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Billey–Postnikov decompositions and the fibre bundle structure of Schubert varieties

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Abstract

A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth Grassmannian Schubert varieties. The proof depends on deep combinatorial results of Billey–Postnikov on Weyl groups. We determine all smooth and rationally smooth Grassmannian Schubert varieties, and give a new proof of Peterson’s theorem that all simply-laced rationally smooth Schubert varieties are smooth. Taken together, our results give a fairly complete geometric description of smooth and rationally smooth Schubert varieties using primarily combinatorial methods.

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Notes

  1. This also follows from the geometric results of [15] together with Peterson’s theorem that all rationally smooth elements in type E are smooth.

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Acknowledgments

We would like to thank Dave Anderson, Sara Billey, Jim Carrell, and Alex Woo for helpful discussions. We thank the anonymous referee for useful suggestions on the manuscript. The first author was partially supported by the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Oklahoma State University, Stillwater, OK, 74074, USA

    Edward Richmond

  2. University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    William Slofstra

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  1. Edward Richmond
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  2. William Slofstra
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Correspondence to Edward Richmond.

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Richmond, E., Slofstra, W. Billey–Postnikov decompositions and the fibre bundle structure of Schubert varieties. Math. Ann. 366, 31–55 (2016). https://doi.org/10.1007/s00208-015-1299-4

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  • DOI: https://doi.org/10.1007/s00208-015-1299-4

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