Split buildings of type $\mathsf {F_4}$ in buildings of type $\mathsf {E_6}$

Abstract

A symplectic polarity of a building $\varDelta$ of type $\mathsf {E_6}$ is a polarity whose fixed point structure is a building of type $\mathsf {F_4}$ containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type $\mathsf {F_4}$). In this paper, we show in a geometric way that every building of type $\mathsf {E_6}$ contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type $\mathsf {F_4}$ fully embedded in the natural point-line geometry of $\varDelta$ arises from a symplectic polarity.

Appendices

Index of Symbols

$x^\perp$ :

The “perp” of the point x: all points equal or collinear to x

$\varDelta$ :

Building of type $\mathsf {E_6}$ or it natural point-line geometry

$\theta$ :

A symplectic polarity in $\varDelta$

$\varGamma$ :

Building of type $\mathsf {F_4}$ or the corresponding symplectic metasymplectic parapolar space

$x\perp y$ :

The point x is collinear to the point y

$x\perp \perp y$ :

The point x is symplectic to the point y

$x\Diamond y$ :

The unique symplecton through the symplectic points x and y

$x\bowtie y$ :

The unique point collinear to both x and y when $\{x,y\}$ is a special pair

$x^{\perp \perp }$ :

All points equal or symplectic to the point x

h(x, y):

The hyperbolic line containing the symplectic pair $\{x,y\}$ of points

S(h):

The unique symplecton containing the hyperbolic line h

$\mathscr {S}_p$ :

The family of symplecta containing the point p

E(p, q):

The equator geometry of the pair $\{p,q\}$ of opposite points

$\widehat{E}=\widehat{E}(p,q)$ :

The extended equator geometry of the pair $\{p,q\}$ of opposite points

$\widehat{T}=\widehat{T}(p,q)$ :

The tropic circle geometry of the pair $\{p,q\}$ of opposite points

$\beta (x)$ :

The unique hyperbolic solid in $\widehat{E}(p,q)$ collinear to $x\in \widehat{T}(p,q)$

$\beta (U)$ :

The unique point collinear to the hyperbolic solid U

$\varTheta (\widehat{T}(p,q))$ :

The imaginary completion of $\widehat{T}(p,q)$ to a half spin $\mathsf {D_5}$

$\widehat{H}(p,q)$ :

The set of point collinear or equal to at least one point of $\widehat{E}(p,q)$

$\mathscr {N}_x$ :

The set of lines of $\varGamma$ through the point x

$\mathsf {D_4}(\mathscr {N}_x)$ :

The point-line geometry of type $\mathsf {D_4}$ defined on $\mathscr {N}_x$

$\mathscr {P}$ :

The point set of the point-line $\mathsf {E_6}$-geometry defined from $\varGamma$

$\mathscr {L}$ :

The line set of the point-line $\mathsf {E_6}$-geometry defined from $\varGamma$

$\mathscr {E}$ :

The family of new points of $(\mathscr {P},\mathscr {L})$, i.e., the family of extended equator geometries of $\varGamma$

$\mathscr {F}$ :

The family of new lines of $(\mathscr {P},\mathscr {L})$, i.e., those containing members of $\mathscr {E}$

$T_{\mathfrak {e}}$ :

The tropic circle geometry of the extended equator geometry $\mathfrak {e}$

$\varSigma (p)$ :

The quad of $(\mathscr {P},\mathscr {L})$ corresponding to the point x

$\varSigma (\widehat{E}(p,q))$ :

The quad of $(\mathscr {P},\mathscr {L})$ corresponding to the new point $\widehat{E}(p,q)$

$\mathscr {Q}$ :

The family of quads of $(\mathscr {P},\mathscr {L})$

$\mathscr {U}$ :

The family of maximal singular 4-spaces of $(\mathscr {P},\mathscr {L})$

U(L):

The projective 4-space associated to the line L of $(\mathscr {P},\mathscr {L})$

$V^+,V^-$ :

Twin hyperbolic cones

$\mathscr {M}$ :

The family of singular 5-spaces of $(\mathscr {P},\mathscr {L})$

$\mathscr {T}$ :

The family of singular planes of $(\mathscr {P},\mathscr {L})$

$\mathfrak {E}$ :

The geometry of type $\mathsf {E_6}$ defined from $\varGamma$

$*$ :

The incidence relation of $\mathfrak {E}$

$\mathscr {U}(\varSigma )$ :

The subset of elements of $\mathscr {U}$ incident with the quad $\varSigma$

$\mathscr {M}(\varSigma )$ :

The set of 4-spaces of the quad $\varSigma$ obtained by intersecting $\varSigma$ with the members of $\mathscr {M}$ that are incident with $\varSigma$

$x^{\perp _{\varGamma }}$,$x^{\perp _{\varDelta }}$:

The perp of x in $\varGamma$ and $\varDelta$, respectively

$Q_x$ :

The unique quad in $\varDelta$ containing all lines of $\varGamma$ through x

Q(x, y):

The unique quad of $\varDelta$ containing the non-collinear points x and y

Index of Notions

$4^{\prime }$-Spaces	Imaginary completion
$\varDelta$-Collinear	Imaginary point
$\varGamma$-Collinear
$\varPi$-Lines	Neighbors
$\varPi$-Regulus	New lines
	New points
Absolute element
	Opposite
Centre of a full pencil
Chamber	Partial linear space
Close (point and symplecton)	Point-line $\mathsf {E_6}$-geometry
Collinear	Point-line-embedded
Complementary regulus	Principle of duality
Deep point	Quad
Dual embedding
	Regulus of lines
Equator geometry	Residue
Extended equator geometry
	Secant quad
Far (point and symplecton	singular geometric hyperplane
Flag	Singular subspace
Full pencil	Special pair of points
	Standard $\mathsf {B_3}$
Geometric line	Standard $\mathsf {D_4}$
Geometric hyperplane	Subspace
Hyperbolic $\mathsf {B_3}$	Symplectic metasymplectic parapolar space
Hyperbolic $\mathsf {B_3}$-cone	Symplectic pair of points
Hyperbolic $\mathsf {D_4}$	Symplectic polarity
Hyperbolic cone	Symplecton
Hyperbolic line
Hyperbolic plane	Tangent quad
Hyperbolic solid	Thick
Hyperbolic space	Tropic circle geometry
Hyperbolic subspace	Twin