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Pfaffian sum formula for the symplectic Grassmannians

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Abstract

We study the torus equivariant Schubert classes of the Grassmannian of non-maximal isotropic subspaces in a symplectic vector space. We prove a formula that expresses each of those classes as a sum of multi Schur-Pfaffians, whose entries are equivariantly modified special Schubert classes. Our result gives a proof to Wilson’s conjectural formula, which generalizes the Giambelli formula for the ordinary cohomology proved by Buch–Kresch–Tamvakis, given in terms of Young’s raising operators. Furthermore we show that the formula extends to a certain family of Schubert classes of the symplectic partial isotropic flag varieties.

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Notes

  1. It is well-known that \(\varGamma \) can be defined as the quotient of the polynomial ring \({\mathbb Z}[Q_1,Q_2,Q_3,\ldots ]\) of the variables \(Q_1,Q_2,\ldots \) by the ideal generated by the following elements

    with \(Q_0=1.\) This fact follows from [17], (8.6)  (ii), \((8.2')\) and Proof of (8.4) in Chap. III].

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Acknowledgments

We are especially grateful to Hiroshi Naruse for explaining his results, and also to Harry Tamvakis for valuable comments to an earlier version of this manuscript. We thank Dave Anderson, Anders Skovsted Buch, Andrew Kresch, Changzheng Li, Leonardo Mihalcea, Masaki Nakagawa for the helpful conversations and their comments. We thank the anonymous referee and Harry Tamvakis for independently pointing out an error of an argument in proving Theorem 4 in a previous version. We also thank Thomas Hudson for carefully reading the manuscript. This paper was written for the most part during the first named author’s stay at KAIST in 2013. The hospitality and perfect working conditions there are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Applied Mathematics, Okayama University of Science, Okayama, 700-0005, Japan

    Takeshi Ikeda

  2. Department of Mathematical Sciences, KAIST, Daehak-ro Yuseong-gu, Daejeon, 305-701, South Korea

    Tomoo Matsumura

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  1. Takeshi Ikeda
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Correspondence to Takeshi Ikeda.

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Dedicated to Hiroshi Naruse on the occasion of his sixtieth birthday.

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Ikeda, T., Matsumura, T. Pfaffian sum formula for the symplectic Grassmannians. Math. Z. 280, 269–306 (2015). https://doi.org/10.1007/s00209-015-1423-x

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