A multiplicative version of the tom Dieck splitting

While the classical tom Dieck splitting in equivariant stable homotopy theory is typically regarded as a formula for suspension spectra in the genuine equivariant stable category, it can be interpreted as a calculation of the fixed points of $G$-spectra that are derived pushforwards from the naive equivariant stable category. We then establish a corresponding multiplicative splitting formula for derived pushforwards of $N_{\infty}$ ring spectra. Just as the usual tom Dieck splitting characterizes the equivariant stable category associated to an $N_{\infty}$ operad $\mathcal{N}$, the multiplicative tom Dieck splitting characterizes the $G$-symmetric monoidal structure on the genuine equivariant stable category associated to $\mathcal{N}$.