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Izv. Akad. Nauk SSSR Ser. Mat., 1974, Volume 38, Issue 2, Pages 294–322 (Mi izv1903)  

This article is cited in 40 scientific papers (total in 40 papers)

Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles

V. L. Popov

Abstract: We construct models of finite-dimensional linear and projective irreducible representations of a connected semisimple group $G$ in linear systems on the variety $G$. We establish an algebro-geometric criterion for the linearizability of an irreducible projective representation of $G$. We explain the algebro-geometric meaning of the numerical characteristic of an arbitrary rational character of a maximal torus of $G$. Using these results we compute the Picard group of an arbitrary homogeneous space of any connected linear algebraic group $H$, prove the homogeneity of an arbitrary one-dimensional algebraic vector bundle over such a space relative to some covering group of $H$, and compute the Chern class of such a bundle.

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English version:
Mathematics of the USSR-Izvestiya, 1974, 8:2, 301–327

Bibliographic databases:

UDC: 519.4
MSC: Primary 14C20, 14M15, 20G05; Secondary 14F05, 32M10
Received: 18.04.1973

Citation: V. L. Popov, “Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles”, Izv. Akad. Nauk SSSR Ser. Mat., 38:2 (1974), 294–322; Math. USSR-Izv., 8:2 (1974), 301–327

Citation in format AMSBIB
\by V.~L.~Popov
\paper Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1974
\vol 38
\issue 2
\pages 294--322
\jour Math. USSR-Izv.
\yr 1974
\vol 8
\issue 2
\pages 301--327

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    This publication is cited in the following articles:
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    3. George Kempf, “The Grothendieck-Cousin complex of an induced representation”, Advances in Mathematics, 29:3 (1978), 310  crossref
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    7. D. N. Akhiezer, “Actions with a finite number of orbits”, Funct. Anal. Appl., 19:1 (1985), 1–4  mathnet  crossref  mathscinet  zmath  isi
    8. D. I. Panyushev, “A problem on the Steinberg conjecture”, Funct. Anal. Appl., 19:2 (1985), 152–153  mathnet  crossref  mathscinet  zmath  isi
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    19. Vladimir L. Popov, “Tensor product decompositions and open orbits in multiple flag varieties”, Journal of Algebra, 313:1 (2007), 392  crossref
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    21. V. S. Zhgoon, “Variation of Mumford quotients by torus actions on full flag varieties. II”, Sb. Math., 199:3 (2008), 341–359  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    23. I. V. Arzhantsev, “On the Factoriality of Cox rings”, Math. Notes, 85:5 (2009), 623–629  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    24. I. V. Arzhantsev, “Projective embeddings of homogeneous spaces with small boundary”, Izv. Math., 73:3 (2009), 437–453  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    25. A. Polishchuk, “K-theoretic exceptional collections at roots of unity”, J K-Theory, 2010, 1  crossref
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    27. R. S. Avdeev, “Extended weight semigroups of affine spherical homogeneous spaces of non-simple semisimple algebraic groups”, Izv. Math., 74:6 (2010), 1103–1126  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    28. Kiumars Kaveh, A. G. Khovanskii, “Newton polytopes for horospherical spaces”, Mosc. Math. J., 11:2 (2011), 265–283  mathnet  mathscinet
    29. Kishimoto T. Prokhorov Yu. Zaidenberg M., “Group Actions on Affine Cones”, Affine Algebraic Geometry: the Russell Festschrift, CRM Proceedings & Lecture Notes, 54, ed. Daigle D. Ganong R. Koras M., Amer Mathematical Soc, 2011, 123–163  isi
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