Abstract

We study the back stable Schubert calculus of the infinite flag variety. Our main results are:–a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;–a novel definition of double and triple Stanley symmetric functions;–a proof of the positivity of double Edelman–Greene coefficients generalizing the results of Edelman–Greene and Lascoux–Schützenberger;–the definition of a new class ofbumplesspipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman–Greene insertion algorithm;–the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;–equivariant Pieri rules for the homology of the infinite Grassmannian;–homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.

Highlights

  • 1.1 Flag varieties and Schubert polynomials The flag variety Fln is the smooth projective algebraic variety classifying full flags inside an n-dimensional complex vector space Cn

  • The flag variety has a distinguished stratification by Schubert varieties, and the cohomology classes of Schubert varieties form a basis of H∗(Fln), called the Schubert basis

  • Under an isomorphism between R and the cohomology of fundamental classes [Xw]: H∗TZ (Fl), we show in Theorem 6.7 that back stable Schubert polynomials represent Schubert classes of Fl

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Summary

Introduction

The flag variety has a distinguished stratification by Schubert varieties, and the cohomology classes of Schubert varieties form a basis of H∗(Fln), called the Schubert basis. Lascoux and Schutzenberger [LS82] defined and studied polynomial representatives for the Schubert classes, called the Schubert polynomials Sw ∈ Q[x1, . This journal is ○c Foundation Compositio Mathematica 2021. Lascoux and Schutzenberger defined the double Schubert polynomials Sw(x; a) that represent Schubert classes in the torus-equivariant cohomology ring HT∗ (Fln). Among the fundamental results crucial to us is the formula of Billey, Jockusch and Stanley [BJS93] for the monomial expansion of Sw

Back stable Schubert polynomials
Coproduct formula
Localization and infinite nilHecke algebra
Schubert polynomials
Back symmetric formal power series
Stanley symmetric functions
Negative Schubert polynomials
Double symmetric functions
Back symmetric double power series
Double Stanley symmetric functions
Negative double Schubert polynomials
Dynkin reversal Extend the Q-algebra automorphism ω
Bumpless pipedreams
Infinite flag variety
Infinite Grassmannian
Equivariant cohomology of infinite flag variety
Localization and GKM rings for infinite flags and infinite Grassmannian
NilHecke algebra
Homology Hopf algebra
Peterson subalgebra
Affine symmetric group
Translation elements
The Peterson subalgebra
Fomin–Stanley algebra
Stability of affine double Edelman–Greene coefficients
10. Back stable triple Schubert polynomials
10.1 Tripling
10.4 Double to triple
11. Affine flag variety
11.1 Affine flag variety and affine Grassmannian
11.2 Equivariant cohomology of affine flag variety
11.3 Presentations
12.1 Schubert varieties and double Schur functions
Molev’s skew double Schur functions
Inverting systems with Schubert polynomials as change-of-basis matrix
Full Text
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