Let X be a smooth projective connected curve over $\mathbb{C}$ and let $n>1$ be an integer. Let $Y= Sym^n\_X$ be the $n$-th symmetric product of $X$.



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If not, what part of the Hodge diamond can be described easily?



Hodge decomposition of the symmetric product of a curve
Look at Example $1.1$ in [this paper](https://www.math.wisc.edu/~maxim/sym_lib60.pdf) for a nice formula. 


You can also compute them by hands (and, hopefully, prove the desired formula) by identifying $\mathrm{H}^{p,q}(\operatorname{Sym}^n X)$ with $S\_n$-invariant part of $\mathrm{H}^{p,q}(X^n)$ (use the Kunneth's formula to compute the latter group).


