Equivariant Chevalley, Giambelli, and Monk Formulae for the Peterson Variety

TL;DR

This paper develops explicit formulas for intersection theory on Peterson varieties, including Giambelli, Chevalley, and Monk formulas, advancing understanding of their equivariant cohomology and intersection multiplicities.

Contribution

It introduces new equivariant formulas for Peterson varieties, including a type-independent Giambelli formula and an equivariant Chevalley formula, with applications to intersection multiplicities.

Findings

Derived a polynomial formula for equivariant fundamental classes

Established a type-independent Giambelli formula for Peterson varieties

Developed an equivariant Chevalley formula and dual Monk rule

Abstract

We present a formula for the Poincar\'e dual in the flag manifold of the equivariant fundamental class of any regular nilpotent or regular semisimple Hessenberg variety as a polynomial in terms of certain Chern classes. We then develop a type-independent proof of the Giambelli formula for the Peterson variety, and use this formula to compute the intersection multiplicity of a Peterson variety with an opposite Schubert variety corresponding to a Coxeter word. Finally, we develop an equivariant Chevalley formula for the cap product of a divisor class with a fundamental class, and a dual Monk rule, for the Peterson variety.