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On Schubert’s Problem of Characteristics

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Schubert Calculus and Its Applications in Combinatorics and Representation Theory (ICTSC 2017)

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Abstract

The Schubert varieties on a flag manifold G/P give rise to a cell decomposition on G/P whose Kronecker duals, known as the Schubert classes on G/P, form an additive base of the integral cohomology \(H^{*}(G/P)\). The Schubert’s problem of characteristics asks to express a monomial in the Schubert classes as a linear combination in the Schubert basis. We present a unified formula expressing the characteristics of a flag manifold G/P as polynomials in the Cartan numbers of the group G. As application we develop a direct approach to our recent works on the Schubert presentation of the cohomology rings of flag manifolds G/P.

The authors are supported by NSFC 11131008, 11431009 and 11661131004.

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Notes

  1. 1.

    This English translation of the profound discovery of van der Waerden in German [59] is due to Schappacher [56].

  2. 2.

    The enumerative results in Schubert [51] were mutually verifiable with the results of other geometers (e.g. Salmon, Clebsch, Chasles and Zeuthen) of the same period, hence were already known to be correct at that time.

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Acknowledgements

The authors would like to thank their referees for valuable suggestions and improvements on the earlier version of the paper.

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Authors and Affiliations

  1. Academy of Mathematics and Systems Sciences, Beijing, 100190, China

    Haibao Duan

  2. School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing, 100049, China

    Haibao Duan

  3. Department of Mathematics, Capital Normal University, Beijing, 100048, China

    Xuezhi Zhao

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  1. Haibao Duan
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  2. Xuezhi Zhao
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Correspondence to Haibao Duan .

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Editors and Affiliations

  1. School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong, China

    Jianxun Hu

  2. School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong, China

    Changzheng Li

  3. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Leonardo C. Mihalcea

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Duan, H., Zhao, X. (2020). On Schubert’s Problem of Characteristics. In: Hu, J., Li, C., Mihalcea, L.C. (eds) Schubert Calculus and Its Applications in Combinatorics and Representation Theory. ICTSC 2017. Springer Proceedings in Mathematics & Statistics, vol 332. Springer, Singapore. https://doi.org/10.1007/978-981-15-7451-1_4

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