<\dots<i_k\leq n}\det\left(A^{i_1,\dots,i_k}_{1,\dots,k}\right)e_{i_1,\dots,i_k}=\sum_{\underline{i}\in I}d^{\underline{i}}e_{\underline{i}}\equiv d^{\underline{i}}e_{\underline{i}}
\end{gather}
where:
\begin{gather}
I=\left\{(i_1,\dots,i_k)\in\{1,\dots,n\}^k\mid1\leq i_1<\dots<i_k\leq n\right\};\\
\forall\underline{i}\in I,\,e_{\underline{i}}=e_{i_1,\dots,i_k}=e_{i_1}\wedge\dots\wedge e_{i_k},\\
d^{\underline{i}}=\det\left(A^{i_1,\dots,i_k}_{1,\dots,k}\right)=\det\left(A^{\underline{i}}\right)=\det\begin{pmatrix}
a_1^{i_1} & a_1^{i_2} & \dots & a_1^{i_k}\\
a_2^{i_1} & a_2^{i_2} & \dots & a_2^{i_k}\\
\vdots & \vdots & \ddots & \vdots\\
a_k^{i_1} & a_k^{i_2} & \dots & a_k^{i_k}
\end{pmatrix}.
\end{gather}
One can show that the charts of $G(k,n)$ are:
\begin{equation}
\forall\underline{i}\in I,\,U_{\underline{i}}=\left\{\left[d^{\underline{i_0}}\right]\in\mathbb{P}^M_{\mathbb{C}}\mid\underline{i_0}\in I,\,d^{\underline{i}}\neq0\right\},
\end{equation}
via the Plücker embedding $p$, and one can identify $[W]=\left[\left\langle v^1,\dots,v^k\right\rangle\right]\in G(k,n)$ on the coordinating open $U_{\underline{i}}$ with the matrix $\left(A^{\underline{i}}\right)^{-1}A$.
Considering the tautological bundle $(T,pr_2)$ of $G(k,n)$, by definition:
\begin{gather}
T=\{(v,[W])\in\mathbb{V}\times G(k,n)\mid v\in W\},\\
pr_2:(v,[W])\in T\to[W]\in G(k,n);
\end{gather}
then:
\begin{gather}
\forall\underline{i}\in I,\,T_{|U_{\underline{i}}}=\{(v,[W])\in\mathbb{V}\times U_{\underline{i}}\mid v\in W\}\cong\mathbb{C}^k\times U_{\underline{i}},\\
\forall\underline{i},\underline{j}\in I,\,g_{\underline{ij}}:A^{\underline{i}}\in U_{\underline{i}}\cap U_{\underline{j}}=U_{\underline{ij}}\to\left(A^{\underline{j}}\right)^{-1}A^{\underline{i}}\in\mathrm{GL}(k,\mathbb{C})\,\text{where:}\,U_{\underline{ij}}\neq\emptyset
\end{gather}
are the transition functions of $T$; therefore $(\det T,pr_2)$ is the line bundle on $G(k,n)$ such that $\{U_{\underline{i}}\}_{\underline{i}\in I}$ is a trivialization for $\det T$ and the transition functions are:
\begin{equation}
\forall\underline{i},\underline{j}\in I,\,\det\circ g_{\underline{ij}}:A^{\underline{i}}\in U_{\underline{i}}\cap U_{\underline{j}}=U_{\underline{ij}}\to\det\left(\left(A^{\underline{j}}\right)^{-1}A^{\underline{i}}\right)\in\mathrm{GL}(1,\mathbb{C})=\mathbb{C}^{\times}
\end{equation}
Obviously, when $k=1$ and $n=M$, $(T,pr_2)$ is the universal bundle $\left(\mathcal{O}_{\mathbb{P}^M_{\mathbb{C}}}(-1),pr_2\right)=(\mathcal{O}(-1),pr_2)$ on $\mathbb{P}^M_{\mathbb{C}}$; defined:
\begin{equation}
\forall\underline{i}\in I,\,V_{\underline{i}}=\left\{\left[x^{\underline{i_0}}\right]\in\mathbb{P}^M_{\mathbb{C}}\mid\underline{i_0}\in I,\,x^{\underline{i}}\neq0\right\}
\end{equation}
one has:
\begin{equation}
\forall\underline{i}\in I,\,p^{-1}\left(V_{\underline{i}}\right)=U_{\underline{i}},
\end{equation}
called $h_{\underline{ij}}$'s the transition functions of $\mathcal{O}(-1)$, one prove that:
\begin{equation}
\forall\underline{i},\underline{j}\in I,\,h_{\underline{ij}}\circ p=\det\circ g_{\underline{ij}};
\end{equation}
that is: the $\det\circ g_{\underline{ij}}$'s are the transition functions of $p^{*}\mathcal{O}(-1)$, the $U_{\underline{i}}$'s trivialize $p^{*}\mathcal{O}(-1)$, then $\det T$ is isomorphic (as line bundle on $G(k,n)$) to $p^{*}\mathcal{O}(-1)$.
I'll take time about the Chern connection on $\det T$.
