Buildings

Buildings are geometric structures created by Jacques Tits (1930– ), and my book Lectures on Buildings gives the basics of the subject. Tits’ original aim was to find analogues of the simple Lie groups over any field, and buildings are closely connected with Lie theory. The simplest ones — those of type \(A_n\) — are obtained as follows.

The A_n buildings

There is one \(A_n\) building for each field or division ring \(k\) — for example \(k\) might be the rational numbers, the real numbers, complex numbers, quaternions, or indeed a finite field. The building is obtained from an \((n+1)\)-dimensional vector space \(V\) whose field of scalars is \(k\). The proper subspaces of \(V\) are the vertices of the building, and when one subspace contains another, the corresponding vertices are joined by an edge. Each nested sequence of proper subspaces forms a face, or simplex, of this building; a sequence of length two gives an edge, a sequence of length three gives a triangular face, and so on. A maximal sequence has length \(n\), and a maximal face is called a chamber. For example if \(n = 3\), then each chamber has three vertices, corresponding to nested subspaces of \(V\) having dimensions 1, 2 and 3 — in this case a chamber is a triangle (or in other words a 2-dimensional simplex).

Apartments

Every building contains important substructures called ‘apartments’. In the \(A_n\) example above, take a basis for the vector space \(V\) — the proper subspaces spanned by subsets of this basis are the vertices of an apartment. For example, when \(\mathrm{dim} V = 4\) a basis has four elements and the vertices of an apartment comprise: four 1-spaces, six 2-spaces, and four 3-spaces. We can represent this geometrically by taking the four 1-spaces as the vertices of a tetrahedron, the six 2-spaces as midpoints of its edges, and the four 3-spaces as the midpoints of its triangular faces. There are 24 chambers — six on each triangular face of the tetrahedron.

When the apartments of a building are tilings of a sphere, as they are in this case, the building is called spherical. Every building is a direct product of irreducible buildings, and the irreducible spherical buildings of rank at least 3 (the rank is 1+dimension, in the spherical case) come in one of the types labelled classically as follows: \(A_n, B_n = C_n, D_n, E_6, E_7, E_8, F_4, H_3, H_4\). The subscript denotes the rank.

Spherical Buildings

Tits determined all irreducible spherical buildings of rank at least 3, assuming a mild non-degeneracy condition. For type \(A_n\) there is one building for each field (possibly non-commutative), as mentioned above. For types \(E\) and \(D_n\) (\(n\) at least 4) there is one for each commutative field. The classification for types \(B_n=C_n\) and \(F_4\) is more complex. For types \(H_3\) and \(H_4\) there are only the apartments themselves.

Tits achieves his classification by first proving that if two spherical buildings have the same local structure, then there is an isomorphism from one to the other. In particular the local structure determines the global structure. When the two buildings are the same, Tits’ results yield automorphisms that impose restrictions on the local structure, and this is a vital feature of the eventual classification.

Affine Buildings

A building is called affine if its apartments can be realised as tilings of Euclidean space — for example a tiling of the Euclidean plane by equilateral triangles. A building having such apartments arises from the group \(\mathrm{SL}_3\) over the \(p\)-adic numbers \(Q_p\). This group also yields a spherical building (of type \(A_2\), in the way described earlier, and there is a relationship between the two: the affine building yields the spherical building as a structure “at infinity”. This is a general feature of affine buildings. Each affine building of rank \(n\) has a spherical building of rank \(n-1\) at infinity, and Tits used this in classifying affine buildings that are irreducible and have rank at least 4.

Unlike the spherical case, the local structure of an affine building does not determine its global structure. For example, in the \(p\)-adic building mentioned above, the local structure is coordinatized by the residue field of \(Q_p\), which is the finite field of \(p\) elements. There are infinitely many fields having this same residue field, so in general there are infinitely many affine buildings having the same local structure.

Other buildings, and (B,N)-pairs

Buildings that are neither spherical nor affine arise from Kac-Moody groups, which are infinite dimensional analogues of simple Lie groups. Such buildings can be constructed directly from the group by using subgroups \(B\) and \(N\), forming what is called a \((B,N)\)-pair, where \(B\) is a chamber stabilizer, and \(N\) stabilizes an apartment. The method of \((B,N)\)-pairs works for all types of buildings, and in the spherical building for the group \(\mathrm{SL}_n\), mentioned earlier, \(B\) is conjugate to the subgroup of upper triangular matrices. This conjugacy class of subgroups is in bijective correspondence with the set of chambers in the building.

In a Kac-Moody group there are two conjugacy classes of subgroups \(B\). These yield two buildings — both of the same type — forming a twin building. The theory of twin buildings is analogous to the theory of spherical buildings, but is less well-developed.