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

Abstract

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

Mixed relations for buildings of type F4

“…Note that this implies that two points of ∆ are ∆ -collinear if and only if they are ∆-collinear. This can also be deduced directly from Proposition 4.5 by noting that this proposition also holds for K = K, where ξ is the image of e 1 under the so-called symplectic polarity of E 6,1 (K) defining ∆, see Lemma 4.19 of [6].…”

“…Remark 4.9. The parabolic intersection mentioned in Lemma 4.8 is called an extended equator geometry in [6].…”

“…It follows that θ does not fix any singular 3-space as, by a dimension argument, each such intersects a ⊥ ∩ b ⊥ in at least a line, and considering the projection of a fixed point collinear to that line onto the fixed 3-space, one obtains a fixed 3-space through a fixed plane, a contradiction. It now follows from Lemma 5.37 of [6] that no point of F 4,4 (L, L) collinear to some point of E is fixed by θ. But then Corollary 5.38 of [6] implies that each point of F 4,4 (L, L) is special to some fixed point in E. Hence θ fixes no point outside E. This now clearly contradicts the assumption that θ pointwise fixes ovoids in fixed symps.…”

“…It now follows from Lemma 5.37 of [6] that no point of F 4,4 (L, L) collinear to some point of E is fixed by θ. But then Corollary 5.38 of [6] implies that each point of F 4,4 (L, L) is special to some fixed point in E. Hence θ fixes no point outside E. This now clearly contradicts the assumption that θ pointwise fixes ovoids in fixed symps.…”

“…We must show nonexistence. We can provide a synthetic argument using the notion of extended equator geometry and its properties proved in [6], see Remark 4.9. Consider two opposite fixed points p, q (one can think of e 1 and e 4 ).…”