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The Beauville-Voisin conjecture for a hyperk\"ahler manifold X states thatthe subring of the Chow ring A^*(X) generated by divisor classes and Cherncharacters of the tangent bundle injects into the cohomology ring of X. Weprove a weak version of this conjecture when X is the Hilbert scheme of pointson a K3 surface, for the subring generated by divisor classes and tautologicalclasses. This in particular implies the weak splitting conjecture of Beauvillefor these geometries. In the process, we extend Lehn's formula and theLi-Qin-Wang W_{1+infinity} algebra action from cohomology to Chow groups, forthe Hilbert scheme of an arbitrary smooth projective surface S

Lehn's formula in Chow and Conjectures of Beauville and Voisin