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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

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Publisher
Springer Journals
Copyright
Copyright © 2018 by The Author(s)
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