Skip to main content
SpringerLink
Log in

Locally convex curves and the Bruhat stratification of the spin group

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study the lifting of the Schubert stratification of the homogeneous space of complete real flags of ℝn+1 to its universal covering group Spinn+1. We call the lifted strata the Bruhat cells of Spinn+1, in keeping with the homonymous classical decomposition of reductive algebraic groups. We present explicit parameterizations for these Bruhat cells in terms of minimal-length expressions \(\sigma = {a_{{i_1}}} \cdots {a_{{i_k}}}\) for permutations σ ∈ Sn+1 in terms of the n generators ai = (i, i + 1). These parameterizations are compatible with the Bruhat orders in the Coxeter—Weyl group Sn+1. This stratification is an important tool in the study of locally convex curves; we present a few such applications.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Alves and N. Saldanha, Results on the homotopy type of the spaces of locally convex curves on \(\mathbb{S}^{3}\), Université de Grenoble. Annales de l’Institut Fourier 69 (2019), 1147–1185.

    Article  MathSciNet  Google Scholar 

  2. T. Ando, Totally positive matrices, Linear Algebra and its Applications 90 (1987), 165–219.

    Article  MathSciNet  Google Scholar 

  3. M. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (1964), 3–38.

    Article  MathSciNet  Google Scholar 

  4. A. Berenstein, S. Fomin and A. Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Advances in Mathematics 122 (1996), 49–149.

    Article  MathSciNet  Google Scholar 

  5. I. Bernstein, I. Gelfand and S. Gelfand, Schubert cells and cohomology of the spaces G/P, Russian Mathematical Surveys 28 (1973), 1–26.

    Article  MathSciNet  Google Scholar 

  6. A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, Vol. 231, Springer, New York, 2005.

    MATH  Google Scholar 

  7. F. Brenti, Combinatorics and total positivity, Journal of Combinatorial Theory. Series A 71 (1995), 175–218.

    Article  MathSciNet  Google Scholar 

  8. C. Chevalley, Sur les décompositions cellulaires des espaces G/B, in Algebraic Groups and their Generalizations: Classical Methods, Proceedings of Symposia in Pure Mathematics, Vol. 56, American Mathematical Society, Providence, RI, 1994, pp. 1–23.

    Book  Google Scholar 

  9. M. Demazure, Désingularization des variétés de Schubert généralisées, Annales Scientifiques de l’École Normale Supérieure 7 (1974), 53–88.

    Article  MathSciNet  Google Scholar 

  10. S. Elnitsky, Rhombic tilings of polygons and classes of reduced words in Coxeter groups, Journal of Combinatorial Theory. Series A 77 (1997), 193–221.

    Article  MathSciNet  Google Scholar 

  11. S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, Journal of the American Mathematical Society 12 (1999), 335–380.

    Article  MathSciNet  Google Scholar 

  12. S. Fomin and A. Zelevinsky, Total positivity: Tests and parametrizations, Mathematical Intelligencer 22 (2000), 23–33.

    Article  MathSciNet  Google Scholar 

  13. F. Gantmacher and M. Krein, Sur les matrices oscillatoires, Comptes rendus hebdomadaires des séances de l’Académie des Sciences 201 (1935), 577–579.

    MATH  Google Scholar 

  14. V. Goulart, Towards a combinatorial approach to the topology of spaces of nondegenerate spherical curves, PhD thesis, Pontifícia Universidade Católica do Rio de Janeiro, 2016.

  15. V. Goulart and N. Saldanha, Combinatorialization of spaces of nondegenerate spherical curves, https://arxiv.org/abs/1810.08632.

  16. V. Goulart and N. Saldanha, Stratification by itineraries of spaces of locally convex curves, https://arxiv.org/abs/1907.01659.

  17. J. Humphreys, Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.

    Book  Google Scholar 

  18. S. Karlin, Total Positivity. Vol. 1, Stanford University Press, Stanford, CA, 1968.

    MATH  Google Scholar 

  19. B. Khesin and V. Ovsienko, Symplectic leaves of the Gelfand—Dickey brackets and homotopy classes of nondegenerate curves, Funktsional’nyi Analiz i ego Prilozheniya 24 (1990), 38–47.

    MATH  Google Scholar 

  20. B. Khesin and B. Shapiro, Homotopy classification of nondegenerate quasiperiodic curves on the 2-sphere, Institut Mathématique. Publications 66 (1999), 127–156.

    MathSciNet  MATH  Google Scholar 

  21. B. Kostant, Lie algebra cohomology and generalized Schubert cells, Annals of Mathematics 77 (1963), 72–144.

    Article  MathSciNet  Google Scholar 

  22. H. Lawson and M. Michelsohn, Spin Geometry, Princeton Mathematical Series, Vol. 38, Princeton University Press, Princeton, NJ, 1989.

    MATH  Google Scholar 

  23. J. Little, Nondegenerate homotopies of curves on the unit 2-sphere, Journal of Differential Geometry 4 (1970), 339–348.

    Article  MathSciNet  Google Scholar 

  24. G. Lusztig, Total positivity in reductive groups, in Lie Theory and Geometry, Progress in Mathematics, Vol. 123, Birkhäuser, Basel, 1994, pp. 531–568.

    Chapter  Google Scholar 

  25. G. Lusztig, Total positivity in partial flag manifolds, Representation Theory 2 (1998), 70–78.

    Article  MathSciNet  Google Scholar 

  26. G. Lusztig, A survey of total positivity, Milan Journal of Mathematics 76 (2008), 125–134.

    Article  MathSciNet  Google Scholar 

  27. A. Postnikov, Positive Grassmannian and polyhedral subdivisions, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited Lectures, World Scientific, Hackensack, NJ, 2018, pp. 3181–3211.

    Google Scholar 

  28. A. Postnikov, Total positivity, Grassmannians, and networks, https://arxiv.org/abs/math/0609764.

  29. K. Rietsch, Intersections of Bruhat cells in real flag varieties, International Mathematics Research Notices 13 (1997), 623–640.

    Article  MathSciNet  Google Scholar 

  30. N. Saldanha, The homotopy type of spaces of locally convex curves in the sphere, Geometry & Topology 19 (2015), 1155–1203.

    Article  MathSciNet  Google Scholar 

  31. N. Saldanha and B. Shapiro, Spaces of locally convex curves in \(\mathbb{S}^{n}\) and combinatorics of the group B +n+1 , Journal of Singularities 4 (2012), 1–22.

    MathSciNet  MATH  Google Scholar 

  32. N. Saldanha, B. Shapiro and M. Shapiro, Grassmann convexity and multiplicative Sturm theory, revisited, https://arxiv.org/abs/1902.09741.

  33. I. Schoenberg, Uber variationsvermindende lineare Transformationen, Mathematische Zeitschrift 32 (1930), 321–328.

    Article  MathSciNet  Google Scholar 

  34. B. Shapiro and M. Shapiro, Projective convexity in ℙ3implies Grassmann convexity, International Journal of Mathematics 11 (2000), 579–588.

    Article  MathSciNet  Google Scholar 

  35. B. Shapiro and M. Shapiro, Linear ordinary differential equations and Schubert calculus, in Proceedings of the Gökova Geometry-Topology Conference. 2010, International Press, Somerville, MA, 2011, pp. 79–87.

    Google Scholar 

  36. B. Shapiro, M. Shapiro and A. Vainshtein, On the number of connected components in the intersection of two open opposite Schubert cells in SLn(ℝ)/B, International Mathematics Research Notices 10(1997), 469–493.

    Article  Google Scholar 

  37. B. Shapiro, M. Shapiro and A. Vainshtein, Skew-symmetric vanishing lattices and intersections of Schubert cells, International Mathematics Research Notices 11 (1998), 563–588.

    Article  MathSciNet  Google Scholar 

  38. D. Verma, Structure of certain induced representations of complex semisimple Lie algebras, Bulletin of the American Mathematical Society 74 (1968), 160–166.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Espírito Santo, Av. Fernando Ferrari 514, Campus de Goiabeiras, Vitória, ES, 29075-910, Brazil

    Victor Goulart

  2. Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, R. Marquês de S. Vicente 255, Rio de Janeiro, RJ, 22451-900, Brazil

    Victor Goulart & Nicolau C. Saldanha

Authors
  1. Victor Goulart
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolau C. Saldanha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Goulart.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Goulart, V., Saldanha, N.C. Locally convex curves and the Bruhat stratification of the spin group. Isr. J. Math. 242, 565–604 (2021). https://doi.org/10.1007/s11856-021-2127-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2127-z

Search

Navigation