For a moduli space M of sheaves on a K3 surface, we propose a series of conjectural identities in the Chow ring of self-products of M which generalize the classic Beauville-Voisin identity for K3 surfaces. We then verify these identities in the case that M is a Hilbert scheme of points on a K3 surface. This is joint work with I. Barros, A. Marian, and R. Silversmith.

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