J. H. Bernstein, I. M. Gel'fand, S. I. Gel'fand, “Schubert cells and cohomology of the spaces $G/P$”, Russian Math. Surveys, 28:3 (1973), 1

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Russian Mathematical Surveys, 1973, Volume 28, Issue 3, Pages 1–26
DOI: https://doi.org/10.1070/RM1973v028n03ABEH001557 (Mi rm4887)

This article is cited in 188 scientific papers (total in 189 papers)

Schubert cells and cohomology of the spaces

$G/P$

J. H. Bernstein, I. M. Gel'fand, S. I. Gel'fand

English version PDF (1188 kB) Citations (189) Russian version article

References:

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DOI: https://doi.org/10.1070/RM1973v028n03ABEH001557

Abstract: We study the homological properties of the factor space

$G/P$ , where

$G$ is a complex semisimple Lie group and

$P$ a parabolic subgroup of

$G$ . To this end we compare two descriptions of the cohomology of such spaces. One of these makes use of the partition of

$G/P$ into cells (Schubert cells), while the other consists in identifying the cohomology of

$G/P$ with certain polynomials on the Lie algebra of the Cartan subgroup

$H$ of

$G$ . The results obtained are used to describe the algebraic action of the Weyl group

$W$ of

$G$ on the cohomology of

$G/P$ .

Received: 13.03.1973

Bibliographic databases:

Document Type: Article

UDC: 519.4

MSC: 22E41, 22E60, 22E46

Language: English

Original paper language: Russian

Citation: J. H. Bernstein, I. M. Gel'fand, S. I. Gel'fand, “Schubert cells and cohomology of the spaces

$G/P$ ”, Russian Math. Surveys, 28:3 (1973), 1–26

Citation in format AMSBIB

\Bibitem{BerGelGel73}

\by J.~H.~Bernstein, I.~M.~Gel'fand, S.~I.~Gel'fand

\paper Schubert cells and cohomology of the spaces $G/P$

\jour Russian Math. Surveys

\yr 1973

\vol 28

\issue 3

\pages 1--26

\mathnet{http://mi.mathnet.ru/eng/rm4887}

\crossref{https://doi.org/10.1070/RM1973v028n03ABEH001557}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=429933}

\zmath{https://zbmath.org/?q=an:0289.57024|0286.57025}

Linking options:

https://www.mathnet.ru/eng/rm4887

https://doi.org/10.1070/RM1973v028n03ABEH001557

https://www.mathnet.ru/eng/rm/v28/i3/p3

This publication is cited in the following 189 articles:

Seginus Mowlavi, “Tangent spaces on the trianguline variety at companion points”, Indagationes Mathematicae, 35:1 (2024), 181
Matthew J. Samuel, “A Molev-Sagan type formula for double Schubert polynomials”, Journal of Pure and Applied Algebra, 228:7 (2024), 107636
You-Hung Hsu, “Categorical Idempotents Via Shifted 0-Affine Algebras”, Algebr Represent Theor, 27:4 (2024), 1735
Martina Lanini, Alexander Pütz, “Permutation actions on Quiver Grassmannians for the equioriented cycle via GKM-theory”, J Algebr Comb, 57:3 (2023), 915
Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann, Changjian Su, “Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells”, Duke Math. J., 172:17 (2023)
H. Duan, X. Zhao, “Schubert calculus and intersection theory of flag manifolds”, Russian Math. Surveys, 77:4 (2022), 729–751
Zerui Zhang, Yuqun Chen, “A weak version of Kirillov's conjecture on Hecke–Grothendieck polynomials”, Journal of Combinatorial Theory, Series A, 186 (2022), 105555
Duan Haibao, Zhao Xuezhi, “Make Schubert calculus rigorous”, Sci. Sin.-Math., 52:8 (2022), 865
ROBERTO MUÑOZ, GIANLUCA OCCHETTA, LUIS E. SOLÁ CONDE, “NESTINGS OF RATIONAL HOMOGENEOUS VARIETIES”, Transformation Groups, 27:1 (2022), 189
Xu-an Zhao, “Computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces”, Front. Math. China, 17:3 (2022), 437
Naoki Fujita, “Schubert calculus from polyhedral parametrizations of Demazure crystals”, Advances in Mathematics, 397 (2022), 108201
E. Yu. Smirnov, A. A. Tutubalina, “Slide polynomials and subword complexes”, Sb. Math., 212:10 (2021), 1471–1490
Chris McDaniel, Shujian Chen, Anthony Iarrobino, Pedro Macias Marques, “Free extensions and Lefschetz properties, with an application to rings of relative coinvariants”, Linear and Multilinear Algebra, 69:2 (2021), 305
Victor Goulart, Nicolau C. Saldanha, “Locally convex curves and the Bruhat stratification of the spin group”, Isr. J. Math., 242:2 (2021), 565
N. I. Shepherd-Barron, “Weyl Group Covers for Brieskorn's Resolutions in All Characteristics and the Integral Cohomology of G/P”, Michigan Math. J., 70:3 (2021)
Chris McDaniel, Larry Smith, “Equivariant coinvariant rings, Bott–Samelson rings and Watanabe's bold conjecture”, Journal of Pure and Applied Algebra, 225:5 (2021), 106524
V. G. Gorbunov, C. Korff, C. Stroppel, “Yang–Baxter algebras, convolution algebras, and Grassmannians”, Russian Math. Surveys, 75:5 (2020), 791–842
Haibao Duan, Xuezhi Zhao, Springer Proceedings in Mathematics & Statistics, 332, Schubert Calculus and Its Applications in Combinatorics and Representation Theory, 2020, 43
G. I. Lehrer, “A factorisation theorem for the coinvariant algebra of a unitary reflection group”, Arch. Math., 114:6 (2020), 631
Cristian Lenart, Kirill Zainoulline, Changlong Zhong, “PARABOLIC KAZHDAN–LUSZTIG BASIS, SCHUBERT CLASSES, AND EQUIVARIANT ORIENTED COHOMOLOGY”, J. Inst. Math. Jussieu, 19:6 (2020), 1889

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Abstract page:	2199
Russian version PDF:	898
English version PDF:	132
References:	143
First page:	5

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