Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 11 August 2012

The concept

A building is an axiomatization of the flag variety G/B of a Kac-Moody group G. The advantages of the building point of view are

  1. a "conceptual picture" of the geometry of the flag variety
  2. a good view of the "independence" of the geometry from the underlying field,
  3. a powerful way to work with tori by viewing them as apartments.

The disadvantage is that the axioms do not allow for certain spaces that ought to be considered as flag varieties of "Lie type" groups and this dichotomy between buildings and Lie type groups is unhealthy.

Let G be a group, B a subgroup of G and let W be an index set for the double cosets of B in G so that G= wW BwB.

Morally, the building of (G,B) is the set =G/B with the function δ: ×W given by δ(g1B, g2B) =w, if Bg1 g21B =BwB. In the case when B is a Borel subgroup of a reductive algebraic group then W is a group with a distinguished set of generators: a Coxeter group. The usual approach to buildings is to axiomatize building in terms of the group W and its special choice of generators.

The definition of a building

A chamber system , or flag system on a set S= {s1,, sn} is a set with given equivalence relations j on indexed by the elements s1, ,sn of S. The set is the set of chambers or flags and the relations j are the adjacency relations. For a fixed chamber c, the chambers j-adjacent to c look like i-adjacent chambers When Card{d dc, dic} is the same for all i and c, q=Card{d dc, dic} is the thickness of . A gallery of type i1, ,i is a sequence c1 i1i c2 i2 il ic of chambers such that ck ck+1 . A Coxeter group is a group W with a given presentation by generators s1,, sn, and relations sj2 =1 and (si sj) mij =1, where mij is the order of sisj (mij = is allowed). Hence, the data of a Coxeter group is the set S={s1, ,sn} and the orders mij of the products sisj . A building of type W is a chamber system over S with a function δ:× W such that
  1. If siS and c then there exists c with ci ,c.
  2. If si1 siW is a reduced expression and there is a gallery of type i1,, i from c to d then δ(c,d) =si1 si .
The relative position of c and d is δ(c, d) and the adjacency relations in are recovered from the fact that δ(c,d) =sj if and only if cij ,d . If W is finite and crystallographic is a spherical building and if W is an affine Weyl group is an affine building.

A geometric realization of is a realization of the simplicial complex which has

where J is a subset of I.

Apartments and retraction

Let W be a Coxeter group. The Coxeter complex of W is the building W given by wj,w sj and δ(u,v) =u-1v . A geometric realization of W is the reflection representation 𝔥* of W where the chambers are the fundamental regions for the action of W.

Let be a building of type W. An apartment is a sub-chamber system of isomorphic to the Coxeter complex W. PICTURE

  1. If c1,c2 then there exists an apartment 𝔱 such that c1,c2 𝔱.
  2. If 𝔱,𝔱 are apartments such that 𝔱𝔱 then there is an isomorphism ψ:𝔱 𝔱 such that ψ𝔱 𝔱 =id.
  3. Apartments are convex: If a chamber c lies on a minimal length gallery joining chambers c and d then c lies in every apartment containing c and d.

Reformulating the axioms of a building in terms of apartment, a building of type W is a simplicial complex with a collection 𝔗 of subcomplexes, the apartments of , such that

  1. ,
  2. If 𝔱𝔗 then 𝔱W,
  3. If c1, c2 then there exists 𝔱𝔗 such that c1𝔱 and c2𝔱,
  4. If 𝔱1, 𝔱2𝔗 and 𝔱1𝔱2 then there exists an isomorphism ψ:𝔱1 𝔱2 such that ψ 𝔱1𝔱2 =id.

Let 𝔱 be an apartment and c a chamber in 𝔱. The retraction onto 𝔱 centered at c is the map ρ𝔱,c :𝔱, given by ρ𝔱,c (d)=ψ(d) , where 𝔱 is an apartment containing both c and d and ψ :𝔱𝔱 is an isomorphism.

Affine buildings

Let be an affine building. An alcove wW is dominant if it is on the positive side of Hα for all αR+. The dominant chamber is C={wIw is dominant} PICTURE A sector is a subchamber system of isomorphic to C.

  1. If C is a sector and c is a chamber in then there exists an apartment 𝔱 containing c and a subsector of C.
  2. If C and D are sectors, then there exist subsectors CC and DD which lie in a common apartment.

Let 𝔱 be an apartment and C a sector in 𝔱. The retraction onto 𝔱 centered at C is the map ρ𝔱,C :𝔱 given by ρ𝔱,C (d) =ψ(d) , where 𝔱 is an apartment containing d and a subsector of C and ψ:𝔱 𝔱 is an isomorphism.

The spherical building at infinity or boundary of is the set of equivalence classes of sectors with respect to the equivalence relation where D1 and D2 are parallel D1D2 if D1D2 contains a sector.

Dictionary to algebraic groups

Let G be a linear algebraic group. Let W be the Weyl group and let B be a Borel subgroup of G. The flag variety ={Borel subgroups of G} G/B is a (spherical) building of type W such that {simplices in } = {proper parabolic subgroups in G} {chambers in } = {minimal parabolic subgroups in G} {vertices in } = {maximal parabolic subgroups in G} {apartments in } {maximal split tori in G} so that {simplices in an apartment } ={parabolics P such that P T}. Let G(𝔽) be the group G over the field 𝔽, W the affine Weyl group, and let I be an Iwahori subgroup of G(𝔽). The affine flag variety ={Iwahori subgroups of G(𝔽)} G(𝔽)/I is an (affine) building of type W with {simplices in } = {proper parahoric subgroups in G(𝔽)} {sectors in } {proper parabolics in G(𝔽)} In G/I our favourite chamber, vertex, apartment and sector are I, 0=K, 𝔥={wIw W}, U- ={wIw is dominant} , respectively. If U- is the favourite sector, its equivalence class [U-] has stabiliser B(𝔽) and gB g[U-] is a bijection between the building = G(𝔽)/ B(𝔽) and .

Let v,w W. Then IwI ={gI δ(I,gI) =w}, U-vI ={hI ρ𝔥, w0C (hI)=vI} , IwI ={gI ρ𝔥,I (gI)=wI} , UwvI ={hI ρ𝔥,wC (gI) =vI}.

On the classification

Most buildings are constructed as in (???) and (???). There are only a few "exotic cases" when the rank is 2 or 3. The classification of spherical buildings of rank ≥ 3 [?] and of affine buildings of rank ≥ 4 [Tits, Como] says that they are the buildings corresponding to BN-pairs in untwisted or twisted Chevalley groups over finite fields, local fields or power series fields.

For m2, a generalised m-gon is a connected graph Γ satisfying

  1. the vertices of Γ can be partitioned into "type 1" and "type 2" such that no two vertices of the same type are connected by an edge,
  2. the maximum distance between two vertices of Γ is m,
  3. the length of the shortest circuit in Γ is m.
Generalised m-gons are the same as buildings of type I2(m) by taking chambers to be the edges of the generalised m-gon, and declaring chambers i-adjacent (i=1,2) if they share a type i vertex (see [Ronan, Proposition 3.2]).

A combinatorial projective plane consists of a set of lines L, a set of points P, and an incidence relation between points and lines (a subset of L×P; write p if p is incident to ) such that

Combinatorial projective planes are the same as generalised 3-gons by setting P to be the vertices of type 1, L to be the vertices of type 2 and letting the edges specify the incidence relation.

Notes and References

This page is the result of joint work with James Parkison in 2006. A significant part of this page overlaps with a file buildings12-18-06.tex in Work2007/Bites2007.


[Br] K. Brown, Buildings, Springer-Verlag, New York, 2002. ISBN: ?????? MR??????.

[Ro] M. Ronan, Lectures on Buildings, Perspectives on Mathematics, Academic Press, 1989 ISBN: ?????? MR??????.

[Ti] J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathematics, Springer-Verlag, volume 386, 1974. ISBN: ?????? MR??????.

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