Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties

Kirillov, A.; Maeno, T.

doi:10.1090/S1061-0022-2011-01151-3

Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties
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by A. N. Kirillov and T. Maeno

St. Petersburg Math. J. 22 (2011), 447-462

DOI: https://doi.org/10.1090/S1061-0022-2011-01151-3

Published electronically: March 18, 2011

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Abstract:

For a root system of type $A$, a certain extension of the quadratic algebra invented by S. Fomin and the first author is introduced and studied, which makes it possible to construct a model for the equivariant cohomology ring of the corresponding flag variety. As an application, a generalization of the equivariant Pieri rule for double Schubert polynomials is described. For a general finite Coxeter system, an extension of the corresponding Nichols–Woronowicz algebra is constructed. In the case of finite crystallographic Coxeter systems, a construction is presented of an extended Nichols–Woronowicz algebra model for the equivariant cohomology of the corresponding flag variety.

References

Nicolás Andruskiewitsch and Hans-Jürgen Schneider, Pointed Hopf algebras, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 1–68. MR 1913436, DOI 10.2977/prims/1199403805
Yuri Bazlov, Nichols-Woronowicz algebra model for Schubert calculus on Coxeter groups, J. Algebra 297 (2006), no. 2, 372–399. MR 2209265, DOI 10.1016/j.jalgebra.2006.01.037
I. Ciocan-Fontanine and W. Fulton, Quantum double Schubert polynomials, Inst. Mittag-Leffler Report no. 6, 1996–1997.
Charles F. Dunkl, Harmonic polynomials and peak sets of reflection groups, Geom. Dedicata 32 (1989), no. 2, 157–171. MR 1029672, DOI 10.1007/BF00147428
Sergey Fomin, Sergei Gelfand, and Alexander Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), no. 3, 565–596. MR 1431829, DOI 10.1090/S0894-0347-97-00237-3
Sergey Fomin and Anatol N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147–182. MR 1667680
A. N. Kirillov, On some quadratic algebras. II, Preprint.
Anatol N. Kirillov and Toshiaki Maeno, Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula, Discrete Math. 217 (2000), no. 1-3, 191–223 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766267, DOI 10.1016/S0012-365X(99)00263-0
Anatol N. Kirillov and Toshiaki Maeno, Noncommutative algebras related with Schubert calculus on Coxeter groups, European J. Combin. 25 (2004), no. 8, 1301–1325. MR 2095483, DOI 10.1016/j.ejc.2003.11.006
Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447–450 (French, with English summary). MR 660739
S. Majid, Free braided differential calculus, braided binomial theorem, and the braided exponential map, J. Math. Phys. 34 (1993), no. 10, 4843–4856. MR 1235979, DOI 10.1063/1.530326
Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow; Cours Spécialisés [Specialized Courses], 3. MR 1852463
Alexander Postnikov, On a quantum version of Pieri’s formula, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 371–383. MR 1667687
Shawn Robinson, A Pieri-type formula for $H^\ast _T(\textrm {SL}_n(\Bbb C)/B)$, J. Algebra 249 (2002), no. 1, 38–58. MR 1887984, DOI 10.1006/jabr.2001.9067
S. Veigneau, Calcul symbolique et calcul distribué en combinatoire algébrique, Thèse, Univ. Marne-la-Valée, 1996.
S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), no. 1, 125–170. MR 994499