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 $Ch\left(F\right)$ of $F$ to be the subring of $H^{\ast }(S;F_p)$ generated by Chern classes of $F$-stable representations of $S$. We show that $Ch\left(F\right)$ is contained in $H^{\ast }(F;F_p)$ and apply a result of Green and the first author to describe its maximal ideal spectrum in terms of a certain category of elementary abelian subgroups. We obtain similar results for various related subrings, including those generated by characteristic classes of $F$-stable $S$-sets.