Skip to main content

Polarized non-abelian representations of slim near-polar spaces

Abstract

In (Bull Belg Math Soc Simon Stevin 4:299–316, 1997), Shult introduced a class of parapolar spaces, the so-called near-polar spaces. We introduce here the notion of a polarized non-abelian representation of a slim near-polar space, that is, a near-polar space in which every line is incident with precisely three points. For such a polarized non-abelian representation, we study the structure of the corresponding representation group, enabling us to generalize several of the results obtained in Sahoo and Sastry (J Algebraic Comb 29:195–213, 2009) for non-abelian representations of slim dense near hexagons. We show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding.

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

Notes

  1. If \(l=0\), then \((C_2)^{-1}\) is not defined. In this case, this sentence should be understood as “Z(R) is isomorphic to \(C_2\)”.

  2. The terms occurring in this sum are Gaussian binomial coefficients.

  3. The map \(\phi _x:R\mapsto R'\) defined by \(\phi _x(r)=[\psi (x),r]\) is a homomorphism (see Corollary 5.7) which is surjective. The kernel of \(\phi _x\) is \(C_R(\psi (x))\) which has index 2 in R by the first isomorphism theorem. Then \(\bar{v}^{\perp _f}\) is precisely the image of \(C_R(\psi (x))\) in V under the canonical homomorphism \(R \rightarrow V; r \mapsto r R'\).

References

  1. Aschbacher, M.: Finite Group Theory. Cambridge Studies in Advanced Mathematics, vol. 10. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  3. Brouwer, A.E., Shpectorov, S.V.: Dimensions of embeddings of near polygons. Unpublished manuscript

  4. Cohen, A.M., Cooperstein, B.N.: A characterization of some geometries of Lie type. Geom. Dedicata 15, 73–105 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  5. De Bruyn, B.: Dual embeddings of dense near polygons. Ars Comb. 103, 33–54 (2012)

    MathSciNet  MATH  Google Scholar 

  6. De Bruyn, B., Sahoo, B.K., Sastry, N.S.N.: Non-abelian representations of the slim dense near hexagons on 81 and 243 points. J. Algebraic Comb. 33, 127–140 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  7. Doerk, K., Hawkes, T.: Finite Soluble Groups. de Gruyter Expositions in Mathematics, vol. 4. Walter de Gruyter & Co., Berlin (1992)

    Book  MATH  Google Scholar 

  8. Gorenstein, D.: Finite Groups. Chelsea Publishing Co., New York (1980)

    MATH  Google Scholar 

  9. Ivanov, A.A.: Non-abelian representations of geometries. “Groups and Combinatorics”—in memory of Michio Suzuki. Adv. Stud. Pure Math. 32, 301–314 (2001). (Math. Soc. Japan, Tokyo)

    Google Scholar 

  10. Ivanov, A.A., Pasechnik, D.V., Shpectorov, S.V.: Non-abelian representations of some sporadic geometries. J. Algebra 181, 523–557 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  11. Patra, K.L., Sahoo, B.K.: A non-abelian representation of the dual polar space \(DQ(2n,2)\). Innov. Incid. Geom. 9, 177–188 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Ronan, M.A.: Embeddings and hyperplanes of discrete geometries. Eur. J. Comb. 8, 179–185 (1987)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  14. Sahoo, B.K., Sastry, N.S.N.: On the order of a non-abelian representation group of a slim dense near hexagon. J. Algebraic Comb. 29, 195–213 (2009)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bart De Bruyn.

Additional information

Binod Kumar Sahoo was partially supported by DST/SERB (SR/FTP/MS-001/2010), Government of India.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

De Bruyn, B., Sahoo, B.K. Polarized non-abelian representations of slim near-polar spaces. J Algebr Comb 44, 59–79 (2016). https://doi.org/10.1007/s10801-015-0653-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-015-0653-7

Keywords

  • Near-polar space
  • Universal/ Polarized non-abelian representation
  • Universal projective embedding
  • Minimal polarized embedding
  • Extraspecial 2-group
  • Combinatorial group theory

Mathematics Subject Classification

  • 05B25
  • 51A45
  • 51A50
  • 20F05