Back stable Schubert calculus. (30th April 2021)

• Record Type: Journal Article
• Authors: Lam, Thomas; Lee, Seung Jin; Shimozono, Mark
• Is Part Of: Compositio mathematica. Volume 157:Number 5(2021)
• Page Start: 883
• Page End: 962
• Publication Date: 2021-04-30
• Subjects: flag variety, -- Grassmannian, -- Schubert polynomial, -- Schur polynomial; 14M15, -- 05E05; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X21007028
• Languages: English
• ISSNs: 0010-437X

Abstract

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 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; – 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.