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Back stable Schubert calculus

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arxiv 1806.11233 v2 pith:VBHXP5KA submitted 2018-06-29 math.CO math.AG

Thomas Lam , Seung Jin Lee , Mark Shimozono This is my paper · ORCID
classification math.COmath.AG
keywords schubertdoublebackinfinitestableedelman-greenehomologycalculus
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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.

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Cited by 3 Pith papers

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  1. Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes

    math.CO 2024-11 conditional novelty 8.0 of 10

    Generic pipe dreams encode the equivariant cohomology classes of lower-upper varieties, unifying classic and bumpless pipe dream formulas and producing new Chern-Schwartz-MacPherson class formulas.

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    Hybrid generic pipe dreams give a hybridization-independent equivariant formula for the classes of lower-upper varieties, proved via Yang-Baxter equations and a degeneration into complete intersections.

  3. Set-valued Rothe Tableaux and Grothendieck Polynomials

    math.CO 2019-08 conditional novelty 7.0 of 10

    A permutation is 1432-avoiding if and only if its double Grothendieck polynomial equals a signed sum over set-valued Rothe tableaux of its Rothe diagram.

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