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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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We study the back stable $K$-theory Schubert calculus of the infinite flag variety. We define back stable (double) Grothendieck polynomials and double $K$-Stanley functions and establish coproduct expansion formulae. Applying work of Weigandt, we extend our previous results on bumpless pipedreams from cohomology to $K$-theory. We study finiteness and positivity properties of the ring of back stable Grothendieck polynomials, and divided difference operators in $K$-homology.
We first define an action of the double coinvariant algebra $DR_n$ on the homology of the affine flag variety $widetilde{Fl}_n$ in type $A$, and use affine Schubert calculus to prove that it preserves the image of the homology of the rational $(n,m)$-affine Springer fiber $H_*(tilde{S}_{n,m})subset H_*(widetilde{Fl}_n)$ under the pushforward of the inclusion map. In our main result, we define a filtration by $mathbb{Q}[mathbf{x}]$-submodules of $DR_ncong H_*(tilde{S}_{n,n+1})$ indexed by compositions, whose leading terms are the Garsia-Stanton descent monomials in the $y$-variables. We find an explicit presentation of the subquotients as submodules of the single-variable coinvariant algebra $R_n(x)cong H_*(Fl_n)$, by identifying the leading torus fixed points with a subset $mathcal{H}subset S_n$ of the torus fixed points of the regular nilpotent Hessenberg variety, and comparing them to a cell decomposition of $tilde{S}_{n,n+1}$ due to Goresky, Kottwitz, and MacPherson. We also discover an explicit monomial basis of $DR_n$, and in particular an independent proof of the Haglund-Loehr formula.
The cohomology of the affine flag variety of a complex reductive group is a comodule over the cohomology of the affine Grassmannian. We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in (torus-equivariant) cohomology and K-theory. As an application, we deduce monomial positivity for the affine Schubert polynomials of the second author.
The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan-Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming from the Chern character.
The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottiles $r$-Bruhat order, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem.
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