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Infinite flags and Schubert polynomials

David Anderson

Abstract

We study Schubert polynomials using geometry of infinite-dimensional flag varieties and degeneracy loci. Applications include Graham-positivity of coefficients appearing in equivariant coproduct formulas and expansions of back-stable and enriched Schubert polynomials. We also construct an embedding of the type C flag variety, and study the corresponding pullback map on (equivariant) cohomology rings.

Infinite flags and Schubert polynomials

Abstract

We study Schubert polynomials using geometry of infinite-dimensional flag varieties and degeneracy loci. Applications include Graham-positivity of coefficients appearing in equivariant coproduct formulas and expansions of back-stable and enriched Schubert polynomials. We also construct an embedding of the type C flag variety, and study the corresponding pullback map on (equivariant) cohomology rings.

Paper Structure

This paper contains 21 sections, 14 theorems, 113 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutations
  4. Vector spaces
  5. Flag varieties
  6. A technical note on limits
  7. Sato Grassmannians and flag varieties
  8. Schubert varieties and Schubert polynomials
  9. Degeneracy loci
  10. Fixed points
  11. Duality, projection, and shift morphisms
  12. Duality
  13. Projections
  14. Shift
  15. Direct sum morphism and coproduct
  16. ...and 6 more sections

Key Result

Proposition 4.2

Suppose $w \in \mathcal{S}_{m'}$. Then where $c^{(m)} = \prod_{i=-m+1}^0 \frac{1+y_i}{1-x_i}$.

Figures (2)

  • Figure 1: The permutation $w$ in $\mathcal{S}_\mathds{Z}$ given in one-line notation as $[2,-2,3,1,0,-3,4,-1]$. The value of the rank function $k_w(3,-1)= 5$ is illustrated as the number of dots enclosed by the dashed line, at left. The diagram and essential set are shown at right.
  • Figure 2: Weights ($\bullet$) on $U^+\times U^+$ and ($\circ$) on $\mathds{U}^+/(U^+\times U^+)$

Theorems & Definitions (30)

  • Remark 2.1
  • Remark 3.1
  • Definition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Theorem 5.1
  • Proposition 6.1
  • Remark 6.2
  • Remark 6.3
  • Proposition 7.1
  • ...and 20 more