Innovations in Incidence Geometry

Embeddings of orthogonal grassmannians

Antonio Pasini

Full-text: Open access

Abstract

In this paper I survey a number of recent results on projective and Veronesean embeddings of orthogonal Grassmannians and propose a few conjectures and problems.

Article information

Source
Innov. Incidence Geom., Volume 13, Number 1 (2013), 107-133.

Dates
Received: 2 February 2012
Accepted: 24 September 2012
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323045

Digital Object Identifier
doi:10.2140/iig.2013.13.107

Mathematical Reviews number (MathSciNet)
MR3173015

Zentralblatt MATH identifier
1318.51001

Subjects
Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx] 17B50: Modular Lie (super)algebras 20C20: Modular representations and characters 20G15: Linear algebraic groups over arbitrary fields 51E24: Buildings and the geometry of diagrams

Keywords
orthogonal polar spaces grassmannians Weyl modules Veronese variety

Citation

Pasini, Antonio. Embeddings of orthogonal grassmannians. Innov. Incidence Geom. 13 (2013), no. 1, 107--133. doi:10.2140/iig.2013.13.107. https://projecteuclid.org/euclid.iig/1551323045


Export citation

References

  • R. J. Blok, Highest weight modules and polarized embeddings of shadow spaces, J. Alg. Combin. 34, (2011), 67–113.
  • R. J. Blok and A. E. Brouwer, Spanning point-line geometries in buildings of spherical type, J. Geometry 62 (1998), 26–35.
  • R. J. Blok, I. Cardinali and B. De Bruyn, On the nucleus of the Grassmann embedding of the symplectic dual polar space $\mathrm{DSp}(2n, \mathbb{F})$, $\mathrm{char}(\mathbb{F})=2$, European J. Combin. 30 (2009), 468–472.
  • R. J. Blok and A. Pasini, Point-line geometries with a generating set that depends on the underlying field, Finite Geometries (A. Blokhuis et al. ed.), Kluwer, Dordrecht 2001, 1–25.
  • ––––, On absolutely universal embeddings, Discrete Math. 267 (2003), 45–62.
  • A. Blokhuis and A. E. Brouwer, The universal embedding dimension of the binary symplectic dual polar space. Discrete Math. 264 (2003), 3–11.
  • F. Buekenhout and P. J. Cameron, Projective and affine geometries over division rings, Handbook of Incidence Geometry, Elsevier, Amsterdam (1995), 27–62.
  • I. Cardinali and G. Lunardon, A geometric description of the spin embedding of symplectic dual polar spaces of rank 3. J. Combin. Theory Ser. A 115 (2008), 1056–1064.
  • I. Cardinali and A. Pasini, Grassmann and Weyl embeddings of orthogonal Grassmannians. J. Alg. Combin. 38 (2013), 863–888.
  • ––––, Veronesean embeddings of dual polar spaces of orthogonal type. J. Combin. Theory Ser. A 120 (2013), 1328-1350.
  • ––––, On certain submodules of Weyl modules for $\mathrm{SO}(2n+1,\mathbb{F})$ with $\mathrm{char}(\mathbb{F}) = 2$. To appear in J. Group Theory.
  • R. W. Carter, Lie Algebras of Finite and Affine Type, Cambridge Univ. Press, Cambridge, 2005.
  • B. N. Cooperstein, On the generation of dual polar spaces of symplectic type over finite fields, J. Combin. Theory Ser. A 83 (1998), 221–232.
  • ––––, Generating long root subgroup geometries of classical groups over finite prime fields, Bull. Belg. Math. Soc. Simon Stevin 5 (1998), 531–548.
  • B. N. Cooperstein and E. E. Shult, Frames and bases of Lie incidence geometries. J. Geometry 60 (1997), 17–46.
  • B. De Bruyn, The structure of the spin embedding of dual polar spaces and related geometries. European J. Combin. 29 (2008), 1242–1256.
  • B. De Bruyn and A. Pasini, On symplectic polar spaces over non-perfect fields of characteristic 2, Linear Multilinear Algebra 57 (2009), 567–575.
  • ––––, Generating symplectic and Hermitian dual polar spaces over arbitrary fields nonisomorphic to $\mathbb{F}_2$, Electronic Journal of Combinatorics 14 (2007), $\#$R54.
  • J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.
  • A. Kasikova and E. E. Shult, Absolute embeddings of point-line geometries, J. Algebra 238 (2001), 265–291.
  • P. Li, On the universal embedding of the $Sp_{2n}(2)$ dual polar space, J. Combin. Theory Ser. A 94 (2001), 100–117.
  • A. Pasini, Embeddings and expansions, Bull. Belg. Math. Soc. Simon Stevin 10 (2003), 585–626.
  • A. Pasini and H. Van Maldeghem, Some constructions and embeddings of the tilde geometry, Note di Matematica 21 (2002/2003), 1–33.
  • S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman, Boston, 1984.
  • M. A. Ronan, Embeddings and hyperplanes of discrete geometries, European J. Combin. 8 (1987), 179–185.
  • J. A. Thas and H. Van Maldeghem, Characterizations of the finite quadric Veroneseans ${{\mathcal{V}}_n}^{2^{n}}$, Q. J. Math. 55 (2004), 99–113.
  • ––––, Generalized Veronesean embeddings of projective spaces, Combinatorica 31 (2011), 615–629.
  • J. Tits, Buildings of Spherical Type and Finite $BN$-pairs, Lect. Notes in Math. 386, Springer, Berlin, 1974.
  • H. Van Maldeghem, Generalized Polygons, Birkhäuser, Basel, 1998.
  • H. Völklein, On the geometry of the adjoint representation of a Chevalley group, J. Algebra 127 (1989), 139–154.