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


Authors: A. N. Kirillov and T. Maeno
Original publication: Algebra i Analiz, tom 22 (2010), nomer 3.
Journal: St. Petersburg Math. J. 22 (2011), 447-462
MSC (2010): Primary 05E15, 14M15
DOI: https://doi.org/10.1090/S1061-0022-2011-01151-3
Published electronically: March 18, 2011
MathSciNet review: 2729944
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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 [Enhancements On Off] (What's this?)

  • 1. N. Andruskiewitsch and H.-J. 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 (2003e:16043)
  • 2. Y. Bazlov, Nichols-Woronowicz algebra model for Schubert calculus on Coxeter groups, J. Algebra 297 (2006), no. 2, 372-399. MR 2209265 (2007b:20083)
  • 3. I. Ciocan-Fontanine and W. Fulton, Quantum double Schubert polynomials, Inst. Mittag-Leffler Report no. 6, 1996-1997.
  • 4. C. Dunkl, Harmonic polynomials and peak sets of reflection groups, Geom. Dedicata 32 (1989), 157-171. MR 1029672 (91h:51015)
  • 5. S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), 565-596. MR 1431829 (98d:14063)
  • 6. S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in Geometry (J.-L. Brylinski, R. Brylinski, V. Nistor, B. Tsygan, and P. Xu, eds.), Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147-182. MR 1667680 (2001a:05152)
  • 7. A. N. Kirillov, On some quadratic algebras. II, Preprint.
  • 8. A. N. Kirillov and T. Maeno, Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula, Discrete Math. 217 (2000), 191-223. MR 1766267 (2001f:05161)
  • 9. -, Noncommutative algebras related with Schubert calculus on Coxeter groups, European J. Combin. 25 (2004), 1301-1325. MR 2095483 (2005j:05097)
  • 10. A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Ser. I Math. 294 (1982), 447-450. MR 0660739 (83e:14039)
  • 11. S. Majid, Free braided differential calculus, braided binomial theorem, and the braided exponential map, J. Math. Phys. 34 (1993), 4843-4856. MR 1235979 (94i:58013)
  • 12. L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monogr., vol. 6, Amer. Math. Soc., Providence, RI, 2001. MR 1852463 (2002h:05161)
  • 13. A. Postnikov, On a quantum version of Pieri's formula, Advances in Geometry (J.-L. Brylinski, R. Brylinski, V. Nistor, B. Tsygan, and P. Xu, eds.), Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 371-383. MR 1667687 (99m:14096)
  • 14. S. Robinson, A Pieri type formula for $ H^*_T({\rm SL}_n(\bf {C})/B)$, J. Algebra 249 (2002), 38-58. MR 1887984 (2003b:14065)
  • 15. S. Veigneau, Calcul symbolique et calcul distribué en combinatoire algébrique, Thèse, Univ. Marne-la-Valée, 1996.
  • 16. S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), 125-170. MR 0994499 (90g:58010)

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Additional Information

A. N. Kirillov
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
Email: kirillov@kurims.kyoto-u.ac.jp

T. Maeno
Affiliation: Department of Electrical Engineering, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan
Email: maeno@kuee.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S1061-0022-2011-01151-3
Keywords: Root system of type $A$, equivariant Pieri rule, Nichols–Woronowicz algebra
Received by editor(s): January 15, 2010
Published electronically: March 18, 2011
Dedicated: To Ludwig Dmitrievich Faddeev on the occasion of his 75th birthday
Article copyright: © Copyright 2011 American Mathematical Society

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