Back stable Schubert calculus

TL;DR

This paper advances the understanding of back stable Schubert calculus on the infinite flag variety by providing new formulas, functions, and algebraic structures, along with positivity results and combinatorial algorithms.

Contribution

It introduces a formula for back stable Schubert classes, defines new symmetric functions, and constructs algebraic and combinatorial tools for infinite flag varieties.

Findings

Derived a formula expressing back stable Schubert classes in terms of symmetric and finite parts.

Proved positivity of double Edelman-Greene coefficients, extending previous results.

Developed new combinatorial models and algebraic structures for infinite Grassmannian homology.

Abstract

We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel definition of double and triple Stanley symmetric functions; 3) a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schutzenberger; 4) the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman-Greene insertion algorithm; 5) the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case; 6) equivariant Pieri rules for the homology of the infinite Grassmannian; 7) homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.