2016/10/11 by Timothy Alland, Edward Richmond · 6 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Bundle #Composite material #Fiber #Fiber bundle #Materials science #Mathematics #Pure mathematics #Schubert polynomial #Schubert variety #math.CO #msc:05A05 #msc:05E05 #msc:14M15 #msc:20F55 #msc:22E40
paper · pdf · doi:10.1016/j.jcta.2017.09.006
published in Journal of Combinatorial Theory Series A 154, 533-550 (Elsevier BV) · 16 pages and 9 Figures
arxiv created 2016/10/11 · openalex publication_date 2017/11/06 · arxiv updated 2018/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.