Given a representation V of a finite group G we can associate a dimension function that to each subgroup H of G assigns the dimension of the fixed point space VH. The dimension functions are "super class functions" that are constant on the conjugacy classes of subgroups in G. For a p-group the list of Borel-Smith conditions characterizes the super class functions that come from real representations.

In a joint paper with Ergün Yalcin we study the representation rings for saturated fusion systems using characteristic bisets and idempotents, and we give a list of Borel-Smith conditions that characterize the dimension functions of the real representations that are stable under the fusion system. Finally, we give applications of these results to the problem of constructing homotopy G-spheres for a finite group G with prime power isotropy and (a multiple of) a predetermined dimension function.