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. Thendefine

$$\widehat{b}:V\to V^{*}:v\mapsto b(-,v):V\to k$$

which is semi-linear, as $b$ is sesquilinear, and bijective, since $b$ isnon-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-degeneratesesquilinear 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 wehave $v,w\in V$ 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 risedo 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 bydimension 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 orderreversing 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 toprovide 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 theDynkin diagram of $A_{d\mathrm{-}\mathrm{1}}$, $d\mathrm{=}\dim \mathit{}V$. When $\dim \mathit{}V\mathrm{=}\mathrm{2}$ there is nograph automorphism, just as there are no dualities (points are hyperplaneswhen $\dim \mathit{}V\mathrm{=}\mathrm{2}$.)