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 $W^{⟂}$ for the image of the induced duality of a non-degenerate sesquilinear form $b$. Notice that $W^{⟂}=\{w\in V:b(v,W)=0\}$. (Although the form may not be reflexive, we still use the $⟂$ notation, but we now demonstrate that we can indeed specialize to the reflexive case.) Notice then that

$$dim{W}^{⟂}=dimV-dimW.$$

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 $⟨v⟩\le {⟨w⟩}^{⟂}=⟨w⟩p$. But $p$ has order 2 so ${⟨v⟩}^{⟂}=⟨v⟩p\ge ⟨w⟩$. 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, $b(W,W^{⟂})=0$ implies $b(W^{⟂},W)=0$ also. As ${(W^{⟂})}^{⟂}=\{v\in V:b(v,{(W^{⟂})}^{⟂})=0\}$ it follows $W\le {(W^{⟂})}^{⟂}$. But by dimension arguments:

$$dim{(W^{⟂})}^{⟂}=dimV-dim{W}^{⟂}=dimV-(dimV-dimW)=dimW$$

we conclude $W={(W^{⟂})}^{⟂}$. 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.

$P\mathrm{\Gamma }{L}^{*}(V)=P\mathrm{\Gamma }L(V)⋊{\mathbb{Z}}_2$, 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 $P\mathrm{\Gamma }{L}^{*}(V)=P\mathrm{\Gamma }L(V)⋊{\mathbb{Z}}_2$. 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 $A_{d\mathrm{-}\mathrm{1}}$, $d\mathrm{=}\dim \mathit{}V$. When $\dim \mathit{}V\mathrm{=}\mathrm{2}$ there is no graph automorphism, just as there are no dualities (points are hyperplanes when $\dim \mathit{}V\mathrm{=}\mathrm{2}$.)

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