The multiplicative structure of the K-theoretical McKay correspondence for the Hilbert scheme of points in the complex plane

We consider the K-theory of the Hilbert scheme of points in the complex plane, which under McKay correspondence is isomorphic to the space of symmetric functions ${\mathrm{\Lambda }}^n$. We prove a formula conjectured by Boissière for the endomorphism of ${\mathrm{\Lambda }}^n$ induced by multiplication by the classes of the Adams powers of the tautological bundle. We describe the structure constants for the multiplication on ${\mathrm{\Lambda }}^n$ induced by the tensor product in K-theory.