polarities and forms
Through out this article we assume $dimV\ne 2$. This is not a true constraint as there are only trivial dualities for $dimV\le 2$.
Proposition 1.
Every duality gives rise to a non-degenerate sesquilinear form^{}, and visa-versa.
Proof.
To see this, let $d:PG(V)\to PG(V)$ be a duality. We may express this as an order preserving map $d:PG(V)\to PG({V}^{*})$. Then by the fundamental theorem of projective geometry^{} it follows $d$ is induced by a bijective^{} semi-linear transformation $\widehat{d}:V\to {V}^{*}$.
An semi-linear isomorphism^{} of $V$ to ${V}^{*}$ is equivalent^{} to specifying a non-degenerate sesquilinear form. In particular, define the form $b:V\times V\to k$ by $b(v,w)=(v)(w\widehat{d})$ (notice $w\widehat{d}\in {V}^{*}$ so $w\widehat{d}:V\to k$).
Now, if $b:V\times V\to k$ is a non-degenerate sesquilinear form. Then define
$$\widehat{b}:V\to {V}^{*}:v\mapsto b(-,v):V\to k$$ |
which is semi-linear, as $b$ is sesquilinear, and bijective, since $b$ is non-degenerate. Therefore $\widehat{b}$ induces an order preserving bijection $PG(V)\to PG({V}^{*})$, that is, a duality. ∎
We write ${W}^{\u27c2}$ for the image of the induced duality of a non-degenerate sesquilinear form $b$. Notice that ${W}^{\u27c2}=\{w\in V:b(v,W)=0\}$. (Although the form may not be reflexive^{}, we still use the $\u27c2$ notation, but we now demonstrate that we can indeed specialize to the reflexive case.) Notice then that
$$dim{W}^{\u27c2}=dimV-dimW.$$ |
Corollary 2.
Every polarity^{} gives rise to a reflexive non-degenerate sesquilinear form, and visa-versa.
Proof.
Let $b$ be the sesquilinear form induced by the polarity $p$. Then suppose we have $v,w\in V$ such that $0=b(v,w)=(v)(w\widehat{p})$. So $\u27e8v\u27e9\le {\u27e8w\u27e9}^{\u27c2}=\u27e8w\u27e9p$. But $p$ has order 2 so ${\u27e8v\u27e9}^{\u27c2}=\u27e8v\u27e9p\ge \u27e8w\u27e9$. But this implies $b(w,v)=0$ so $b$ is reflexive.
Likewise, given a reflexive non-degenerate sesquilinear form $b$ it gives rise do a duality $d$ induced by $\widehat{b}$. By the reflexivity, $b(W,{W}^{\u27c2})=0$ implies $b({W}^{\u27c2},W)=0$ also. As ${({W}^{\u27c2})}^{\u27c2}=\{v\in V:b(v,{({W}^{\u27c2})}^{\u27c2})=0\}$ it follows $W\le {({W}^{\u27c2})}^{\u27c2}$. But by dimension^{} arguments:
$$dim{({W}^{\u27c2})}^{\u27c2}=dimV-dim{W}^{\u27c2}=dimV-(dimV-dimW)=dimW$$ |
we conclude $W={({W}^{\u27c2})}^{\u27c2}$. Thus $d$ is a polarity. ∎
From the fundamental theorem of projective geometry it follows if $dimV\ne 2$ then every order preserving map is induced by a semi-linear transformation of $V$. In similar^{} fashion we have
Proposition 3.
$P\mathrm{\Gamma}{L}^{*}(V)=P\mathrm{\Gamma}L(V)\u22ca{\mathbb{Z}}_{2}$, meaning that every order reversing map $f\mathrm{:}P\mathit{}G\mathit{}\mathrm{(}V\mathrm{)}\mathrm{\to}P\mathit{}G\mathit{}\mathrm{(}V\mathrm{)}$ can be decomposed as a $f\mathrm{=}s\mathit{}r$ where $s$ is induced from a semi-linear transformation and $r$ is a polarity.
Proof.
Let $d$ be any duality of $PG(V)$. Then ${d}^{2}$ is order preserving. Thus ${d}^{2}$ is a projectivity^{} so by the fundamental theorem of projective geometry ${d}^{2}$ is induced by a semi-linear transformation $s$. Therefore $P\mathrm{\Gamma}L(V)$ has index 2 in $P\mathrm{\Gamma}{L}^{*}(V)$. Finally it suffices to provide any polarity of $PG(V)$ to prove $P\mathrm{\Gamma}{L}^{*}(V)=P\mathrm{\Gamma}L(V)\u22ca{\mathbb{Z}}_{2}$. For this use any reflexive non-degenerate sesquilinear form. ∎
Remark 4.
The group $P\mathit{}\mathrm{\Gamma}\mathit{}{L}^{\mathrm{*}}\mathit{}\mathrm{(}V\mathrm{)}$ is the automorphism group^{} of $P\mathit{}S\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$. In particular, the polarities account for the graph automorphisms^{} of the Dynkin diagram of ${A}_{d\mathrm{-}\mathrm{1}}$, $d\mathrm{=}\mathrm{dim}\mathit{}V$. When $\mathrm{dim}\mathit{}V\mathrm{=}\mathrm{2}$ there is no graph automorphism, just as there are no dualities (points are hyperplanes^{} when $\mathrm{dim}\mathit{}V\mathrm{=}\mathrm{2}$.)
References
- 1 Gruenberg, K. W. and Weir, A.J. Linear Geometry^{} 2nd Ed. (English) [B] Graduate Texts in Mathematics. 49. New York - Heidelberg - Berlin: Springer-Verlag. X, 198 p. DM 29.10; $ 12.80 (1977).
- 2 Kantor, W. M. Lectures notes on Classical Groups.
Title | polarities and forms |
Canonical name | PolaritiesAndForms |
Date of creation | 2013-03-22 15:58:13 |
Last modified on | 2013-03-22 15:58:13 |
Owner | Algeboy (12884) |
Last modified by | Algeboy (12884) |
Numerical id | 7 |
Author | Algeboy (12884) |
Entry type | Topic |
Classification | msc 51A05 |
Related topic | polarity |
Related topic | Projectivity |
Related topic | ProjectiveGeometry |
Related topic | Isometry2 |
Related topic | ProjectiveGeometry3 |
Related topic | ClassicalGroups |
Related topic | Polarity2 |
Related topic | DualityWithRespectToANonDegenerateBilinearForm |