Affine pavings of regular nilpotent Hessenberg varieties and intersection theory of the Peterson variety

Abstract

This paper describes a paving by affines for regular nilpotent Hessenberg varieties in all Lie types, namely a kind of cell decomposition that can be used to compute homology despite its weak closure conditions. Precup recently proved a stronger result; we include ours because we use different methods. We then use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explicitly identify some of the coefficients of Schubert classes in this expansion, which is a problem of independent interest in Schubert calculus.

36 pages, 3 figures

References in corpus (1)

• Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

Cited by in corpus (4)

• The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: a research announcement
• The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\dots ,n)$
• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
• Affine Schubert calculus and double coinvariants