central collineations

Definitions and general properties

Definition 1.

A collineation of a finite dimensional projective geometry is a central collineationif there is a hyperplane of points fixed by the collineation.

Recall that collineations send any three collinear points to three collinear points. Thusif a collineation fixes more than a hyperplane of points then it in fact fixes all the pointsof the geometry and so it is the identity map. Therefore a central collineation can beviewed the simplest of the non-identity collineations.

Theorem 2.

Every collineation of a finite dimensional projective geometry of dimension $n\mathrm{>}\mathrm{1}$ isa product of at most $n$ central collineations. In particular, the automorphism groupof a projective geometry of dimension $n\mathrm{>}\mathrm{1}$ is generated by central collineations.

Suppose that a central collineation is not the identity. Then the hyperplane of fixed pointsis unique and receives the title of the axis of the central collineation. There isone further important result which justifies the name “central”.

Proposition 3.

Given a non-identity central collineation $f$, there is aunique point $C$ such that for all other points $P$, it follows that $C$, $P$ and $P\mathit{}f$ arecollinear.

The point $C$ determined by Proposition 3 is called the centerof the non-identity central collineation. It is possible for the center to lie on the axis.

Central collineations in coordinates

Suppose we have a projective geometry of dimension $n>2$, that is, we exclude nowthe case of projective lines and planes. The the geometry can be coordinatized throughso that we may regard the projective geometry as the latticeof subspaces of a vector space $V$ of dimension $n+1$ over a division ring $\mathrm{\Delta }$.Following the fundamental theorem of projective geometry we further know thatevery collineation is induced by a semi-linear transformation of $V$. So it is possibleto explore central collineations as semi-linear transformations.

Every hyperplane is a kernel of some linear functional, so we let $\phi :V\to \mathrm{\Delta }$ be a linear functional of $V$ with $H\phi =0$. Furthermore, we fix $v\in V$ so that$v\phi =1$ (which implieas also that $v\notin H$). Hence, for each $u\in V$,$u=(u-(u\phi )v)+(u\phi )v$ where $u-(u\phi )v\in H$ and $(u\phi )v\in ⟨v⟩$.

Let $f\in {\mathrm{GL}}_{\mathrm{\Delta }}(V)$ such that $f$ induces a central collineation $\tilde{f}$ on $PG(V)$ withaxis $H\le V$. As every scalar multiple of $f$ induces the same collineation of $PG(V)$, wemay assume that $f$ is the identity on $H$. Using the decomposition given by $\phi$ we have

$$uf=((u-(u\phi )v)+(u\phi )v)f=(u-(u\phi )v)+(u\phi )vf,u\in V.$$

Hence

$$uf=u+(u\phi )\widehat{v},\widehat{v}:=vf-v.$$

Suppose instead that $\phi$ is any linear functional of $V$. Then select some $\widehat{v}\in V$such that $\widehat{v}\phi \ne -1$. Then

$$ug:=u+(u\phi )\widehat{v}$$

fixes all the points of $\ker \phi$ so $g$ induces a central collineation.

If we wish to do the same without appealing to linear functionals, we may select a basis$\{v_1,\mathrm{\dots },v_{n+1}\}$ such that $H=⟨v_1,\mathrm{\dots },v_n⟩$ and$v_{n+1}\phi =1$. As $f$ is selected to be the identity on $H$ we have so far specified$f$ by the matrix:

in the basis $\{v_1,\mathrm{\dots },v_n,v_{n+1}\}$.