Smooth Schubert varieties in the affine flag variety of type [formula omitted

Abstract

We show that every smooth Schubert variety of affine type à 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 à 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 Ã.