This paper describes a paving by affines for regular nilpotent Hessenbergvarieties in all Lie types, namely a kind of cell decomposition that can beused to compute homology despite its weak closure conditions. Precup recentlyproved a stronger result; we include ours because we use different methods. Wethen use this paving to prove that the homology of the Peterson variety injectsinto the homology of the full flag variety. The proof uses intersection theoryand expands the class of the Peterson variety in the homology of the flagvariety in terms of the basis of Schubert classes. We explicitly identify someof the coefficients of Schubert classes in this expansion, which is a problemof independent interest in Schubert calculus.
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