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Structure constants for Chern classes of Schubert cells

Abstract

A formula for the structure constants of the multiplication of Schubert classes is obtained in (Rebecca and Allen. arXiv preprint arXiv:1909.05283, 2019). In this note, we prove analogous formulae for the Chern–Schwartz–MacPherson (CSM) classes and Segre–Schwartz–MacPherson (SSM) classes of Schubert cells in the flag variety. By the equivalence between the CSM classes and the stable basis elements for the cotangent bundle of the flag variety, a formula for the structure constants for the latter is also deduced.

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Notes

  1. If X is not smooth, we can embed X into a smooth ambient space, and use the total Chern class of the ambient space to define the SSM classes, see [2].

  2. These operators are the \(\hbar =1\) specialization of \(L_i\) and \(L^\vee _i\) in [3, Section 5.2].

  3. Here we have used the fact \((-1)^{\dim G/B}e^{T\times {\mathbb C}^*}(T^*(G/B))|_{\hbar =1}=c^T(T(G/B))\), see the proof of Corollary 7.4 in loc. cit..

  4. The author thanks L. Mihalcea for pointing out this proof.

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Acknowledgements

The author thanks A. Knutson and A. Yong for discussions. Special thanks go to L. Mihalcea for providing the proof of Theorem 2.5. The author also thanks the anonymous referees for useful comments.

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Su, C. Structure constants for Chern classes of Schubert cells. Math. Z. 298, 193–213 (2021). https://doi.org/10.1007/s00209-020-02595-5

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Keywords

  • MacPherson classes
  • Flag variety
  • Bott–Samelson variety

Mathematics Subject Classification

  • 57T15
  • 05E15