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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 $tilde{A}$.
We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumars work on the equivariant coho
We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine FK algeb
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 s
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
We discuss a relationship between Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized nil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillov algebra implies the nonne