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Polarized non-abelian representations of slim near-polar spaces


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.

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  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 \(R \rightarrow V; r \mapsto r R'\).


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Correspondence to Bart De Bruyn.

Additional information

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

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De Bruyn, B., Sahoo, B.K. Polarized non-abelian representations of slim near-polar spaces. J Algebr Comb 44, 59–79 (2016).

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  • 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