# Constructing a field from a spherical building

Tits proved that (sufficiently high rank) spherical buildings arise from an algebraic group and a field, so any building is some $\Delta(G, F)$. He also showed that a building isomorphism $\Delta(G,F)\simeq\Delta(G',F')$ induces a field isomorphism $F\to F'$.

This shows that the field is somehow coded up in the isomorphism type of the building. I'm wondering whether a construction is floating around anywhere that shows how to construct the field from the combinatorics of the building.

To help clarify what I'm after: I've seen a construction of the real field from the incidence structure or geometry of the real projective plane (pick a pair of lines, prove they're bijective, define addition via some more lines and unique intersection points which must exist, etc.). In the end you have parametrized a projective line by matching up points on it with the underlying field. The construction requires a choice of a few points in general position and so is given sort of "up to collineation". I'm under the impression that I should view those results on spherical buildings as generalizations of results in projective geometry, and I'm looking for an analogous construction. That is, a construction of a field in terms of the apartments, relations between them, etc. In the end presumably some collection of objects is parametrized by the underlying field. Or is this not the case?

The building in question must be of sufficiently high rank. Projective planes give rise to buildings (of type $A_2$), but need not come from groups or fields. There is plenty of examples of this sort. Same applies to generalized quadrangles (buildings of type $C_2$). And there are also "trivial" rank 2 buildings of type $A_1\times A_1$, which need not come from fields.

Thus if you want to reconstruct a field from a rank 2 building, you will need an extra geometric structure; e.g. in the case of $A_2$, you need e.g. a Pappian axiom to hold.
It gets better with rank greater than 2. E.g. if you have an $A_3$ as a sub-building, then one can use a standard projective geometry argument, which shows that one has a projective space over a division ring. (Yes, you must allow for such a possibility, commutativity would not come from building axioms alone).

• Yep. Notice that Tits classified thick irreducible spherical buildings of rank at least 3; see for example Weiss's The Structure of Spherical Buildings. In particular, projective planes seem, at least for the moment, to be hopelessly wild. – Todd Trimble Nov 15 '12 at 15:55
• Thanks for the response. Sorry if this is basic, but could you comment on what objects end up being parametrized when we construct the field in the building case? The usual proj. geom. argument parametrizes a line in a projective space; its points are field elements. Say we start with a high rank euclidean building, take its building at infinity, and construct the field, what are we parametrizing? We can't be parametrizing things inside a given apartment, that seems like it would always give $\mathbb R$. Is the correspondence that apartments are field elements? – user18087 Nov 15 '12 at 18:49
• It's probably the easiest to look at the associated incidence geometry (the same as the sets of cosets of the maximal parabolic subgroups, with incidence given by intersection). E.g. in the case of $A_n$ this gives you the projective geometry (points, lines,..., hyperplanes). It's been a while, and I don't recall how exactly this works for the building at infinity in the affine case, but it should be similar... – Dima Pasechnik Nov 16 '12 at 2:51

There is no uniform definition for the field of definition. In particular this means that recovering $F$ (somewhat) depends on the classification of spherical buildings of rank at least three.

The basic method to recover a field is to find a projective line embedded in a rank one residue of the building. Such a projective line then has a field associated to it, which you then call the field of definition.

If you want to recover the field for a spherical building of rank at least three combinatorially, one can do the following. Every such building contains a rank two residue isomorphic to a Moufang projective plane (the projective plane is possible because of the classification of spherical coxeter diagrams, its residue is Moufang because of results of Tits), from which you can extract a field, skew field or octonion algebra somewhat combinatorially.

In general it is possible to extract a (skew) field from a rank two Moufang spherical building (also known as a Moufang generalized polygon) in a combinatorial way, but the algorithm is different for each class coming from the classification of Moufang polygons by J. Tits and R. Weiss.

I hope this helps.

• @Koen This is helpful. Do you happen to know a good reference for getting at the Moufang projective plane mentioned your 3rd paragraph? If not, thanks for pointing out that path anyway. – user18087 Nov 15 '12 at 23:33
• Good references are the addendum to "Moufang buildings" of Tits' book "Buildings of spherical type and finite BN-pairs", The book "Moufang generalized polygons" of Tits and Weiss, or the book "The structure of spherical buildings" by Weiss. – Koen S Nov 16 '12 at 12:44

If you want to see explicitly how a field (or division ring) arises from combinatorics of the building (say, in the $A_2$ case, provided that enough extra axioms are satisfied), you can use Von Staudt's construction: He encodes algebraic operations into certain arrangements of points and lines in the abstract projective plane. You can find this construction, say, in Hartshorne's "Foundations of Projective Geometry". Von Staudt used this construction to prove his "fundamental theorem of projective geometry". Tits' theorem is a far-reaching generalization of this result, but the idea is clearest in the $A_2$ case.

Incidentally, this encoding procedure has lead to other interesting mathematical constructions, like Mnev's Universality theorem and its generalizations.