  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.
