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Equivariant quantum cohomology of the odd symplectic Grassmannian

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Abstract

The odd symplectic Grassmannian \(\mathrm {IG}:=\mathrm {IG}(k, 2n+1)\) parametrizes k dimensional subspaces of \({\mathbb {C}}^{2n+1}\) which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on \(\mathrm {IG}\) with two orbits, and \(\mathrm {IG}\) is itself a smooth Schubert variety in the submaximal isotropic Grassmannian \(\mathrm {IG}(k, 2n+2)\). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of \(\mathrm {IG}\), i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case \(k=2\), and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring.

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Notes

  1. This can be explained by using that the localized equivariant cohomology is generated by divisors; see [4, Sect. 5].

  2. However, Gelfand and Zelevinsky [14] defined another group \(\widetilde{{{\mathrm{Sp}}}}_{2n+1}\) closely related to \({{\mathrm{Sp}}}_{2n+1}\) such that \({{\mathrm{Sp}}}_{2n} \subset \widetilde{{{\mathrm{Sp}}}}_{2n+1} \subset {{\mathrm{Sp}}}_{2n+2}\).

  3. Another way to see that \(F_c \subsetneq \overline{F^\circ }\) is to notice that every line in \(F^\circ \) has isotropic span, therefore any line in the closure must satisfy the same property. But we have seen that there exist lines in \(X_c\) with non isotropic span.

  4. One word of caution: the Bruhat order does not translate into partition inclusion. For example, \((2n+2-k, 0, \ldots , 0) \leqslant (1, 1, \ldots 1) \) in the Bruhat order for \(k < n+1\).

  5. But the Schubert divisor generates the ring \(\mathrm {QH}^*_T(\mathrm {IG})\)localized at the equivariant parameters. We refer to [4, Sect. 5] for details.

  6. The proof of [32, Lemma 7.1] is given in the Appendix of [32]. We comment that the full details proving the non-emptiness statement from loc.cit. can be found in [2, Theorem 2.5.5].

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Acknowledgements

We would like to thank Dan Orr and Mark Shimozono for discussions and valuable suggestions and to Pierre-Emmanuel Chaput, Changzheng Li, and Nicolas Perrin for discussions and collaborations on related projects. Special thanks are due to Anders Buch for encouragement and interest in this project. We thank the referee for carefully reading our paper.

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Authors and Affiliations

  1. Department of Mathematics, 460 McBryde Hall, Virginia Tech, Blacksburg, VA, 24061, USA

    Leonardo C. Mihalcea

  2. Department of Mathematics and Computer Science, Henson Science Hall, Salisbury University, Salisbury, MD, 21801, USA

    Ryan M. Shifler

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  1. Leonardo C. Mihalcea
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Correspondence to Ryan M. Shifler.

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L.M. was supported in part by NSA Young Investigator Award 98320-16-1-0013 and a Simons Collaboration grant.

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Mihalcea, L.C., Shifler, R.M. Equivariant quantum cohomology of the odd symplectic Grassmannian. Math. Z. 291, 1569–1603 (2019). https://doi.org/10.1007/s00209-018-2120-3

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