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I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such as $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$ over any field.

Now, Chevalley managed to construct the corresponding groups of Lie type over finite fields; but is there a way one can instead first construct the buildings and then define the group as a particular subgroup of the automorphism group of the building? What do these buildings look like? The building for $\mathrm{SL}_3(\mathbb{F}_2)$, for instance, is the Heawood graph (I can't embed images yet); one can count the 28 apartments making up this complex, which are just hexagons with alternating colors. This graph is the incidence graph for the Fano plane $ \mathbb{P}^2/\mathbb{F}_2 $. One recovers the full group $ \mathrm{SL}_2(\mathbb{F}_2) $ as the group of type-preserving automorphisms of this graph (preserving the distinction points vs lines of the geometry, i.e. the colours in the above graph).
One also recovers the corresponding BN-pair: pick any chamber (any edge) and any apartment containing it (a hexagon with alternating colours); then B consists of the subgroup of automorphisms that fix the chosen chamber, and N consists of the subgroup of automorphism preserving the chosen apartment. Obviously, one could use Chevalley's construction to get a BN-pair and then get back the corresponding building, but I'm really looking for something that starts with the building and gets back the corresponding group.

Are there similar descriptions for the exceptional groups and their corresponding buildings?

(I'm aware that such neat pictorial representations aren't really going to be possible, given that already $G_2(3)$ has order of several million.)

(Crossposted from Math Stackexchange.)

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the link to the MS post is broken –  José Figueroa-O'Farrill Dec 8 '11 at 2:00
    
Sorry; fixed now. –  Will Dec 8 '11 at 2:04
    
The lines leading up to the question here are not quite correct as written. Chevalley's Tohoku paper already gave a uniform way to study simple (adjoint type) algebraic groups of all types over an arbitrary field (not just a finite field). I think the study of the groups in terms of their actions on buildings evolved from his earlier interest in geometries arising from group structure and how the groups act on them. This was exploited for instance in classifying simple groups. Soon the buildings took on a life of their own, especially over local fields. –  Jim Humphreys Dec 19 '11 at 0:24
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2 Answers 2

In the case of groups of rank 2, such as your examples $\mathrm{SL}_3(\mathbb{F}_2)$ or $\mathsf{G}_2(3)$, the building is rather easy to describe (either as an incidence geometry or as a bipartite graph); it is a so-called generalized polygon.

Spherical buildings of higher rank are completely determined by their rank $2$ residues, but perhaps that point of view is not explicit enough for your purposes. There exist several more "geometric" characterizations of the exceptional buildings in the literature, notably by people like Arjeh Cohen and Bruce Cooperstein; see for instance here.

The split buildings of type $\mathsf{G}_2$, $\mathsf{F}_4$ and $\mathsf{E}_6$ can also be described in terms of algebraic structures (but then the corresponding groups can also more easily be described in terms of this structure). For instance, the split groups of type $\mathsf{F}_4$ are precisely the automorphism groups of split Albert algebras (these are certain $27$-dimensional non-associative algebras); the corresponding building can be described in terms of isotropic subspaces of this Albert algebra. A similar description is possible for $\mathsf{E}_6$, but is much more difficult for $\mathsf{E}_7$ and to the best of my knowledge unknown for $\mathsf{E}_8$.

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I knew you were going to answer, Tom! (:-) –  Alain Valette Dec 8 '11 at 10:57
    
@Alain: Indeed, I felt obligated to answer this question ;-) –  Tom De Medts Dec 8 '11 at 14:57
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For spherical buildings of type $F_4$, a complete description appears in J. Tits, ``Buildings of spherical type and finite BN-pairs'', Springer LNM 386.

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