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

Bart De Bruyn UGent (2009) 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)\$.
author
organization
year
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
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.