×

zbMATH — the first resource for mathematics

Kostant polynomials and the cohomology ring for \(G/B\). (English) Zbl 0980.22018
Kostant polynomials play a crucial role in the Schubert calculus on \(G/B\) for a semisimple Lie group \(G\) and its Borel subgroup \(B\). These polynomials which are characterized by vanishing properties on the orbits of a regular point under the action of the Weyl group have nonzero values on the corresponding certain elements of higher Bruhat order. The author succeeded in giving explicit forms of these values. It should be emphasized that his description is very minute.

MSC:
22E46 Semisimple Lie groups and their representations
14M17 Homogeneous spaces and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] I. N. Bernšteĭ n, I. M. Gelfand, and S. I. Gelfand, Schubert cells, and the cohomology of the spaces \(G/P\) , Uspehi Mat. Nauk 28 (1973), no. 3(171), 3-26. · Zbl 0289.57024 · doi:10.1070/rm1973v028n03ABEH001557
[2] Sara C. Billey, Kostant polynomials and the cohomology ring for \(G/B\) , Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 1, 29-32. JSTOR: · Zbl 0878.17002 · doi:10.1073/pnas.94.1.29 · links.jstor.org
[3] Sara Billey and Mark Haiman, Schubert polynomials for the classical groups , J. Amer. Math. Soc. 8 (1995), no. 2, 443-482. JSTOR: · Zbl 0832.05098 · doi:10.2307/2152823 · links.jstor.org
[4] James B. Carrell, Some remarks on regular Weyl group orbits and the cohomology of Schubert varieties , Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989), Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 33-41. · Zbl 0807.14040
[5] Michel Demazure, Désingularisation des variétés de Schubert généralisées , Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88. · Zbl 0312.14009 · numdam:ASENS_1974_4_7_1_53_0 · eudml:81930
[6] Sergey Fomin and Anatol N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials , Discrete Math. 153 (1996), no. 1-3, 123-143, in Proceedings of the Fifth Conference on Power Series and Algebraic Combinatorics (Florence, 1993), North-Holland, Amsterdam. · Zbl 0852.05078 · doi:10.1016/0012-365X(95)00132-G
[7] Sergey Fomin and Anatol N. Kirillov, Combinatorial \(B_ n\)-analogues of Schubert polynomials , Trans. Amer. Math. Soc. 348 (1996), no. 9, 3591-3620. JSTOR: · Zbl 0871.05060 · doi:10.1090/S0002-9947-96-01558-9 · links.jstor.org
[8] Sergey Fomin and Anatol N. Kirillov, Universal exponential solution of the Yang-Baxter equation , Lett. Math. Phys. 37 (1996), no. 3, 273-284. · Zbl 0867.17009
[9] William Fulton, Determinantal formulas for orthogonal and symplectic degeneracy loci , J. Differential Geom. 43 (1996), no. 2, 276-290. · Zbl 0911.14001
[10] William Graham, The class of the diagonal in flag bundles , J. Differential Geom. 45 (1997), no. 3, 471-487. · Zbl 0935.14015
[11] James E. Humphreys, Introduction to Lie algebras and representation theory , Springer-Verlag, New York, 1972, Grad. Texts in Math. 9. · Zbl 0254.17004
[12] James E. Humphreys, Reflection groups and Coxeter groups , Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. · Zbl 0725.20028
[13] Victor G. Kac, Infinite-dimensional Lie algebras , Cambridge University Press, Cambridge, 1990, 3d ed. · Zbl 0716.17022
[14] Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of \(G/P\) for a Kac-Moody group \(G\) , Adv. in Math. 62 (1986), no. 3, 187-237. · Zbl 0641.17008 · doi:10.1016/0001-8708(86)90101-5
[15] Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Flag varieties and the Yang-Baxter equation , Lett. Math. Phys. 40 (1997), no. 1, 75-90. · Zbl 0918.17013 · doi:10.1023/A:1007307826670
[16] Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert , C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447-450. · Zbl 0495.14031
[17] A. Lascoux and M.-P. Schützenberger, Interpolation de Newton à plusieurs variables , Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin, 36ème année (Paris, 1983-1984), Lecture Notes in Math., vol. 1146, Springer, Berlin, 1985, pp. 161-175. · Zbl 0584.41002
[18] A. Lasku and M.-P. Shyuttsenberzhe, Symmetrization operators in polynomial rings , Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 77-78, in Russian. · Zbl 0659.13008 · doi:10.1007/BF01077811
[19] Alain Lascoux and Marcel-Paul Schützenberger, Décompositions dans l’algèbre des différences divisées , Discrete Math. 99 (1992), no. 1-3, 165-179. · Zbl 0784.05060 · doi:10.1016/0012-365X(92)90372-M
[20] I. Macdonald, Notes on Schubert Polynomials , Publ. LACIM, vol. 6, Université du Québec, Montréal, 1991. · Zbl 0784.05061
[21] P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; \(\~Q\)-polynomial approach , Compositio Math. 107 (1997), no. 1, 11-87. · Zbl 0916.14026 · doi:10.1023/A:1000182205320
[22] M. Shimozono, private communication . · Zbl 1151.17009
[23] J. Stembridge, Coxeter-Yang-Baxter equations , manuscript, February, 1993.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.