As to your question about the linear vs. set-theoretic interpretation of G-reps i cannot remember a dialogue between 2 persons. But, there’s a fun lecture 1 by John Baez starting with this. Here’s the streaming video.]]>

I’ll be certainly interested on reading the whole story, so if you find time please write it!

]]>at the danger of spoiling some of the fun, here’s the key argument. what is a vector-space over F_un? a set. what is a n-dml grouprepresentation over F_un : a group-morphism G –> S(n) so a G-action on the set. so what are simple F_un G-representations? transitive G-actions or if you want G-sets G/H where H is a subgroup of finite index in G.

in particular, if you start with the modular group PSL(2,Z), the simple representations are precisely Grothendieck’s dessins d’enfants…

in fact, i claim that Grothendieck’s anabelian geometry is an arche-typical example of a noncommutative F_un scheme, the obvious noncommutative extension of C-C is to consider functors from groups –> sets from all finite groups (not necessarely Abelian) to (graded) sets.

If anyone is still interested in this, i’ll write a proper post on it. if not,not.]]>