  For any field $k$, we give an algebraic description of the category
$\mathrm{Perv}_\mathscr{S}(S^n (\mathbb{C}^2),k)$ of perverse sheaves on the
$n$-fold symmetric product of the plane $S^n(\mathbb{C}^2)$ constructible with
respect to its natural stratification and with coefficients in $k$. In
particular, we show that it is equivalent to the category of modules over a new
algebra that is closely related to the Schur algebra. As part of our
description we obtain an analogue of modular Springer theory for the Hilbert
scheme $\mathrm{Hilb}^n(\mathbb{C}^2)$ of $n$ points in the plane with its
Hilbert-Chow morphism.
