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Collineations of polar spaces with restricted displacements

Beukje Temmermans (UGent) , Joseph Thas (UGent) and Hendrik Van Maldeghem (UGent)
(2012) DESIGNS CODES AND CRYPTOGRAPHY. 64(1-2). p.61-80
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Abstract
Let J be a set of types of subspaces of a polar space. A collineation (which is a type-preserving automorphism) of a polar space is called J-domestic if it maps no flag of type J to an opposite one. In this paper we investigate certain J-domestic collineations of polar spaces. We describe in detail the fixed point structures of collineations that are i-domestic and at the same time (i + 1)-domestic, for all suitable types i. We also show that {point, line}-domestic collineations are either point-domestic or line-domestic, and then we nail down the structure of the fixed elements of point-domestic collineations and of line-domestic collineations. We also show that {i, i + 1}-domestic collineations are either i-domestic or (i + 1)-domestic (under the assumption that i + 1 is not the type of the maximal subspaces if i is even). For polar spaces of rank 3, we obtain a full classification of all chamber-domestic collineations. All our results hold in the general case (finite or infinite) and generalize the full classification of all domestic collineations of polar spaces of rank 2 performed in Temmermans et al. (to appear in Ann Comb).
Keywords
Domestic collineations, Polar spaces, Opposition

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Citation

Please use this url to cite or link to this publication:

Chicago
Temmermans, Beukje, Joseph Thas, and Hendrik Van Maldeghem. 2012. “Collineations of Polar Spaces with Restricted Displacements.” Designs Codes and Cryptography 64 (1-2): 61–80.
APA
Temmermans, B., Thas, J., & Van Maldeghem, H. (2012). Collineations of polar spaces with restricted displacements. DESIGNS CODES AND CRYPTOGRAPHY, 64(1-2), 61–80.
Vancouver
1.
Temmermans B, Thas J, Van Maldeghem H. Collineations of polar spaces with restricted displacements. DESIGNS CODES AND CRYPTOGRAPHY. 2012;64(1-2):61–80.
MLA
Temmermans, Beukje, Joseph Thas, and Hendrik Van Maldeghem. “Collineations of Polar Spaces with Restricted Displacements.” DESIGNS CODES AND CRYPTOGRAPHY 64.1-2 (2012): 61–80. Print.
@article{2486789,
  abstract     = {Let J be a set of types of subspaces of a polar space. A collineation (which is a type-preserving automorphism) of a polar space is called J-domestic if it maps no flag of type J to an opposite one. In this paper we investigate certain J-domestic collineations of polar spaces. We describe in detail the fixed point structures of collineations that are i-domestic and at the same time (i + 1)-domestic, for all suitable types i. We also show that \{point, line\}-domestic collineations are either point-domestic or line-domestic, and then we nail down the structure of the fixed elements of point-domestic collineations and of line-domestic collineations. We also show that \{i, i + 1\}-domestic collineations are either i-domestic or (i + 1)-domestic (under the assumption that i + 1 is not the type of the maximal subspaces if i is even). For polar spaces of rank 3, we obtain a full classification of all chamber-domestic collineations. All our results hold in the general case (finite or infinite) and generalize the full classification of all domestic collineations of polar spaces of rank 2 performed in Temmermans et al. (to appear in Ann Comb).},
  author       = {Temmermans, Beukje and Thas, Joseph and Van Maldeghem, Hendrik},
  issn         = {0925-1022},
  journal      = {DESIGNS CODES AND CRYPTOGRAPHY},
  language     = {eng},
  number       = {1-2},
  pages        = {61--80},
  title        = {Collineations of polar spaces with restricted displacements},
  url          = {http://dx.doi.org/10.1007/s10623-011-9509-x},
  volume       = {64},
  year         = {2012},
}

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