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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.
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.
We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(lambda_bullet)$ in the orthogonal Grassmannian $mathrm{OG}_n$, which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A. First, using the type B Wronski map, we show that the real locus of the Schubert curve has a natural covering map to $mathbb{RP}^1$, with monodromy operator $omega$ defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute $omega$ without rectifying the skew tableau. The first uses recently-developed shifted tableau crystal operators, while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yongs shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves.
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.
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.
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