The 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: $$ \binom{ a }{ b}_{\!\!q} $$

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", 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.