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Schur polynomials and weighted Grassmannians

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Abstract

In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. We show that it represents the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by Corti–Reid, and we regard it as an analogue of the Schur polynomials. Furthermore, we prove that these polynomials are the characters of certain representations, and hence, we give an interpretation of the Schubert structure constants of the weighted Grassmannians as the (rational) multiplicities of the tensor products of the representations. We also derive two determinantal formulas for the weighted Schubert classes: One is in terms of the special weighted Schubert classes, and the other is in terms of the Chern classes of the tautological orbi-bundles.

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Acknowledgments

The authors would like to thank the organizers of “MSJ Seasonal Institute 2012 Schubert calculus” for providing us an excellent environment for the discussions on the topic of this paper. The authors also would like to show their gratitude to Takeshi Ikeda and Tatsuya Horiguchi for many useful discussions. The first author is particularly grateful to Takashi Otofuji for many helpful comments. The first author was supported by JSPS Research Fellowships for Young Scientists. The second author was supported by the National Research Foundation of Korea (NRF) Grants funded by the Korea government (MEST) (No. 2012-0000795, 2011-0001181). He also would like to express his gratitude to the Algebraic Structure and its Application Research Institute at KAIST for providing him an excellent research environment in 2011–2012.

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Authors and Affiliations

  1. Osaka City University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Hiraku Abe

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

    Tomoo Matsumura

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  1. Hiraku Abe
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  2. Tomoo Matsumura
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Correspondence to Tomoo Matsumura.

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Abe, H., Matsumura, T. Schur polynomials and weighted Grassmannians. J Algebr Comb 42, 875–892 (2015). https://doi.org/10.1007/s10801-015-0608-z

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  • DOI: https://doi.org/10.1007/s10801-015-0608-z

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