polarities and forms

Through out this article we assume dimV2. This is not a true constraint as there are only trivial dualities for dimV2.

Proposition 1.

Every duality gives rise to a non-degenerate sesquilinear formPlanetmathPlanetmath, and visa-versa.


To see this, let d:PG(V)PG(V) be a duality. We may express this as an order preserving map d:PG(V)PG(V*). Then by the fundamental theorem of projective geometryMathworldPlanetmath it follows d is induced by a bijectiveMathworldPlanetmathPlanetmath semi-linear transformation d^:VV*.

An semi-linear isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of V to V* is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to specifying a non-degenerate sesquilinear form. In particular, define the form b:V×Vk by b(v,w)=(v)(wd^) (notice wd^V* so wd^:Vk).

Now, if b:V×Vk is a non-degenerate sesquilinear form. Then define


which is semi-linear, as b is sesquilinear, and bijective, since b is non-degenerate. Therefore b^ induces an order preserving bijection PG(V)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={wV:b(v,W)=0}. (Although the form may not be reflexiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath, we still use the notation, but we now demonstrate that we can indeed specialize to the reflexive case.) Notice then that

Corollary 2.

Every polarityMathworldPlanetmathPlanetmath gives rise to a reflexive non-degenerate sesquilinear form, and visa-versa.


Let b be the sesquilinear form induced by the polarity p. Then suppose we have v,wV such that 0=b(v,w)=(v)(wp^). So vw=wp. But p has order 2 so v=vpw. 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 b^. By the reflexivity, b(W,W)=0 implies b(W,W)=0 also. As (W)={vV:b(v,(W))=0} it follows W(W). But by dimensionPlanetmathPlanetmath arguments:


we conclude W=(W). Thus d is a polarity. ∎

From the fundamental theorem of projective geometry it follows if dimV2 then every order preserving map is induced by a semi-linear transformation of V. In similarMathworldPlanetmathPlanetmath fashion we have

Proposition 3.

PΓL*(V)=PΓL(V)2, meaning that every order reversing map f:PG(V)PG(V) can be decomposed as a f=sr where s is induced from a semi-linear transformation and r is a polarity.


Let d be any duality of PG(V). Then d2 is order preserving. Thus d2 is a projectivityMathworldPlanetmath so by the fundamental theorem of projective geometry d2 is induced by a semi-linear transformation s. Therefore PΓL(V) has index 2 in PΓL*(V). Finally it suffices to provide any polarity of PG(V) to prove PΓL*(V)=PΓL(V)2. For this use any reflexive non-degenerate sesquilinear form. ∎

Remark 4.

The group PΓL*(V) is the automorphism groupMathworldPlanetmath of PSL(V). In particular, the polarities account for the graph automorphismsMathworldPlanetmath of the Dynkin diagram of Ad-1, d=dimV. When dimV=2 there is no graph automorphism, just as there are no dualities (points are hyperplanesMathworldPlanetmathPlanetmath when dimV=2.)


  • 1 Gruenberg, K. W. and Weir, A.J. Linear GeometryMathworldPlanetmath 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