### On trivectors and hyperplanes of symplectic dual polar spaces

(2012)- 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.

Please use this url to cite or link to this publication:
http://hdl.handle.net/1854/LU-3052055

- author
- Mariusz Kwiatkowski UGent
- promoter
- Bart De Bruyn UGent
- organization
- year
- 2012
- type
- dissertation (monograph)
- subject
- keyword
- exterior algebra, Grassmann embedding, Symplectic group
- pages
- VI, 200 pages
- publisher
- Ghent University. Faculty of Sciences
- place of publication
- Ghent, Belgium
- defense location
- Gent : Campus Sterre (S25, auditorium Emmy Noether)
- defense date
- 2012-11-09 16:00
- language
- English
- UGent publication?
- yes
- classification
- D1
- copyright statement
*I have transferred the copyright for this publication to the publisher*- id
- 3052055
- handle
- http://hdl.handle.net/1854/LU-3052055
- date created
- 2012-11-13 12:54:59
- date last changed
- 2012-11-13 17:01:31

@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}, }

- 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.