  Peterson varieties are a special class of Hessenberg varieties that have been
extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with
the quantum cohomology of the flag variety. In this manuscript, we develop a
generalized Schubert calculus, and in particular a positive Chevalley-Monk
formula, for the ordinary and Borel-equivariant cohomology of the Peterson
variety $Y$ in type $A_{n-1}$, with respect to a natural $S^1$-action arising
from the standard action of the maximal torus on flag varieties. As far as we
know, this is the first example of positive Schubert calculus beyond the realm
of Kac-Moody flag varieties $G/P$.
  Our main results are as follows. First, we identify a computationally
convenient basis of $H^*_{S^1}(Y)$, which we call the basis of Peterson
Schubert classes. Second, we derive a manifestly positive, integral
Chevalley-Monk formula for the product of a cohomology-degree-2 Peterson
Schubert class with an arbitrary Peterson Schubert class. Both $H^*_{S^1}(Y)$
and $H^*(Y)$ are generated in degree 2. Finally, by using our Chevalley-Monk
formula we give explicit descriptions (via generators and relations) of both
the $S^1$-equivariant cohomology ring $H^*_{S^1}(Y)$ and the ordinary
cohomology ring $H^*(Y)$ of the type $A_{n-1}$ Peterson variety. Our methods
are both directly from and inspired by those of GKM
(Goresky-Kottwitz-MacPherson) theory and classical Schubert calculus. We
discuss several open questions and directions for future work.
