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On trivectors and hyperplanes of symplectic dual polar spaces

(2012)
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Abstract
The motivation for the work in this thesis is the study and the classification of the hyperplanes of the symplectic dual polar space DW (5, F) that arise from the Grassmann embedding. This dual polar space is related to a 6-dimensional vector space V over the field F which is equipped with a nondegenerate alternating bilinear form f . The hyperplanes of DW (5, F) are related to vectors of the third exterior power of V , i.e. to trivectors of V . We give a complete classification of the Sp(V, f)-equivalence classes of trivectors of V which is valid for any field F and apply this classification to obtain classification results for hyperplanes of DW (5, F) arising from the Grassmann embedding. As a base of our work we use Revoy’s classification of the GL(V )- equivalence classes of trivectors of V . In this classification five families of trivectors can be distinguished, the so-called trivectors of Type (A), (B), (C), (D) and (E). We classify the Sp(V, f )-equivalence classes of trivectors of all those types in Chapters 2 − 5. The classification of the Sp(V, f )-equivalence classes of trivectors for algebraically closed fields of characteristic distinct from 2 was done earlier by different means by Popov. We present his results and identify the trivectors of his classification with those of our classification in Chapter 6. In Chapter 7 we invoke the classification of all Sp(V, f)-equivalence classes of trivectors of V to obtain classification results for hyperplanes of DW (5, F) arising from the Grassmann embedding. The notion of quasi-Sp(V, f)-equivalence plays an important role here, and we will also obtain classification results for the quasi-Sp(V, f )-equivalence classes of trivectors of V . In Chapter 7, we obtain a complete classification of all the hyperplanes of DW (5, F) arising from the Grassmann embedding in case the field F has characteristic distinct from 2. In case the characteristic of F is 2, we obtain a partial classification. In Chapter 8, we present a method for determining all deep quads of a given hyperplane of DW (5, F) arising from the Grassmann embedding.
Keywords
exterior algebra, Grassmann embedding, Symplectic group

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

Chicago
Kwiatkowski, Mariusz. 2012. “On Trivectors and Hyperplanes of Symplectic Dual Polar Spaces”. Ghent, Belgium: Ghent University. Faculty of Sciences.
APA
Kwiatkowski, M. (2012). On trivectors and hyperplanes of symplectic dual polar spaces. Ghent University. Faculty of Sciences, Ghent, Belgium.
Vancouver
1.
Kwiatkowski M. On trivectors and hyperplanes of symplectic dual polar spaces. [Ghent, Belgium]: Ghent University. Faculty of Sciences; 2012.
MLA
Kwiatkowski, Mariusz. “On Trivectors and Hyperplanes of Symplectic Dual Polar Spaces.” 2012 : n. pag. Print.
@phdthesis{3052055,
  abstract     = {The motivation for the work in this thesis is the study and the classi\unmatched{fb01}cation of the hyperplanes of the symplectic dual polar space DW (5, F) that arise from the Grassmann embedding. This dual polar space is related to a 6-dimensional vector space V over the \unmatched{fb01}eld F which is equipped with a nondegenerate alternating bilinear form f . The hyperplanes of DW (5, F) are related to vectors of the third exterior power of V , i.e. to trivectors of V . We give a complete classi\unmatched{fb01}cation of the Sp(V, f)-equivalence classes of trivectors of V which is valid for any \unmatched{fb01}eld F and apply this classi\unmatched{fb01}cation to obtain classi\unmatched{fb01}cation results for hyperplanes of DW (5, F) arising from the Grassmann embedding.
As a base of our work we use Revoy{\textquoteright}s classi\unmatched{fb01}cation of the GL(V )- equivalence classes of trivectors of V . In this classi\unmatched{fb01}cation \unmatched{fb01}ve families of trivectors can be distinguished, the so-called trivectors of Type (A), (B), (C), (D) and (E). We classify the Sp(V, f )-equivalence classes of trivectors of all those types in Chapters 2 \ensuremath{-} 5. The classi\unmatched{fb01}cation of the Sp(V, f )-equivalence classes of trivectors for algebraically closed \unmatched{fb01}elds of characteristic distinct from 2 was done earlier by di\unmatched{fb00}erent means by Popov. We present his results and identify the trivectors of his classi\unmatched{fb01}cation with those of our classi\unmatched{fb01}cation in Chapter 6.
In Chapter 7 we invoke the classi\unmatched{fb01}cation of all Sp(V, f)-equivalence classes of trivectors of V to obtain classi\unmatched{fb01}cation results for hyperplanes of DW (5, F) arising from the Grassmann embedding. The notion of quasi-Sp(V, f)-equivalence plays an important role here, and we will also obtain classi\unmatched{fb01}cation results for the quasi-Sp(V, f )-equivalence classes of trivectors of V . In Chapter 7, we obtain a complete classi\unmatched{fb01}cation of all the hyperplanes of DW (5, F) arising from the Grassmann embedding in case the \unmatched{fb01}eld F has characteristic distinct from 2. In case the characteristic of F is 2, we obtain a partial classi\unmatched{fb01}cation.
In Chapter 8, we present a method for determining all deep quads of a given hyperplane of DW (5, F) arising from the Grassmann embedding.},
  author       = {Kwiatkowski, Mariusz},
  keyword      = {exterior algebra,Grassmann embedding,Symplectic group},
  language     = {eng},
  pages        = {VI, 200},
  publisher    = {Ghent University. Faculty of Sciences},
  school       = {Ghent University},
  title        = {On trivectors and hyperplanes of symplectic dual polar spaces},
  year         = {2012},
}