Monk's rule and Giambelli's formula for Peterson varieties of all Lie types

A Peterson variety is a subvariety of the flag variety which appears in the construction of the quantum cohomology of partial flag varieties. Each Peterson variety has a one-dimensional torus acting on it. We give a basis of Peterson Schubert classes for and identify the ring generators. In type , Harada-Tymoczko gave a positive Monk formula (arXiv:0908.3517, 2009), and Bayegan-Harada gave Giambelli's formula (arXiv:1012.4053, 2010) for multiplication in the cohomology ring. This paper gives Monk's rule and Giambelli's formula for all Lie types.