Rigidity of Bott–Samelson–Demazure–Hansen variety for \({\varvec{F}}_{\varvec{4}}\) and \({\varvec{G}}_{\varvec{2}}\)

Abstract

Let G be a simple algebraic group of adjoint type over \({\mathbb {C}},\) whose root system is of type \(F_{4}.\) Let T be a maximal torus of G and B be a Borel subgroup of G containing T. Let w be an element of the Weyl group W and X(w) be the Schubert variety in the flag variety G/B corresponding to w. Let \(Z(w, {\underline{i}})\) be the Bott–Samelson–Demazure–Hansen variety (the desingularization of X(w)) corresponding to a reduced expression \({\underline{i}}\) of w. In this article, we study the cohomology modules of the tangent bundle on \(Z(w_{0}, {\underline{i}}),\) where \(w_{0}\) is the longest element of the Weyl group W. We describe all the reduced expressions of \(w_{0}\) in terms of a Coxeter element such that \(Z(w_{0}, {\underline{i}})\) is rigid (see Theorem 7.1). Further, if G is of type \(G_{2},\) there is no reduced expression \({\underline{i}}\) of \(w_{0}\) for which \(Z(w_{0}, {\underline{i}})\) is rigid (see Theorem 8.2).

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. 1.

    Balaji V, Senthamarai Kannan S and Subrahmanyam K V, Cohomology of line bundles on Schubert varieties – I, Transformation Groups 9(2) (2004) 105–131

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bott R and Samelson H, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958) 964–1029

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brion M and Kumar S, Frobenius Splitting Methods in Geometry and Representation theory, Progress in Mathematics, vol. 231 (2005) (Boston, MA: Birkhäuser, Boston Inc.)

  4. 4.

    Demazure M, Desingularisation des varieties de Schubert generalisees, Ann. Sci. Ecole Norm. Sup. 7 (1974) 53–88

    Article  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    Hansen H C, On cycles on flag manifolds, Math. Scand. 33 (1973) 269–274

    MathSciNet  Article  Google Scholar 

  7. 7.

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

  8. 8.

    Humphreys J E, Linear Algebraic Groups (1975) (Berlin: Springer)

  9. 9.

    Humphreys J E, Reflection Groups and Coxeter Groups, vol. 29 (1992) (Cambridge: Cambridge University Press)

  10. 10.

    Humphreys J E, Conjugacy classes in semisimple algebraic groups, Math. Surveys Monographs, vol. 43 (1995) (Amer. Math. Soc.)

  11. 11.

    Huybrechts D, Complex Geometry: An Introduction (2005) (Berlin: Springer)

  12. 12.

    Jantzen J C, Representations of Algebraic Groups, second edition, Mathematical Surveys and Monographs, vol. 107 (2003)

  13. 13.

    Narasimha Chary B, Senthamarai Kannan S and Parameswaran A J, Automorphism group of a Bott–Samelson–Demazure–Hansen variety, Transformation Groups 20(3) (2015) 665–698

    MathSciNet  Article  Google Scholar 

  14. 14.

    Narasimha Chary B and Senthamarai Kannan S, Rigidity of Bott–Samelson–Demazure–Hansen variety for $PSp(2n,{\mathbb{C}}),$ J. Lie Theory 27(2) (2017) 435–468

    MathSciNet  Google Scholar 

  15. 15.

    Senthamarai Kannan S, On the automorphism group of a smooth Schubert variety, Algebr. Represent. Theory 19(4) (2016) 761–782.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Senthamarai Kannan S and Saha Pinakinath, Rigidity of Bott–Samelson–Demazure–Hansen variety for $PSO(2n+1,{\mathbb{C}})$, preprint

  17. 17.

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

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Infosys Foundation for the partial financial support.

Author information

Affiliations

Authors

Corresponding author

Correspondence to S Senthamarai Kannan.

Additional information

Communicating Editor: Nitin Nitsure

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kannan, S.S., Saha, P. Rigidity of Bott–Samelson–Demazure–Hansen variety for \({\varvec{F}}_{\varvec{4}}\) and \({\varvec{G}}_{\varvec{2}}\). Proc Math Sci 130, 19 (2020). https://doi.org/10.1007/s12044-019-0535-3

Download citation

Keywords

  • Bott–Samelson–Demazure–Hansen variety
  • coexeter element
  • tangent bundle

2000 Mathematics Subject Classification

  • 14M15