Smooth Schubert varieties in the affine flag variety of type $\tilde{A}$

Abstract

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 $\tilde{A}$.

17 pages; as indicated in the paper, some results (including Theorem 1.1) originally appeared in arXiv:1408.0084v1, but were moved to this paper during the publication process