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Structure Constants in equivariant oriented cohomology of flag varieties

Rebecca Goldin, Changlong Zhong

Abstract

We obtain a formula for structure constants of certain variant form of Bott-Samelson classes for equivariant oriented cohomology of flag varieties. Specializing to singular cohomology/K-theory, we recover formulas of structure constants of Schubert classes of Goldin-Knutson, and that of structure constants of Segre-Schwartz-MacPherson classes of Su. We also obtain a formula for K-theoretic stable basis. Our method comes from the study of formal affine Demazure algebra, so is purely algebraic, while the above mentioned results are geometric.

Structure Constants in equivariant oriented cohomology of flag varieties

Abstract

We obtain a formula for structure constants of certain variant form of Bott-Samelson classes for equivariant oriented cohomology of flag varieties. Specializing to singular cohomology/K-theory, we recover formulas of structure constants of Schubert classes of Goldin-Knutson, and that of structure constants of Segre-Schwartz-MacPherson classes of Su. We also obtain a formula for K-theoretic stable basis. Our method comes from the study of formal affine Demazure algebra, so is purely algebraic, while the above mentioned results are geometric.

Paper Structure

This paper contains 10 sections, 21 theorems, 123 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The operator algebras $Q_W$ and ${\mathbf D}_F$
  4. The dual operator algebras
  5. Generalized Demazure operators and the generalized Leibniz rule
  6. The structure constants of equivariant oriented cohomology of flag varieties
  7. Structure constants in singular cohomology and K-theory
  8. Structure constants of cohomological stable bases
  9. Structure constants for K-theoretic stable bases
  10. The restriction formula

Key Result

Lemma 2.2

Z15 Let $\alpha$ and $\beta$ be simple roots. The following identities hold in $Q_W$:

Theorems & Definitions (58)

  • Example 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Example 2.4
  • Example 2.5
  • Proposition 2.6
  • Example 2.7
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 48 more