On Positivity for the Peterson Variety

TL;DR

This paper explores a specific form of geometric positivity in the cohomology rings of flag varieties and Peterson varieties, demonstrating positivity phenomena similar to Schubert calculus and discussing potential extensions to other Hessenberg varieties.

Contribution

It extends known positivity results from Schubert calculus to Peterson varieties and discusses open questions for broader Hessenberg varieties.

Findings

Positivity in equivariant cohomology of Peterson varieties

Extension of positivity phenomena to certain Hessenberg varieties

Open questions on generalizing positivity results

Abstract

We aim in this manuscript to describe a specific notion of geometric positivity that manifests in cohomology rings associated to the flag variety $G/B$ and, in some cases, to subvarieties of $G/B$. We offer an exposition on the the well-known geometric basis of the homology of $G/B$ provided by Schubert varieties, whose dual basis in cohomology has nonnegative structure constants. In recent work [22] we showed that the equivariant cohomology of Peterson varieties satisfies a positivity phenomenon similar to that for Schubert calculus for $G/B$. Here we explain how this positivity extends to this particular nilpotent Hessenberg variety, and offer some open questions about the ingredients for extending positivity results to other Hessenberg varieties.