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

Rebecca Goldin, Rahul Singh

Abstract

We present a formula for the Poincaré 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.

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

Abstract

We present a formula for the Poincaré 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.
Paper Structure (20 sections, 26 theorems, 93 equations, 3 tables)

This paper contains 20 sections, 26 theorems, 93 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Equivariant (Co)Homology
  3. The integration pairing
  4. Spaces with affine paving
  5. Chern classes and Euler classes
  6. Flag Manifolds
  7. Flag manifolds and Schubert varieties
  8. Line bundles on the flag manifold
  9. Stability of Dynkin diagrams
  10. The height function
  11. Hessenberg varieties and Poincaré duals
  12. Hessenberg Varieties
  13. The Peterson variety
  14. The Equivariant Giambelli formula and intersection multiplicities
  15. Cohomology of regular Hessenberg varieties
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose $H_0\subset H$. We have the following equalities in the equivariant homology of the flag manifold $G/B$: where the product is over the set $\left\{\alpha\in\Phi\,\middle\vert\,\mathfrak g_{-\alpha}\not\subset H\right\}$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2: Giambelli formula
  • Theorem 1.3: Multiplicity Formula
  • Theorem 1.4: Equivariant Chevalley formula
  • Theorem 1.5: Equivariant Monk Rule
  • Lemma 2.1: cf. graham:positivity
  • Lemma 2.2
  • Corollary 2.3
  • Corollary 2.4
  • proof
  • ...and 35 more