The filtration on the infinite symmetric product of spheres by number of
factors provides a sequence of spectra between the sphere spectrum and
the integral Eilenberg-Mac Lane spectrum. This filtration has received a
lot of attention and the subquotients are interesting stable homotopy
types.
In this talk I will discuss the equivariant stable homotopy types, for
finite groups, obtained from this filtration for the infinite symmetric
product of representation spheres. The filtration is more complicated
than in the non-equivariant case, and already on the zeroth homotopy
groups an interesting filtration of the augmentation ideal of the Burnside
rings arises. Our method is by `global' homotopy theory, i.e., we study
the simultaneous behaviour for all finite groups at once. In this context,
the equivariant subquotients are no longer rationally trivial, nor even
concentrated in dimension 0.