Pattern avoidance criteria for fiber bundles on Schubert varieties

Abstract

We show a permutation pattern avoidance criteria for when the natural projection from a flag variety to the Grassmannian, restricted to a Schubert variety, is a fiber bundle structure. We define the concept of split patterns and show that this projection is a fiber bundle if and only if the corresponding permutation avoids the split patterns 3|12 and 23|1. Continuing, we show that a Schubert variety has an iterated fiber bundle structure of Schubert varieties in the Grassmannian if and only if the corresponding permutation avoids the patterns 3412, 52341, and 635241. This extends the findings of Lakshmibai-Sandhya, Ryan, and Wolper, who's combined results show that a Schubert variety has such an iterated fiber bundle structure if it is smooth i.e. the corresponding permutation avoids the patterns 3412 and 4231.