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 $\mathbb{C}\mathbb{P}^n.$
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$\begingroup$ please give more context. What is $\tilde{K}$? $\endgroup$– Dima PasechnikCommented Apr 1, 2015 at 9:03
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$\begingroup$ @ Dima Pasenchnik, I thought these are commonly used notations, that's why I didn't give the meanings of them. $\endgroup$– PrateepCommented Apr 1, 2015 at 9:20
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$\begingroup$ Complex conjugation is the Adams operation $\Psi^{-1}$. This is determined for complex projective space in Theorem 7.2 of Adams' paper "Vector fields on spheres", Ann of Math 75 (1962), 603-632. $\endgroup$– Matthias WendtCommented Apr 1, 2015 at 10:28
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$\begingroup$ The conjugate of $H$ is $H^{-1}$. The conjugate of $\eta$ follows from that: it is $(1+\eta)^{-1}-1$. $\endgroup$– Neil StricklandCommented Apr 1, 2015 at 11:38
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$\begingroup$ @Neil Strickland, I missed that point. Thank you. $\endgroup$– PrateepCommented Apr 1, 2015 at 11:58
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