# Split buildings of type \$\$\mathsf {F_4}\$\$ F 4 in buildings of type \$\$\mathsf {E_6}\$\$ E 6

Split buildings of type \$\$\mathsf {F_4}\$\$ F 4 in buildings of type \$\$\mathsf {E_6}\$\$ E 6 A symplectic polarity of a building \$\$\varDelta \$\$ Δ of type \$\$\mathsf {E_6}\$\$ E 6 is a polarity whose fixed point structure is a building of type \$\$\mathsf {F_4}\$\$ F 4 containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type \$\$\mathsf {F_4}\$\$ F 4 ). In this paper, we show in a geometric way that every building of type \$\$\mathsf {E_6}\$\$ 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}\$\$ F 4 fully embedded in the natural point-line geometry of \$\$\varDelta \$\$ Δ arises from a symplectic polarity. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

# Split buildings of type \$\$\mathsf {F_4}\$\$ F 4 in buildings of type \$\$\mathsf {E_6}\$\$ E 6

, Volume 88 (1) – Jan 16, 2018
64 pages

/lp/springer-journals/split-buildings-of-type-mathsf-f-4-f-4-in-buildings-of-type-mathsf-e-6-srM74gqscB
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/s12188-017-0190-5
Publisher site
See Article on Publisher Site

### Abstract

A symplectic polarity of a building \$\$\varDelta \$\$ Δ of type \$\$\mathsf {E_6}\$\$ E 6 is a polarity whose fixed point structure is a building of type \$\$\mathsf {F_4}\$\$ F 4 containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type \$\$\mathsf {F_4}\$\$ F 4 ). In this paper, we show in a geometric way that every building of type \$\$\mathsf {E_6}\$\$ 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}\$\$ F 4 fully embedded in the natural point-line geometry of \$\$\varDelta \$\$ Δ arises from a symplectic polarity.

### Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Jan 16, 2018

### References

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