The [q-Vandermonde identity](https://en.wikipedia.org/wiki/Q-Vandermonde_identity) reads: 


$$ \binom{m + n}{k}\_{\!\!q} =\sum\_{j} \binom{m}{k - j}\_{\!\!q} \binom{n}{j}\_{\!\!q} q^{j(m-k+j)} $$


The [q-binomial coefficients](https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient): 
$$ \binom{ a }{ b}\_{\!\!q} $$ 


are known to count the number of points over the Grassmannians $Gr(a,b)$ over $F\_q$, and $q^{j(m-k+j)}$ is the number of points over $A^{j(m-k+j)}$ over $F\_q$.


So the identity above states that number of points of $Gr(k, m+n)$ can be written as the sum of numbers of points of smaller Grassmannians and $A^l$.
That would be naturally explained if there would be some geometric relation between these manifolds, like fiber bundles.


**Question 1** What could be the geometric interpretation of the identity above, if any? If yes, what about other fields, e.g. $C$? 


**Question 2** Is the identity above true on the level of motives? 


**Question 2b** If yes, then is it true for arbitrary fields?




---


For the case of q=1 - the hypothetical ["field with one element"](https://en.wikipedia.org/wiki/Field_with_one_element),
everything is "Okay": the $Gr(a,b,F\_1)$ are just the sets of all a-combinations out of b and so the usual Vandermonde identity implies that $Gr(m+n,k)$ can be factorized in terms of smaller Grassmannians. That is, however, a kind of cheating, since over $F\_1$, geometry disappears and counting is enough to give a "geometric" identity.



Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?
Assume $V$ is a vector space of dimension $m+n$, $M \subset V$ is a subspace of dimension $m$, and $N = V/M$. Let $p:V \to N$ be the projection. Consider the Grassmannian $X = Gr(k,V)$ and its stratification by the dimension of intersection with $M$, i.e., set
$$
X\_j = \{ U \in Gr(k,V) | \dim(U \cap M) = k - j \}.
$$
If $U \in X\_k$ then the projection of $U$ to $N$ has dimension $j$. Thus, we have a natural map
$$
\pi \colon X\_j \to Gr(k-j,M) \times Gr(j,N),\qquad U \mapsto (U \cap M,p(U)).
$$
Finally, for any $U\_M \in Gr(k-j,M)$ and $U\_N \in Gr(j,N)$ we have
$$
\pi^{-1}(U\_M,U\_N) \cong \{U' \in Gr(j,p^{-1}(U\_N)/U\_M) | U' \cap (U/U\_M) = 0\}.
$$ 
This is an open Schubert cell, hence is isomorphic to $\mathbb{A}^{j(m-k-j)}$. Moreover, the map $\pi$ is locally trivial, hence 
$$
[X\_j] = [Gr(k-j,m)][Gr(j,n)][\mathbb{A}^{j(m-k-j)}]
$$
in the Grothendieck ring of varieties. Summing up over $j$, we obtain
$$
[Gr(k,m+n)] = \sum\_j [Gr(k-j,m)][Gr(j,n)][\mathbb{A}^{j(m-k-j)}],
$$
a motivic version of the formula..


