Have there been any attempts to extend the "F_un" analogy to the representation theory of finite groups?
If I might be allowed some speculation:
If combinatorics can be regarded as analagous to linear algebra over the "field with one element", then are characters of finite groups over the "field with one element" simply permutation characters? Perhaps the analogous notion of irreducibility in this case would be transitivity?
In the complex character theory of finite groups, two irreducible characters are identical if and only if representations that afford them are equivalent. Perhaps by developing a representation theory of finite groups over the field with one element, we would find some analogous way of comparing "F_un characters" to check if two permutation representations of a given group are equivalent.
