Is the conjugate operation on $\overset{\sim}{K}(\mathbb{C}\mathbb{P}^n)$ known? If so, can I get the full formula at least in terms of the basis $\eta^i$? Here $\overset{\sim}{K}(X)$ denotes the reduced stable isomorphic classes of complex vector bundles over X. And conjugation operation on $\overset{\sim}{K}(x)$ is induced from the conjugate operation on complex vector bundles. And $\eta=H-1$ where $H$ is the canonical complex line bundle over $\mathbb{C}\mathbb{P}^n.$