Let $X$ be a smooth projective algebraic variety and $D^b(X)$ be the derived category of coherent sheaves on $X$. Denote by $Sym^nX$ the $n$-th symmetric product of $X$. Can we describe the derived category $D^b(Sym^nX)$ in terms of $D^b(X)$. If so, how are they related? Is there any reference? 


This question is intrigued by the question [Hilbert schemes of points and exceptional collections](https://mathoverflow.net/questions/34479/hilbert-schemes-of-points-and-exceptional-collections) asked by Cat.



Derived categories of symmetric products
There is a category closely related to $D^b(Sym^n X)$ which can be described. Namely, the $S\_n$-equivariant derived category of coherent sheaves on $X^n$. This category can be considered as a noncommutative resolution of singularities of $D^b(Sym^n X)$ (the latter category is singular when $\dim X > 1$). The description of the $S\_n$-equivariant derived category I have in mind is the following. First, one can consider the $n$-th tensor power'' of $D^b(X)$ --- if one fixes a DG-enhancement for $D^b(X)$ then it is the derived category of DG-modules over the $n$-th tensor power of the underlying DG-category. The symmetric group acts naturally on this category, so one can consider the corresponding equivariant category. That's it. 


