The toric variety of the symmetric group

The toric variety T_n associated to the reflecting hyperplanes of the symmetric group S_n is a nonsingular projective variety which occurs in several different contexts. One way to define it is as an iterated blow-up of (n-1)-dimensional projective space along coordinate subspaces. The cohomology of the complex locus T_n(C) is well understood; less so the (rational) cohomology of the real locus T_n(R). I will present some recent results on the representation of S_n on these cohomologies.