### On symplectic polar spaces over non-perfect fields of characteristic 2

(2009) Linear and Multilinear Algebra. 57(6). p.567-575- abstract
- Given a field $ of characteristic 2 and an integer \geq 2$, let (2n-1,K)$ be the symplectic polar space defined in $\PG(2n-1,K)$ by a non-degenerate alternating form of (2n,K)$ and let (2n,K)$ be the quadric of $\PG(2n,K)$ associated to a non-singular quadratic form of Witt index $. In the literature it is often claimed that (2n-1,K) \cong Q(2n,K)$. This is true when $ is perfect, but false otherwise. In this paper we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that (2n-1,K)$ is indeed isomorphic to a non-singular quadric $, but when $ is non-perfect the nucleus of $ has vector dimension greater than 1. So, in this case, (2n,K)$ is a proper subgeometry of (2n-1,K)$. We show that, in spite of this fact, (2n-1,K)$ can be embedded in (2n,K)$ as a subgeometry and that this embedding induces a full embedding of the dual (2n-1,K)$ of (2n-1,K)$ into the dual (2n,K)$ of (2n,K)$.

Please use this url to cite or link to this publication:
http://hdl.handle.net/1854/LU-879255

- author
- Bart De Bruyn UGent and Antonio Pasini
- organization
- year
- 2009
- type
- journalArticle (original)
- publication status
- published
- subject
- keyword
- orthogonal (dual) polar space, symplectic (dual) polar space
- journal title
- Linear and Multilinear Algebra
- Linear Multilinear Algebra
- volume
- 57
- issue
- 6
- pages
- 567 - 575
- Web of Science type
- Article
- Web of Science id
- 000268752300004
- JCR category
- MATHEMATICS
- JCR impact factor
- 0.74 (2009)
- JCR rank
- 92/251 (2009)
- JCR quartile
- 2 (2009)
- ISSN
- 0308-1087
- DOI
- 10.1080/03081080802012623
- language
- English
- UGent publication?
- yes
- classification
- A1
- copyright statement
*I don't know the status of the copyright for this publication*- id
- 879255
- handle
- http://hdl.handle.net/1854/LU-879255
- date created
- 2010-02-24 15:15:56
- date last changed
- 2010-03-05 10:43:54

@article{879255, abstract = {Given a field \$ of characteristic 2 and an integer {\textbackslash}geq 2\$, let (2n-1,K)\$ be the symplectic polar space defined in \${\textbackslash}PG(2n-1,K)\$ by a non-degenerate alternating form of (2n,K)\$ and let (2n,K)\$ be the quadric of \${\textbackslash}PG(2n,K)\$ associated to a non-singular quadratic form of Witt index \$. In the literature it is often claimed that (2n-1,K) {\textbackslash}cong Q(2n,K)\$. This is true when \$ is perfect, but false otherwise. In this paper we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that (2n-1,K)\$ is indeed isomorphic to a non-singular quadric \$, but when \$ is non-perfect the nucleus of \$ has vector dimension greater than 1. So, in this case, (2n,K)\$ is a proper subgeometry of (2n-1,K)\$. We show that, in spite of this fact, (2n-1,K)\$ can be embedded in (2n,K)\$ as a subgeometry and that this embedding induces a full embedding of the dual (2n-1,K)\$ of (2n-1,K)\$ into the dual (2n,K)\$ of (2n,K)\$.}, author = {De Bruyn, Bart and Pasini, Antonio}, issn = {0308-1087}, journal = {Linear and Multilinear Algebra}, keyword = {orthogonal (dual) polar space,symplectic (dual) polar space}, language = {eng}, number = {6}, pages = {567--575}, title = {On symplectic polar spaces over non-perfect fields of characteristic 2}, url = {http://dx.doi.org/10.1080/03081080802012623}, volume = {57}, year = {2009}, }

- Chicago
- De Bruyn, Bart, and Antonio Pasini. 2009. “On Symplectic Polar Spaces over Non-perfect Fields of Characteristic 2.”
*Linear and Multilinear Algebra*57 (6): 567–575. - APA
- De Bruyn, B., & Pasini, A. (2009). On symplectic polar spaces over non-perfect fields of characteristic 2.
*Linear and Multilinear Algebra*,*57*(6), 567–575. - Vancouver
- 1.De Bruyn B, Pasini A. On symplectic polar spaces over non-perfect fields of characteristic 2. Linear and Multilinear Algebra. 2009;57(6):567–75.
- MLA
- De Bruyn, Bart, and Antonio Pasini. “On Symplectic Polar Spaces over Non-perfect Fields of Characteristic 2.”
*Linear and Multilinear Algebra*57.6 (2009): 567–575. Print.