Tokyo Journal of Mathematics

Construction of Double Grothendieck Polynomials of Classical Types using IdCoxeter Algebras

Anatol N. KIRILLOV and Hiroshi NARUSE

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We construct double Grothendieck polynomials of classical types which are essentially equivalent to but simpler than the polynomials defined by A.~N. Kirillov in arXiv:1504.01469 and identify them with the polynomials defined by T.~Ikeda and H.~Naruse in Adv. Math. (2013) for the case of maximal Grassmannian permutations. We also give geometric interpretation of them in terms of algebraic localization map and give explicit combinatorial formulas.

Article information

Tokyo J. Math., Volume 39, Number 3 (2017), 695-728.

First available in Project Euclid: 6 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05E05: Symmetric functions and generalizations


KIRILLOV, Anatol N.; NARUSE, Hiroshi. Construction of Double Grothendieck Polynomials of Classical Types using IdCoxeter Algebras. Tokyo J. Math. 39 (2017), no. 3, 695--728. doi:10.3836/tjm/1491465733.

Export citation


  • N. Bergeron and S. Billey, RC-Graphs and Schubert Polynomials, Experiment. Math. 2 (1993), 257–269.
  • I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Schubert cells and cohomology of the spaces $G/P,$ Russian Math. Surveys 28 (1973), no. 3, 1–26.
  • S. Billey and M. Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), no. 2, 443–482.
  • A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115–207.
  • A. Buch, A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), 2633–2640.
  • M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. 7 (1974), 53–88.
  • S. Fomin and A. N. Kirillov, Grothendieck polynomials and the Yang-Baxter equation, Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183–190.
  • S. Fomin and A. N. Kirillov, Combinatorial $B_n$-analogues of Schubert polynomials, Trans. Amer. Math. Soc. 348 (1996), no. 9, 3591–3620.
  • S. Fomin and A. N. Kirillov, Yang-Baxter equation, symmetric functions and Grothendieck polynomials, preprint arXiv:hep-th/9306005.
  • S. V. Fomin and R. Stanley, Schubert polynomials and the nilCoxeter algebra, Adv. Math. 103 (1994), no. 2, 196–207.
  • R. Goldin, The cohomology ring of weight varieties and polygon spaces, Adv. Math. 160 (2001), no. 2, 175–204.
  • M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25–83.
  • W. Graham and S. Kumar, On positivity in $T$-equivariant $K$-theory of flag varieties, Int. Math. Res. Not. IMRN 2008, Art. ID rnn 093.
  • T. Hudson, T. Ikeda, T. Matsumura and H. Naruse, Degeneracy Loci Classes in K-theory - Determinantal and Pfaffian Formula -, arXiv:1504.02828v3 [math.AG].
  • T. Ikeda, L. Mihalcea and H. Naruse, Double Schubert polynomials for the classical groups, Adv. Math. 226 (2011), 840–886.
  • T. Ikeda and H. Naruse, Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc. 361 (2009), no. 10, 5193–5221.
  • T. Ikeda and H. Naruse, $K$-theory analogue of factorial Schur $P$-,$Q$- functions, Adv. Math. 243 (2013), 22–66.
  • A. N. Kirillov, On Double Schubert and Grothendieck polynomials for classical groups, preprint (1999); update version arXiv:1504.01469.
  • A. N. Kirillov, Notes on Schubert, Grothendieck and Key polynomials, SIGMA 12 (2016), 034, 57 pages.
  • B. Kostant and S. Kumar, $T$-equivariant $K$-theory of generalized flag varieties, J. Differential Geom. 32 (1990), no. 2, 549–603.
  • T. Lam, A. Schilling and M. Shimozono, $K$-Theory Schubert calculus of the affine Grassmannian, Compositio Math. 146 (2010), 811–852.
  • A. Lascoux, Classes de Chern des variétés de drapeaux, C.R. Acad. Sci. Paris Sér. I Math. 295 (1982), 393–398.
  • A. Lascoux, Anneau de Grothendieck de la variété de drapeaux, The Grothendieck Festschrift, Vol. III, 1–34, Progr. Math., 88, Birkhüser Boston, Boston, MA, 1990.
  • A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert, C.R. Acad. Sci. Paris Sér. I Math. 294 (1982), 447–450.
  • A. Lascoux and M.-P. Schützenberger, Symmetry and flag mainfolds, Lecture Notes 996 (1983), 118–144.
  • M. Levin and F. Morel, Algebraic cobordism, Springer Monograph (2008).
  • I. G. Macdonald, Notes on Schubert polynomials, Laboratoire de combinatoire et d'informatique mathématique (LACIM), Univ. du Québec à Montréal, Montréal, 1991.
  • M. Nakagawa and H. Naruse, Generalized (co)homology of the loop spaces of classical groups and the universal factorial Schur $P$- and $Q$-functions, Adv. Study in Pure Math., vol. 71, 2016 Schubert Calculus–Osaka 2012, pp. 337–417.
  • E. K. Sklyanin, L. A. Takhtadzhyan and L. D. Faddeev, Quantum inverse problem method. I, Theor. Math. Phys. 40 (1979), Issue 2, 688–706.