Monk's Rule and Giambelli's Formula for Peterson Varieties of All Lie Types

A Peterson variety is a subvariety of the flag variety $G/B$ which appears in the construction of the quantum cohomology of partial flag varieties. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type $A$ Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives Monk's rule and Giambelli's formula for all Lie types.