Pattern avoidance and fiber bundle structures on Schubert varieties

TL;DR

This paper characterizes when Schubert varieties form fiber bundles based on permutation pattern avoidance, extending previous results to include split and non-split patterns, and providing new criteria for their fiber bundle structures.

Contribution

It introduces split pattern avoidance criteria and characterizes Schubert varieties as iterated fiber bundles using specific pattern avoidance conditions.

Findings

Fiber bundle structures correspond to avoiding split patterns 3|12 and 23|1.

Schubert varieties are iterated fiber bundles if they avoid patterns 3412, 52341, and 635241.

Extends previous results by including non-split pattern avoidance criteria.

Abstract

We give a permutation pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. In particular, we introduce the notion of a split pattern and show that a Schubert variety has such a fiber bundle structure if and only if the corresponding permutation avoids the split patterns 3|12 and 23|1. Continuing, we show that a Schubert variety is an iterated fiber bundle of Grassmannian Schubert varieties if and only if the corresponding permutation avoids (non-split) patterns 3412, 52341, and 635241. This extends a combined result of Lakshmibai-Sandhya, Ryan, and Wolper who prove that Schubert varieties whose permutation avoids the "smooth" patterns 3412 and 4231 are iterated fiber bundles of smooth Grassmannian Schubert varieties.