Advertisement

Journal of Algebraic Combinatorics

, Volume 45, Issue 4, pp 931–963 | Cite as

A geometric approach to alternating k-linear forms

  • Ilaria Cardinali
  • Luca Giuzzi
  • Antonio Pasini
Article
  • 176 Downloads

Abstract

Denote by \({{\mathcal {G}}}_k(V)\) the Grassmannian of the k-subspaces of a vector space V over a field \({\mathbb {K}}\). There is a natural correspondence between hyperplanes H of \({\mathcal {G}}_k(V)\) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of \({{\mathcal {G}}_k}(V)\), we define a subspace \(R^{\uparrow }(H)\) of \({{\mathcal {G}}_{k-1}}(V)\) whose elements are the \((k-1)\)-subspaces A such that all k-spaces containing A belong to H. When \(n-k\) is even, \(R^{\uparrow }(H)\) might be empty; when \(n-k\) is odd, each element of \({\mathcal {G}}_{k-2}(V)\) is contained in at least one element of \(R^{\uparrow }(H)\). In the present paper, we investigate several properties of \(R^{\uparrow }(H)\), settle some open problems and propose a conjecture.

Keywords

Grassmann geometry Hyperplane Multilinear form Alternating form 

Mathematics Subject Classification

15A75 14M15 15A69 

References

  1. 1.
    Brown, R.B., Gray, A.: Vector cross products. Comment. Math. Helv. 42, 222–236 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buekenhout, F., Cohen, A. M., Diagram Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 57. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-34453-4
  3. 3.
    Cardinali, I., Giuzzi, L.: Geometries arising from trilinear forms on low-dimensional vector spaces, in preparationGoogle Scholar
  4. 4.
    Cohen, A.M., Helminck, A.G.: Trilinear alternating forms on a vector space of dimension 7. Commun. Algebra 16(1), 1–25 (1988). doi: 10.1080/00927878808823558 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cooperstein, B.N.: External flats to varieties in \(\text{ PG }(\wedge ^{2}(V))\) over finite fields. Geom. Dedicata 69(3), 223–235 (1998). doi: 10.1023/A:1005053409486 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Bruyn, B.: Hyperplanes of embeddable Grassmannians arise from projective embeddings: a short proof. Linear Algebra Appl. 430(1), 418–422 (2009). doi: 10.1016/j.laa.2008.08.003 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Bruyn, B.: On polyvectors of vector spaces and hyperplanes of projective Grassmannians. Linear Algebra Appl. 435(5), 1055–1084 (2011). doi: 10.1016/j.laa.2011.02.031 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    De Wispelaere, A., Van Maldeghem, H.: A distance-2-spread of the generalized hexagon H(3). Ann. Comb. 8(2), 133–154 (2004). doi: 10.1007/s00026-004-0211-9 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Djoković, D.Ž.: Classification of trivectors of an eight-dimensional real vector space. Linear Multilinear Algebra 13, 3–39 (1988)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Draisma, J., Shaw, R.: Singular lines of trilinear forms. Linear Algebra Appl. 433(3), 690–697 (2010). doi: 10.1016/j.laa.2010.03.040 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Draisma, J., Shaw, R.: Some noteworthy alternating trilinear forms. J. Geom. 105(1), 167–176 (2014). doi: 10.1007/s00022-013-0202-2 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gille, P., Szamuely, T.: Central Simple Algebras and Galois cohomology Cambridge Studies in Advanced Mathematics, vol. 101. Cambridge University Press, Cambridge (2006). doi: 10.1017/CBO9780511607219 CrossRefzbMATHGoogle Scholar
  13. 13.
    Gurevich, G.B.: Foundations of the theory of algebraic invariants, Translated by J.R. M. Radok, A.J.M. Spencer, P. Noordhoff Ltd., Groningen (1964)Google Scholar
  14. 14.
    Hall, J.I., Shult, E.E.: Geometric hyperplanes of nonembeddable Grassmannians. Eur. J. Comb. 14(1), 29–35 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Harris, J.: Algebraic Geometry, Graduate Texts in Mathematics, vol. 133. Springer, New York (1995)Google Scholar
  16. 16.
    Jenks, R.D., Sutor, R.S.: AXIOM. Numerical Algorithms Group Ltd., Oxford; Springer, New York (1992)Google Scholar
  17. 17.
    Lounesto, P.: Clifford Algebras and Spinors, London Mathematical Society Lecture Note Series, vol. 286, 2nd edn. Cambridge University Press, Cambridge (2001). doi: 10.1017/CBO9780511526022
  18. 18.
    Revoy, P.: Trivecteurs de rang 6. Bull. Soc. Math. France Mém. 59, 141–155 (1979). (French)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Serre, J.: Galois Cohomology. Springer Monographs in Mathematics. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  20. 20.
    Shult, E.E.: Geometric hyperplanes of embeddable Grassmannians. J. Algebra 145(1), 55–82 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shult, E.E.: Points and Lines. Universitext. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-15627-4 CrossRefzbMATHGoogle Scholar
  22. 22.
    Van Maldeghem, H.: Generalized Polygons. Monographs in Mathematics, vol. 93. Birkhäuser, Basel (1998)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Information Engineering and MathematicsUniversity of SienaSienaItaly
  2. 2.D.I.C.A.T.A.M. — Section of Mathematics Università di BresciaBresciaItaly

Personalised recommendations