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
Summary
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
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