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Representation rings for fusion systems and dimension functions

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.