  This paper defines and studies permutation representations on the equivariant
cohomology of Schubert varieties, as representations both over C and over
C[t_1, t_2,...,t_n]. We show these group actions are the same as an action of
simple transpositions studied geometrically by M. Brion, and give topological
meaning to the divided difference operators studied by
Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these
representations using the combinatorial approach to equivariant cohomology
introduced by Goresky-Kottwitz-MacPherson. We find that each permutation
representation on equivariant cohomology produces a representation on ordinary
cohomology that is trivial, though the equivariant representation is not.
