Skip to main content
SpringerLink
Log in

GIT Related Problems of the Flag Variety for the Action of a Maximal Torus

  • Conference paper
  • First Online:
Groups of Exceptional Type, Coxeter Groups and Related Geometries

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 82))

Abstract

Let \(G\) be a simple adjoint group over the field \(\mathbb C\) of complex numbers. Let \(T\) be a maximal torus of \(G\). Let \(P\) be a parabolic subgroup of \(G\). In this article, we give a survey on the Geometric Invariant Theory related problems for the left action of \(T\) on \(G/P\).

Dedicated to Professor N.S.N. Sastry on the occasion of his 60th birth day.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Andersen, H.H.: Schubert varieties and Demazure’s character formula. Invent. Math. 79, 611–618 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Balaji, V., Kannan, S.S., Subrahmanyam, K.V.: Cohomology of line bundles on Schubert varieties-I. Transform. Groups 9(2), 105–131 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Billey, S.C., Lakshmibai, V.: Singular Loci of Schubert Varieties, Progress in Mathematics, vol. 182. Birkhäuser, Boston (2000)

    Google Scholar 

  4. Borel, A.: Linear representations of semi-simple algebraic groups. Proc. Symp. Pure Math. 29, 421–440 (1975)

    Article  MathSciNet  Google Scholar 

  5. Bott, R.: Homogeneous vector bundles. Ann. Math. Ser. 2(66), 203–248 (1957)

    Google Scholar 

  6. Chevalley, C.: Invariants of finite groups generated by reflections. Amer. J. Math. 77, 778–782 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dabrowski, R.: On normality of the closure of a generic torus orbit in \(G/P\). Pacific J. Math. 172(2), 321–330 (1996)

    MATH  MathSciNet  Google Scholar 

  8. Demazure, M.: A very simple proof of Bott’s theorem. Invent. Math. 33, 271–272 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  9. Howard, B.J.: Matroids and geometric invariant theory of torus actions on flag spaces. J. Algebra 312, 527–541 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  11. Humphreys, J.E.: Linear Algebraic Groups. Springer, New York (1975)

    Google Scholar 

  12. Humphreys, J.E.: Conjugacy classes in semisimple algebraic groups, Math. Surveys Monographs. Amer. Math. Soc. 43, 2961–2988 (1995)

    Google Scholar 

  13. Jantzen, J.C.: Representations of Algebraic Groups, Pure and Applied Mathematics. Academic Press, Orlando (1987)

    Google Scholar 

  14. Kannan, S.S.: Torus quotients of homogeneous spaces. Proc. Indian Acad. Sci. (Math. Sci.) 108(1), 1–12 (1998)

    Google Scholar 

  15. Kannan, S.S.: Torus quotients of homogeneous spaces-II, Proc. Indian Acad. Sci. (Math. Sci.) 109(1), 23–39 (1999)

    Google Scholar 

  16. Kannan, S.S.: Cohomology of line bundles on Schubert varieties in the Kac-Moody setting. J. Algebra 310, 88–107 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kannan, S.S., Sardar, P.: Torus quotients of homogeneous spaces of the general linear group and the standard representation of certain symmetric groups. Proc. Indian Acad. Sci. (Math. Sci.) 119(1), 81–100 (2009)

    Google Scholar 

  18. Kannan, S.S., Pattanayak, S.K.: Torus quotients of homogeneous spaces-minimal dimensional Schubert varieties admitting semi-stable points. Proc. Indian Acad. Sci. (Math. Sci.) 119(4), 469–485 (2009)

    Google Scholar 

  19. Kannan, S.S., Chary, B.N., Pattanayak, S.K.: Torus invariants of the homogeneous coordinate ring of \(G/B\)—connection with Coxeter elements. Commun. Algebra 42(5), 1880–1895 (2014)

    Google Scholar 

  20. Kannan, S.S.: On the automorphism group of a smooth Schubert variety (2014, preprint), http://www.arxiv.org/abs/1312.7066

  21. Kannan, S.S., Chary, B.N.: Top cohomology of a Schubert variety with trivial \(B\)-module. preprint (2014)

    Google Scholar 

  22. Kumar, S.K.: Descent of line bundles to GIT quotients of flag varieties by maximal torus. Transform. Groups 13(3–4), 757–771 (2008)

    Google Scholar 

  23. Littelmann, P.: Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras. J. Amer. Math.Soc. 11(3), 551–567 (1998)

    Google Scholar 

  24. Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. Math. (2) 122(1), 27–40 (1985)

    Google Scholar 

  25. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Springer, New York (1994)

    Book  Google Scholar 

  26. Newstead, P.E.: Introduction to Moduli Problems and Orbit Spaces, TIFR Lecture Notes, Tata Institute of Fundamental Research, Bombay (1978)

    Google Scholar 

  27. Onishchik, A.L.: On compact Lie groups transitive on certain manifolds. Sov. Math. Dokl. 1, 1288–1291 (1961)

    MATH  Google Scholar 

  28. Ramanan, S., Ramanathan, A.: Projective normality of flag varieties and Schubert varieties. Invent. Math. 79(2), 217–224 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. Serre, J.P.: Groupes finis d’automorphisms d’anneaux locaux reguliers, pp. 1–11. Colloq. d’Alg. Ecole Norm. de Jeunes Filles, Paris (1967)

    Google Scholar 

  30. Seshadri, C.S.: Mumford’s conjecture for \(GL(2)\) and applications, International Colloquim on Algebraic Geometry, Bombay, (1968)

    Google Scholar 

  31. Seshadri, C.S.: Quotient Spaces modulo reductive algebraic groups. Ann. Math. 95, 511–556 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  32. Seshadri, C.S.: Introduction to the Theory of Standard Monomials, Texts and Readings in Mathematics 46. Hindustan Book Agency, New Delhi (2007)

    Google Scholar 

  33. Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274–304 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  34. Skorobogatov, A.N.: Automorphisms and forms of toric quotients of homogeneous spaces (Russian). Mat. Sb. 200(10), 107–122 (2009); translation in Sb. Math. 200(9–10), 1521–1536 (2009)

    Google Scholar 

  35. Steinberg, R.: Regular elements of semisimple algebraic groups. Inst. des Hautes Etudes Sci. Publ. Math. 25, 49–80 (1965)

    Article  Google Scholar 

  36. Strickland, E.: Quotients of flag varieties by a maximal torus. Math. Z. 234(1), 1–7 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  37. Tits, J.: Espaces homogenes Complexes Compact. Comm. Helv. Math. 37, 111–120 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  38. Wehlau, D.: When is a ring of torus invariants a polynomial ring. Manuscripta Math. 82, 161–170 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  39. Yang, S.W., Zelevinsky, A.: Cluster algebras of finite type via Coxeter elements and principal minors. Transform. Groups 13(3–4), 855–895 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zhgoon, V.S.: Variation of Mumford quotients by torus actions on full flag varieties. I (Russian). Izv. Ross. Akad. Nauk Ser. Mat. 71(6), 29–46 (2007); translation in Izv. Math. 71(6), 1105–1122 (2007)

    Google Scholar 

  41. Zhgoon, V.S.: Variation of Mumford quotient by torus actions on full flag varieties. II (Russian). Mat. Sb. 199(3), 25–44 (2008); translation in Sb. Math. 199(3–4), 341–359 (2008)

    Google Scholar 

Download references

Acknowledgments

We thank the referees for the useful comments. We thank Professor C.S. Seshadri for useful discussions

Author information

Authors and Affiliations

  1. Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Siruseri, Kelambakkam, 603103, Tamilnadu, India

    S. Senthamarai Kannan

Authors
  1. S. Senthamarai Kannan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Senthamarai Kannan .

Editor information

Editors and Affiliations

  1. Stat Math Unit, Indian Statistical Institute, Bangalore, Karnataka, India

    N.S. Narasimha Sastry

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer India

About this paper

Cite this paper

Kannan, S.S. (2014). GIT Related Problems of the Flag Variety for the Action of a Maximal Torus. In: Sastry, N. (eds) Groups of Exceptional Type, Coxeter Groups and Related Geometries. Springer Proceedings in Mathematics & Statistics, vol 82. Springer, New Delhi. https://doi.org/10.1007/978-81-322-1814-2_10

Download citation

Publish with us

Policies and ethics

Search

Navigation