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One of the constructions of the group $^2E_6(q)$ was presented by Tits in his paper "Les «formes réelles» des groupes de type $E_6$". It is being constructed by looking at the action of $^2E_6(q)$ on the plane $\Pi$, consisting of points and lines and satisfying certain properties. However, when it comes to the case of a finite field, some mystery happens. Of course, Tits gives us the orbit sizes of the described action, but he completely avoids the details leaving us with pure numbers. In fact, there is at least one typo in the given values, so having the "counting" method would be a really good idea. The typo is in the number of points in $\Pi$: there must be $(q^8-1)$ instead of $(q^4-1)$ in the expression for $N$.

After some more research it seems that the plane $\Pi$ consists of so-called white points, defined in the paper by Cohen and Cooperstein, but again, there is no clue on how to describe the action of $^2E_6(q)$ on these "white" points.

Does anyone have any clues or ideas on how to find those orbit sizes and how to describe the action in general? Any help would be very appreciated.

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It is an early paper by Tits; later on he generalised all these things into the theory of buildings. You can certainly find this construction in one form or another in many books on this subject, e.g. as discussed in this mathoverflow question.

$^2E_6(q)$ has two natural combinatorial structures of points and lines, one with lines of size $q+1$, and another one with lines of size $q^2+1$. It terms of diagram geometries, one has diagram $$o_q-o_q=o_{q^2}-o_{q^2}$$ and points are either leftmost or rightmost type. (This is an attempt to draw Dynkin digram $F_4$ with labelled nodes...)

My guess (mt French reading skills are very bad) is that Tits considers one (or both) of these structures.

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  • $\begingroup$ Thanks for the reference! But are there any explanations regarding precisely these formulas (I mean, the orbit size formulas from the paper mentioned)? $\endgroup$
    – AlexCon
    Jul 26, 2016 at 20:37
  • $\begingroup$ These numbers are sort of easy to figure out. IIRC you deal with a permutation representation of rank 5, with well-understood combinatorics. $\endgroup$
    – Dima Pasechnik
    Jul 26, 2016 at 20:40
  • $\begingroup$ Not yet that easy for me, sorry :) $\endgroup$
    – AlexCon
    Jul 26, 2016 at 20:42
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Please read Robert Wilson paper "ALBERT ALGEBRAS AND CONSTRUCTION OF THE FINITE SIMPLE GROUPS $F_4(q)$, $E_6(q)$ AND $^2E_6(q)$ AND THEIR GENERIC COVERS".

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