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. |