Skip to main content
SpringerLink
Log in

An Outline of Polar Spaces: Basics and Advances

  • 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))

  • 922 Accesses

Abstract

This paper is an extended version of a series of lectures on polar spaces given during the workshop and conference ‘Groups and Geometries’  held at the Indian Statistical Institute in Bangalore in December 2012. We firstly give a concise exposition of the theory of polar spaces, ending up with the classification of polar spaces of rank at least \(3\). Then we present a few related research topics, as polar spaces of infinite rank, non-linear embeddings of polar spaces, projective embeddings of dual polar spaces and polar grassmannians.

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. Adamovich, A.M.: The submodule lattice of Weyl modules for symplectic groups with fundamental highest weights. Mosc. Univ. Math. Bull. 41, 6–9 (1986)

    MATH  Google Scholar 

  2. Blok, R.J.: The generating rank of the symplectic line-Grassmannian. Beiträge Algebra Geom. 44, 575–580 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Blok, R.J.: The generating rank of the symplectic Grassmannians: hyperbolic and isotropic geometry. European J. Combin. 28, 1368–1394 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Blok, R.J., Brouwer, A.E.: The geometry far from a residue.In: Groups and Geometries (Siena: (1996), pp. 29–38. Trends Math , Birkhäuser, Basel, (1998)

    Google Scholar 

  5. Blok, R.J., Brouwer, A.E.: Spanning point-line geometries in buildings of spherical type. J. Geom. 62, 26–35 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Blok, R.J., Cooperstein, B.N.: Projective subgrassmannians of polar Grassmannian. Bull. Belg. Math. Soc. Simon Stevin 17(4), 675–691 (2010)

    Google Scholar 

  7. Blok, R.J., Cooperstein, B.N.: The generating rank of the unitary and symplectic grassmannians. J. Combin. Theory Ser. A119(1), 1–13 (2012)

    Google Scholar 

  8. Blok, R.J., Cardinali, I., De Bruyn, B.: On the nucleus of the Grassmann embedding of the symplectic dual polar space \(DSp(2n,\mathbb{F})\), \(\text{ char }(\mathbb{F})=2\). European J. Combin. 30, 468–472 (2009)

    Google Scholar 

  9. Blok, R.J., Cardinali, I., De Bruyn, B., Pasini, A.: Polarized and homogeneous embeddings of dual polar spaces. J. Alg. Combin. 30, 381–399 (2009)

    Article  MATH  Google Scholar 

  10. Blokhuis, A., Brouwer, A.E.: The universal embedding dimension of the binary symplectic dual polar space. Discrete Math. 264, 3–11 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bourbaki, N., Algèbre, Chp. 9: Formes sesquilinéares et formes quadratiques. Actu. Sci. Ind. n 1272, Hermann, Paris, (1959)

    Google Scholar 

  12. Brouwer, A.E., Wilbrink, H.A.: The structure of near polygons with quads. Geom. Dedicata 14, 145–176 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  13. Buekenhout, F.: On the foundations of polar geometry II. Geom. Dedicata 33, 21–26 (1990)

    Google Scholar 

  14. Buekenhout, F., Cameron, P.J.: Projective and affine geometry over division rings. (Chapter 2) of the In: F. Buekenhout, F.(ed. ) “Handbook of Incidence Geometry" Elsevier, Amsterdam (1995)

    Google Scholar 

  15. Buekenhout, F., Cohen, A.: Diagram geometry: related to classical groups and buildings. Springer, Berlin (2013)

    Book  Google Scholar 

  16. Buekenhout, F., Lefèvre-Percsy, C.: Generalized quadrangles in projective spaces. Arch. Math. (Basel) 25, 540–552 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  17. F. Buekenhout and E. E. Shult. On the foundations of polar geometry. Geom. Dedicata 3, 155–170 (1974)

    Google Scholar 

  18. Buekenhout, F., Sprague, A.: Polar spaces having some line of cardinality two. J. Combin. Theory Ser. A33(2), 223–228 (1982)

    Google Scholar 

  19. Cameron, P.J.: Projective and polar spaces. QMW Maths Notes, 13. Queen Mary and Westfield College, School of Mathematical Sciences, London, viii + 147pp (1990)

    Google Scholar 

  20. Cardinali, I., Lunardon, G.: A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3. J. Combin. Theory Ser. A 115, 1056–1064 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Cardinali, I., Pasini, A.: On Weyl modules for the symplectic group. Innov. Incidence Geom. 12, 85–110 (2011)

    MATH  MathSciNet  Google Scholar 

  22. Cardinali, I., Pasini, A.: Veronesean embeddings of dual polar spaces of orthogonal type. J. Combin. Th. A 120, 1328–1350 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Cardinali, I., Pasini, A.: Grassmann and Weyl embeddings of orthogonal grassmannians. J. Algebraic Combin. 38,863–888 (2013)

    Google Scholar 

  24. Cardinali, I., Pasini, A.: Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties. (this volume)

    Google Scholar 

  25. Cardinali, I., Payne, S.E.: \(q\)-Clan Geometries in Characteristic 2. Birkhäuser Verlag, Basel (2007)

    MATH  Google Scholar 

  26. Cardinali, I., De Bruyn, B., Pasini, A.: Minimal full polarized embeddings of dual polar spaces. J. Algebraic Combin. 25, 7–23 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Cardinali, I., De Bruyn, B., Pasini, A.: On the simple connectedness of hyperplane complements in dual polar spaces. Discrete Math. 309, 294–303 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Cooperstein, B.N.: On the generation of dual polar spaces of unitary type over finite fields. European J. Combin. 18, 849–856 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  29. Cooperstein, B.N.: On the generation of dual polar spaces of symplectic type over finite fields. J. Combin. Th. A83, 221–232 (1998)

    Google Scholar 

  30. Cooperstein, B.N.: Generation of Lie Incidence Geometries: a Survey (this volume)

    Google Scholar 

  31. Cooperstein, B.N., Shult, E.E.: Frames and bases of Lie incidence geometries. J. Geometry 60, 17–46 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  32. Cooperstein, B.N., Shult, E.E.: A note on embedding and generating dual polar spaces. Adv. Geom. 1, 37–48 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Cuypers, H., Pasini, A.: Locally polar geometries with affine planes. European J. Combin. 13, 39–57 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  34. Cuypers, H., Johnson, P.M., Pasini, A.: On the embeddability of polar spaces. Geom. Dedicata 44, 349–358 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  35. De Bruyn, B.: A decomposition of the natural embedding spaces for the symplectic dual polar spaces. Linear Algebra Appl. 426, 462–477 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. De Bruyn, B.: On the Grassmann-embeddings of the hermitian dual polar spaces. Linear Multilinear Algebra 56, 665–677 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. De Bruyn, B.: The structure of the spin-embeddings of dual polar spaces and related geometries. European J. Combin. 29, 1242–1256 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  38. De Bruyn, B.: On the Grassmann modules for the symplectic groups. J. Algebra 324(2), 218–230 (2010)

    Google Scholar 

  39. De Bruyn, B.: A note on the spin-embedding of the dual polar space \(DQ^-(2n+1,\mathbb{K})\). Ars Combin. 99, 365–375 (2011)

    MATH  MathSciNet  Google Scholar 

  40. De Bruyn, B., Pasini, A.: Generating symplectic and Hermitian dual polar spaces over arbitrary fields nonisomorphic to \(\mathbb{F}_2\). Electron. J. Combin. 14 R54, pp.1–17 (2007)

    Google Scholar 

  41. De Bruyn, B. Pasini, A.: Minimal scattered sets and polarized embeddings of dual polar spaces. European J. Combin. 28(7):1890–1909 (2007)

    Google Scholar 

  42. De Bruyn, B., Pasini, A.: On symplectic polar spaces over non-perfect fields of characteristic 2. Linear and Multilinear Algebra 57, 567–575 (2009)

    Google Scholar 

  43. De Bruyn, B., Van Maldeghem, H.: Non-embeddable polar spaces. Münster J. Math. (To appear)

    Google Scholar 

  44. De Bruyn, B., Van Maldeghem, H.: Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras. J. Reine Angew. (To appear)

    Google Scholar 

  45. Dieudonné, J.: Sur les groupes classiques. Actualits Sci. Ind., no. 1040 = Publ. Inst. Math. Univ. Strasbourg (N.S.) 1, (1945)

    Google Scholar 

  46. J. Dieudonné. La géométrie des groupes classiques. Ergebnisse der Math. N.F. Bd.5, 2de éd, Springer, Berlin (1963)

    Google Scholar 

  47. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer-Verlag, New York (1972)

    Book  MATH  Google Scholar 

  48. Ivanov, A.A.: Non-abelian representation of geometries. Groups and combinatorics-In: memory of Michio Suzuki, Adv. Stud. Pure Math., 32, Math. Soc. 301–314, Japan, Tokyo (2001)

    Google Scholar 

  49. Ivanov, A.A., Shpectorov, S.V.: Geometry of sporadic groups. II. Representations and amalgams. Encyclopedia of Mathematics and its Applications, 91, Cambridge University Press, Cambridge (2002)

    Google Scholar 

  50. Johnson, P.M.: Polar spaces of arbitrary rank. Geom. Dedicata 35, 229–250 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  51. Kasikova, A., Shult, E.E.: Absolute embeddings of point-line geometries. J. Algebra 238, 265–291 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  52. Li, P.: On the universal embedding of the \(Sp_{2n}(2)\) dual polar space. J. Combin. Theory Ser. A94, 100–117 (2001)

    Google Scholar 

  53. Li, P.: On the universal embedding of the \(U_{2n}(2)\) dual polar space. J. Combin. Theory Ser. A98, 235–252 (2002)

    Google Scholar 

  54. Mühlherr, B.: A geometric approach to non-embeddable polar spaces of rank 3. Algebra, groups and geometry. Bull. Soc. Math. Belg. Sr.A42(3):577–594 (1990)

    Google Scholar 

  55. Pankov, M.: Grassmannians of classical buildings. Algebra and Discrete Mathematics, 2. World Scientific Publishing Co., Pte. Ltd., Hackensack, NJ (2010)

    Google Scholar 

  56. Pasini, A.: Diagram Geometries. Oxford University Press, Oxford (1994)

    MATH  Google Scholar 

  57. Pasini, A.: Embeddings and expansions. Bull. Belg. Math. Soc. Simon Stevin 10, 585–626 (2003)

    MATH  MathSciNet  Google Scholar 

  58. Pasini, A.: On polar spaces of infinite rank. J. Geom. 91, 1–2, 84–118 (2009)

    Google Scholar 

  59. Pasini, A.: Embeddings of Orthogonal Grassmannians. Innov. Incidence Geom. 13, 107–133 (2013)

    Google Scholar 

  60. Payne, S.E., Thas, J.A.: Finite Generalized quadrangles, 2nd edn. EMS Series of Lectures in Mathematics, Zürich (2009)

    Book  MATH  Google Scholar 

  61. Premet, A.A., Suprunenko, I.D.: The Weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights. Comm. Algebra 11, 1309–1342 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  62. Ronan, M.A.: Embeddings and hyperplanes of discrete geometries. European J. Combin. 8, 179–185 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  63. Sahoo, B.K., Sastry, N.S.N.: A characterization of finite symplectic polar spaces of odd prime order. J. Combin. Theory Ser. A 114, 52–64 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  64. Shult, E.E.: Points and lines. Characterizing the classical geometries. Universitext. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  65. Shult, E.E.: Embeddings and hyperplanes of Lie incidence geometries. In: Di Martino, L., Kantor, W. (eds.) Groups of Lie Type and their Geometries, pp. 215–232 Cambridge Univ. Press, Cambridge (1995)

    Google Scholar 

  66. Shult, E.E.: On Veldkamp lines. Bull. Belg. Math. Soc. Simon Stevin 4, 299–316 (1997)

    MATH  MathSciNet  Google Scholar 

  67. Shult, E.E., Yanushka, A.: Near \(n\)-gons and line systems. Geom. Dedicata 9, 1–72 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  68. Thas, K.: A course on elation quadrangles. E.M.S. Publishing House, Zürich (2012)

    Book  MATH  Google Scholar 

  69. Thas, J.A. Van Maldeghem, H.: Embeddings of small generalized polygons. Finite Fields Appl. 12, 4, 565–594 (2006)

    Google Scholar 

  70. Thas, J.A., Thas, K., Van Maldeghem, H.: Translation generalized quadrangles. Ser. Pure Math. 26, World Scientific, Singapore (2006)

    Google Scholar 

  71. Tits, J.: Les groupes de Lie exceptionnels et leur interprétation géométrique. Bull. Soc. Math. Belg. 8, 48–81 (1956)

    MATH  MathSciNet  Google Scholar 

  72. Tits. J.: Buildings of Spherical type and Finite \(BN\)-pairs. In: Lecture Notes in Mathematics, vol. 386 Springer, Berlin (1974)

    Google Scholar 

  73. Tits, J., Weiss, R.M.: Moufang Polygons. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  74. Ueberberg, J.: Foundations of incidence geometry. Projective and polar spaces. Springer Monographs in Mathematics. Springer, Heidelberg (2011)

    Google Scholar 

  75. Van Maldeghem, H.: Generalized polygons. In: Monographs in Mathematics, vol. 93. Birkhäuser, Basel (1998)

    Google Scholar 

  76. Veldkamp. F.D.: Polar geometry. I, II, III, IV, V. Nederl. Akad. Wetensch. In: Proc. Ser. A 62; 63 = Indag. Math. 21, (1959)

    Google Scholar 

  77. Weiss, R.M.: The Structure of Spherical Buildings. Princeton Univ. Press, Princeton (2003)

    Google Scholar 

  78. Wells, A.: Universal projective embeddings of the Grassmannian, half spinor, and dual orthogonal geometries. Quart. J. Math. Oxford 34, 375–386 (1983)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, 53100, Siena, Italy

    Ilaria Cardinali

Authors
  1. Ilaria Cardinali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilaria Cardinali .

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

Cardinali, I. (2014). An Outline of Polar Spaces: Basics and Advances. 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_3

Download citation

Publish with us

Policies and ethics

Search

Navigation