Spectra of subrings of cohomology generated by characteristic classes for fusion systems

If F is a saturated fusion system on a finite p-group S, we define the Chern subring Ch(F) of F to be the subring of the mod-p cohomology H^*(S) of S generated by the Chern classes of F-stable representations of S. We show that Ch(F) is contained in H^*(F) and apply a result of Green and the first author to describe its spectrum in terms of a certain category of elementary abelian subgroups of S. We obtain similar results for various related subrings, including those generated by characteristic classes of F-stable S-sets.