Proceedings of the Japan Academy, Series A, Mathematical Sciences

The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials

Tatsuya Horiguchi

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Abstract

In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. We show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 94, Number 9 (2018), 87-92.

Dates
First available in Project Euclid: 1 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.pja/1541059248

Digital Object Identifier
doi:10.3792/pjaa.94.87

Mathematical Reviews number (MathSciNet)
MR3871391

Subjects
Primary: 14N15: Classical problems, Schubert calculus

Keywords
Flag varieties Hessenberg varieties Schubert polynomials

Citation

Horiguchi, Tatsuya. The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials. Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 9, 87--92. doi:10.3792/pjaa.94.87. https://projecteuclid.org/euclid.pja/1541059248


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References

  • H. Abe, M. Harada, T. Horiguchi and M. Masuda, The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A, Int. Math. Res. Not. IMRN., DOI: https://doi.org/10.1093/imrn/rnx275.
  • T. Abe, T. Horiguchi, M. Masuda, S. Murai and T. Sato, Hessenberg varieties and hyperplane arrangements, arXiv:1611.00269.
  • M. Brion and J. B. Carrell, The equivariant cohomology ring of regular varieties, Michigan Math. J. 52 (2004), no. 1, 189–203.
  • P. Brosnan and T. Y. Chow, Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties, Adv. Math. 329 (2018), 955–1001.
  • F. De Mari, C. Procesi and M. A. Shayman, Hessenberg varieties, Trans. Amer. Math. Soc. 332 (1992), no. 2, 529–534.
  • F. De Mari and M. A. Shayman, Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix, Acta Appl. Math. 12 (1988), no. 3, 213–235.
  • E. Drellich, Monk's rule and Giambelli's formula for Peterson varieties of all Lie types, J. Algebraic Combin. 41 (2015), no. 2, 539–575.
  • W. Fulton, Young tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997.
  • M. Guay-Paquet, A second proof of the Shareshian-Wachs conjecture, by way of a new Hopf algebra, arXiv:1601.05498.
  • M. Harada, T. Horiguchi and M. Masuda, The equivariant cohomology rings of Peterson varieties in all Lie types, Canad. Math. Bull. 58 (2015), no. 1, 80–90.
  • M. Harada and J. Tymoczko, A positive Monk formula in the $S^{1}$-equivariant cohomology of type $A$ Peterson varieties, Proc. Lond. Math. Soc. (3) 103 (2011), no. 1, 40–72.
  • E. Insko and A. Yong, Patch ideals and Peterson varieties, Transform. Groups 17 (2012), 1011–1036.
  • B. Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight $\rho$, Selecta Math. (N.S.) 2 (1996), no. 1, 43–91.
  • D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. (3) 9 (1959), 253–286.
  • M. Precup, Affine pavings of Hessenberg varieties for semisimple groups, Selecta Math. (N.S.) 19 (2013), no. 4, 903–922.
  • K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties, J. Amer. Math. Soc. 16 (2003), no. 2, 363–392.
  • J. Shareshian and M. L. Wachs, Chromatic quasisymmetric functions, Adv. Math. 295 (2016), 497–551.
  • E. Sommers and J. Tymoczko, Exponents for $B$-stable ideals, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3493–3509.
  • T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207.
  • T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279–293.
  • J. S. Tymoczko, Linear conditions imposed on flag varieties, Amer. J. Math. 128 (2006), no. 6, 1587–1604.
  • J. S. Tymoczko, Paving Hessenberg varieties by affines, Selecta Math. (N.S.) 13 (2007), no. 2, 353–367.