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

## TL;DR

This paper proves that smooth Schubert varieties in affine type A are iterated Grassmannian bundles, confirming a conjecture about pattern avoidance characterizing smoothness and providing a generating function for their count.

## Contribution

It extends finite type A results to affine type A, establishing the fiber bundle structure and confirming the pattern avoidance criterion for smoothness.

## Key findings

- Smooth affine Schubert varieties are iterated Grassmannian bundles.
- A conjecture linking smoothness to pattern avoidance is confirmed.
- A generating function for counting smooth affine Schubert varieties is derived.

## 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 $\tilde{A}$.