Complex Projective Spaces in Rational $SO(2)$-Equivariant Stable Homotopy Theory

We show that for almost every finite dimensional complex representation $W$ of $SO(2)$ the associated complex projective space $\mathbb{C}P(W{)}_{+}$ does not have the rational stable homotopy type of a specific wedge of representation spheres constructed from a chosen filtration of $\mathbb{C}P(W{)}_{+}$. This contrasts with the case for representations of finite Abelian groups where such an equivalence does hold. In the finite group case, the splitting result provides us with a classification result for projective spaces up to rational stable equivalence. In light of the non-splitting as a particular wedge of spheres in the case of projective spaces of $SO(2)$-representations, we also prove a separate classification theorem.