Timeline for Stabilizer of a nonsingular vector in a quadratic space (char (k)=2)
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 13, 2014 at 12:35 | vote | accept | César Galindo | ||
| Mar 13, 2014 at 9:09 | answer | added | Derek Holt | timeline score: 1 | |
| Mar 13, 2014 at 5:30 | comment | added | user76758 | For any $k$ of char. 2, $B_q$ on the hyperplane $H=v^{\perp}$ has 1-dimensional defect space $L$, with symplectic form $\overline{B}$ on $H/L$. Also, $G:={\rm{Stab}}_v(O(q))$ has 2 connected components, each geometrically connected over $k$, and $G^0 \rightarrow{\rm{Sp}}(\overline{B})\simeq{\rm{Sp}}_{2n-2}$ is a $k$-group isomorphism. Indeed, WLOG $k = \overline{k}$, so WLOG $q(v)=1$, and then use the self-contained proof of Prop. C.3.1 of math.stanford.edu/~conrad/papers/luminysga3.pdf. For finite $k$ this proof gives $G(k)={\rm{Sp}}_{2n-2}(k)\times\mathbf{Z}/(2)$ by Lang's theorem. | |
| Mar 13, 2014 at 1:48 | comment | added | Will Jagy | Not sure he does your question, though, after reading those bits. | |
| Mar 13, 2014 at 1:39 | comment | added | Will Jagy | I like Classical Groups and geometric Algebra by Larry C. Grove, chapters devoted to characteristic 2 | |
| Mar 13, 2014 at 1:22 | history | asked | César Galindo | CC BY-SA 3.0 |