Back stable Schubert calculus

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.

Double Grothendieck Polynomials and Colored Lattice Models

We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of Weigandt relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of Motegi and Sakai to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by McNamara. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström–Gessel–Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.

Double Schubert polynomials for the classical Lie groups

International audience For each infinite series of the classical Lie groups of type $B$, $C$ or $D$, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur $Q$- or $P$-functions defined earlier by Ivanov. Pour chaque série infinie des groupe de Lie classiques de type $B$,$C$ ou $D$, nous présentons une famille de polynômes indexées par de éléments de groupe de Weyl correspondant de rang infini. Ces polynômes représentent des classes de Schubert dans la cohomologie équivariante des variétés de drapeaux. Ils ont une certain propriété de stabilité, et ils étendent naturellement des polynômes Schubert (simples) de Billey et Haiman, que représentent des classes de Schubert dans la cohomologie non-équivariante. Quand ils sont indexées par des éléments Grassmanniennes de groupes de Weyl, ces polynômes sont égaux à des analogues factorielles de fonctions $Q$ et $P$ de Schur, étudiées auparavant par Ivanov.

QUANTUM SCHUBERT POLYNOMIALS AND QUANTUM SCHUR FUNCTIONS

We introduce the quantum multi–Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations, the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nägelsbach–Kostka and Jacobi–Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove also an analog of the Giambelli and the Billey–Jockusch–Stanley formula for quantum Schubert polynomials. Finally we formulate two conjectures about the structure of quantum double and quantum Schubert polynomials for 321–avoiding permutations.