The canonical class and Chern character operators on Hilbert schemes of points

Let $X$ be a smooth projective complex surface. We obtain explicit formulas for the Chern character operators on the Hilbert schemes of points $X^{[n]}$ in terms of Nakajima operators, determining in particular the coefficients of the terms involving the canonical class $K_X$ and its square. Consequently, the cohomology ring structure of the Hilbert scheme is determined. As an application, we give an alternative proof of the action of the affine Yangian $Y_{t_1,t_2}({\widehat{\mathfrak{g}\mathfrak{l}}}_1)$ on the cohomological Fock space ${\mathbb{H}}_X={⨁}_{n\ge 0}{H}^{\ast }(X^{[n]},\mathbb{Q})$.