Higher Goodwillie approximations of symmetric products

Determine the higher Goodwillie approximations of the symmetric product functor Sp^m.

Background

The paper studies the first derivatives of the symmetric product functors, which produce the symmetric product spectra. Their filtration and quotients are closely related to the Goodwillie tower of the identity. The higher Goodwillie approximations of Spm itself are not described.

References

What do the higher Goodwillie approximations of $Spm$ look like?

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 2, subsection “Symmetric products”