Perverse filtration on Hilbert schemes via upward flow

We explicitly compute the perverse Leray filtration on the top cohomology of the Hilbert scheme of points on $\mathrm{\Sigma }\times \mathbb{C}$, for any connected smooth projective curve $\mathrm{\Sigma }$. The computation is carried out in the natural basis given by the ${\mathbb{C}}^{\ast }$-upward-flow cycles. The result is described by a simple symmetric-function dictionary: upward-flow classes correspond to products of complete homogeneous symmetric functions, while the perverse-homogeneous basis corresponds to products of power-sum symmetric functions. This gives an explicit triangular change-of-basis between the two bases.