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Characteristic Classes of Orbit Stratifications, the Axiomatic Approach

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

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Abstract

Consider a complex algebraic group G acting on a smooth variety M with finitely many orbits, and let \(\Omega \) be an orbit. The following three invariants of \(\Omega \subset M\) can be characterized axiomatically: (1) the equivariant fundamental class \([\overline{\Omega }, M]\in H^*_G(M)\), (2) the equivariant Chern–Schwartz–MacPherson class \({{\,\mathrm{c^{sm}}\,}}(\Omega , M)\in H^*_G(M)\), and (3) the equivariant motivic Chern class \({{\,\mathrm{mC}\,}}(\Omega , M) \in K_G(M)[y]\). The axioms for Chern–Schwartz–MacPherson and motivic Chern classes are motivated by the axioms for cohomological and K-theoretic stable envelopes of Okounkov and his coauthors. For M a flag variety and \(\Omega \) a Schubert cell—an orbit of the Borel group acting—this implies that CSM and MC classes coincide with the weight functions studied by Rimányi–Tarasov–Varchenko. In this paper we review the general theory and illustrate it with examples.

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Notes

  1. 1.

    Since \(u^k=[\mathcal O_{\mathbb {P}^{n-1-k}}]\in K(\mathbb {P}^{n-1})\) thus \(p_*(u^k)=\chi (\mathbb {P}^{n-1-k};\mathcal O)=1\).

  2. 2.

    This idea appeared already in an early version of [26].

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Acknowledgements

L.F. is supported by NKFI grants K 112735 and KKP 126683. R.R. is supported by the Simon Foundation grant 523882. A.W. is supported by NCN grants 2013/08/A/ST1/00804 and 2016/23/G/ST1/04282.

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Authors and Affiliations

  1. Institute of Mathematics, Eötvös University Budapest, Budapest, Hungary

    László M. Fehér

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    László M. Fehér

  3. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    Richárd Rimányi

  4. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    Andrzej Weber

Authors
  1. László M. Fehér
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  2. Richárd Rimányi
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  3. Andrzej Weber
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Correspondence to Richárd Rimányi .

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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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Fehér, L.M., Rimányi, R., Weber, A. (2020). Characteristic Classes of Orbit Stratifications, the Axiomatic Approach. 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_9

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