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We enumerate smooth and rationally smooth Schubert varieties in the classical finite types A, B, C, and D, extending Haimans enumeration for type A. To do this enumeration, we introduce a notion of staircase diagrams on a graph. These combinatorial structures are collections of steps of irregular size, forming interconnected staircases over the given graph. Over a Dynkin-Coxeter graph, the set of nearly-maximally labelled staircase diagrams is in bijection with the set of Schubert varieties with a complete Billey-Postnikov (BP) decomposition. We can then use an earlier result of the authors showing that all finite-type rationally smooth Schubert varieties have a complete BP decomposition to finish the enumeration.
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We show that every smooth Schubert variety of affine type $tilde{A}$ is an iterated fibre bundle of Grassmannians, extending an analogous result by Ryan and Wolper for Schubert varieties of finite type $A$. As a consequence, we finish a conjecture of Billey-Crites that a Schubert variety in affine type $tilde{A}$ is smooth if and only if the corresponding affine permutation avoids the patterns $4231$ and $3412$. Using this iterated fibre bundle structure, we compute the generating function for the number of smooth Schubert varieties of affine type $tilde{A}$.
This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over C and over C[t_1, t_2,...,t_n]. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators studied by Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial, though the equivariant representation is not.
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 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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