  The filtration on the infinite symmetric product of spheres by the 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. While the
symmetric product filtration has been a major focus of research since the
1980s, essentially nothing was known when one adds group actions into the
picture.
  We investigate the equivariant stable homotopy types, for compact Lie groups,
obtained from this filtration of infinite symmetric products of representation
spheres. The situation differs from the non-equivariant case, for example the
subquotients of the filtration are no longer rationally trivial and on the
zeroth equivariant homotopy groups an interesting filtration of the
augmentation ideals of the Burnside rings arises. Our method is by global
homotopy theory, i.e., we study the simultaneous behavior for all compact Lie
groups at once.
