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polarities and forms (Topic)

Through out this article we assume $ \dim V\neq 2$. This is not a true constraint as there are only trivial dualities for $ \dim V\leq 2$.

Proposition 1   Every duality gives rise to a non-degenerate sesquilinear form, and visa-versa.
Proof. To see this, let $ d:PG(V)\rightarrow PG(V)$ be a duality. We may express this as an order preserving map $ d:PG(V)\rightarrow PG(V^*)$. Then by the fundamental theorem of projective geometry it follows $ d$ is induced by a bijective semi-linear transformation $ \hat{d}:V\rightarrow 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\rightarrow k$ by $ b(v,w)=(v)(w\hat{d})$ (notice $ w\hat{d}\in V^*$ so $ w\hat{d}:V\rightarrow k$).

Now, if $ b:V\times V\rightarrow k$ is a non-degenerate sesquilinear form. Then define

$\displaystyle \hat{b}:V\rightarrow V^*:v\mapsto b(-,v):V\rightarrow k$
which is semi-linear, as $ b$ is sesquilinear, and bijective, since $ b$ is non-degenerate. Therefore $ \hat{b}$ induces an order preserving bijection $ PG(V)\rightarrow PG(V^*)$, that is, a duality. $ \qedsymbol$

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

$\displaystyle \dim W^\perp=\dim V-\dim W.$
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\hat{p})$. So $ \langle v\rangle \leq \langle w\rangle^\perp=\langle w\rangle p$. But $ p$ has order 2 so $ \langle v\rangle^\perp=\langle v\rangle p\geq \langle w\rangle$. 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 $ \hat{b}$. By the reflexivity, $ b(W,W^\perp)=0$ implies $ b(W^\perp, W)=0$ also. As $ (W^\perp)^\perp=\{v\in V:b(v,(W^\perp)^\perp)=0\}$ it follows $ W\leq (W^\perp)^\perp$. But by dimension arguments:

$\displaystyle \dim (W^\perp)^\perp =\dim V-\dim W^\perp=\dim V-(\dim V-\dim W)=\dim W$
we conclude $ W=(W^\perp)^\perp$. Thus $ d$ is a polarity. $ \qedsymbol$

From the fundamental theorem of projective geometry it follows if $ \dim V\neq 2$ then every order preserving map is induced by a semi-linear transformation of $ V$. In similar fashion we have

Proposition 3   $ P\Gamma L^*(V)=P\Gamma L(V)\rtimes \mathbb{Z}_2$, meaning that every order reversing map $ f:PG(V)\rightarrow PG(V)$ can be decomposed as a $ f=sr$ 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\Gamma L(V)$ has index 2 in $ P\Gamma L^*(V)$. Finally it suffices to provide any polarity of $ PG(V)$ to prove $ P\Gamma L^*(V)=P\Gamma L(V)\rtimes \mathbb{Z}_2$. For this use any reflexive non-degenerate sesquilinear form. $ \qedsymbol$
Remark 4   The group $ P\Gamma L^*(V)$ is the automorphism group of $ PSL(V)$. In particular, the polarities account for the graph automorphisms of the Dynkin diagram of $ A_{d-1}$, $ d=\dim V$. When $ \dim V=2$ there is no graph automorphism, just as there are no dualities (points are hyperplanes when $ \dim V=2$.)

Bibliography

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.



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See Also: polarity, projectivity, projective geometry, isometry, projective geometry, classical groups, polarity, duality with respect to a non-degenerate bilinear form
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Cross-references: hyperplanes, points, Dynkin diagram, graph automorphisms, automorphism group, group, index, projectivity, order reversing, similar, arguments, dimension, reflexivity, implies, reflexive non-degenerate sesquilinear, polarity, Reflexive, image, bijection, induces, equivalent, isomorphism, semi-linear transformation, bijective, induced, fundamental theorem of projective geometry, map, order, sesquilinear form, non-degenerate, dualities
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This is version 4 of polarities and forms, born on 2006-06-09, modified 2006-06-19.
Object id is 7986, canonical name is PolaritiesAndForms.
Accessed 841 times total.

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AMS MSC: 51A05 (Geometry :: Linear incidence geometry :: General theory and projective geometries)
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