Being a physicist, I think it'd be cool to have Coxeter plane projections of the root systems of the symmetry groups associated with the fundamental forces hanging on my walls (example for E8: http://en.wikipedia.org/wiki/Coxeter_element#mediaviewer/File:E8Petrie.svg.) For the non-physicists in the room:
Electromagnetic -> U(1)
Weak -> SU(2)
Strong -> SU(3)
Gravity -> Diffeomorphisms of Minkowski Space, usually denoted Diff(M)
The first 3 seem simple enough, but what about gravity? I asked a related question a few months ago, and got a few answers which may be helpful: https://physics.stackexchange.com/questions/138544/generators-of-the-diffeomorphism-group. I'm not clear on the relationship between generators and roots. According to TwoBs' answer in that link, Diff(M) has infinitely many generators, so would Diff(M) have infinitely many roots as well?
My guess would be, then, that the Coxeter plane projection of Diff(M) would be something like a projection of a 4-dimentional grid of points (since Minkowski space is 4D) onto a 2-dimentional plane? Is that right or wrong? Not even wrong?
Anyone know? Help!