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arXiv · 1610.03535

Pattern avoidance and fiber bundle structures on Schubert varieties

Timothy Alland·Edward Richmond
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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.

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Timothy Alland, Edward Richmond. 2016-10-11. Pattern avoidance and fiber bundle structures on Schubert varieties. https://doi.org/10.1016/j.jcta.2017.09.006

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