Smooth Schubert varieties in the affine flag variety of type $\tilde{A}$

Edward Richmond; William Slofstra

Smooth Schubert varieties in the affine flag variety of type $\tilde{A}$

Combinatorics 2017-02-09 v1 Algebraic Geometry

Authors: Edward Richmond , William Slofstra

Abstract

We show that every smooth Schubert variety of affine type $\tilde{A}$ is an iterated fibre bundle of Grassmannians, extending an analogous result by Ryan and Wolper for Schubert varieties of finite type $A$ . As a consequence, we finish a conjecture of Billey-Crites that a Schubert variety in affine type $\tilde{A}$ is smooth if and only if the corresponding affine permutation avoids the patterns $4231$ and $3412$ . Using this iterated fibre bundle structure, we compute the generating function for the number of smooth Schubert varieties of affine type $\tilde{A}$ .

Keywords

Schubert variety affine flag variety smooth variety Grassmannian fibre bundle affine permutation pattern avoidance

Cite

@article{arxiv.1702.02236,
  title  = {Smooth Schubert varieties in the affine flag variety of type $\tilde{A}$},
  author = {Edward Richmond and William Slofstra},
  journal= {arXiv preprint arXiv:1702.02236},
  year   = {2017}
}

Comments

17 pages; as indicated in the paper, some results (including Theorem 1.1) originally appeared in arXiv:1408.0084v1, but were moved to this paper during the publication process

Smooth Schubert varieties in the affine flag variety of type $\tilde{A}$

Abstract

Keywords

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