Back stable Schubert calculus

By Thomas Lam, Seung Jin Lee, Mark Shimozono. Compositio Mathematica, 2018

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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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/eb42af10ff7bac9c967de839fc195b37d8998a7d