Mathematische Zeitschrift

, Volume 280, Issue 1–2, pp 269–306 | Cite as

Pfaffian sum formula for the symplectic Grassmannians



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.


Schubert classes Symplectic Grassmannians Torus equivariant cohomology Giambelli type formula Wilson’s conjecture Double Schubert polynomials 

Mathematics Subject Classification

Primary 14M15 Secondary 05E05 



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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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Applied MathematicsOkayama University of ScienceOkayamaJapan
  2. 2.Department of Mathematical SciencesKAISTDaejeonSouth Korea

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