Sune Precht Reeh

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 $V^H$. 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 project with Ergün Yalcin we show that while we cannot lift Borel-Smith functions to real representations for a general group $G$, we can lift a multiple of any Borel-Smith function to an action of $G$ on a finite homotopy sphere (which would be the unit sphere if we had a representation).
To prove this we localize at each prime p and study dimension functions for saturated fusion systems. That is, we give a list of Borel-Smith conditions for a fusion system that characterize the dimension functions of the fusion stable real representations. The proof for fusion systems involves biset functors for saturated fusion systems.