Let $X$ be a smooth projective variety, and $Quot$ some quot scheme, i.e. it parametrizes quotients of some fixed sheaf $F$ on $X$ of some fixed Hilbert polynomial. There is a universal quotient sheaf $\mathcal{Q}$ on $X \times Quot$ such that $\mathcal{Q}|_{X \times [E]} = E$. Because $X$ is smooth and $F$ is flat over $Quot$, there is a finite resolution of the universal quotient $\mathcal{Q}$ by finite dimensional locally free sheaves on $X \times Quot$.

Recall that by construction, $Quot$ enjoys a Plucker embedding into a Grassmannian of sections of $F(n)$ for some large $n$.

(How) can the resolution of $\mathcal{Q}$ be described explicitly in terms of restrictions of tautological bundles on the Grassmannian?