Weak buildings of spherical type

SARAH REES

W E A K B U I L D I N G S O F S P H E R I C A L T Y P E

ABSTRACT. In this article constructions are given for weak buildings of spherical type. The buildings are constructed as point-line geometries according to a pattern found within the appropriate Coxeter complexes. In all the examples constructed here there is at least one point of the geometry on no lines of more than two points. We show that this is true in all weak spherical buildings, and discuss to what further extent all weak buildings follow the given construction.

O. I N T R O D U C T I O N

The purpose of this article is to provide a workable description of the weak (that is, essentially, the non-thick) buildings of spherical type. The thick ones of rank 3 or more are, of course, classified by Tits in [10] and are the buildings associated with groups with BN-pairs.

This article complements independent work of Scharlau [7] who shows that from any non-thick building of spherical type a lower rank thick building can be found as a building whose chambers are certain equivalence classes of chambers of the original. The possible types of lower rank buildings associated in this way to any particular spherical building can be calculated from a knowledge of the associated Coxeter group, and this is done by Scharlau in his article. Scharlau's work was motivated by similar work of Tits in [11].

This article adopts a different approach. I show how to build a non-thick building via an associated point-line geometry which is formed by glueing together certain geometries associated with a particular building of lower rank (not, in fact, the building of Scharlau's construction; our situation might be said to be dual to his), according to a pattern found within the appropriate Coxeter complex. Infinitely many concrete examples of spherical buildings of all types can be built in this way, and all non-thick spherical buildings must arise in essentially this way.

To be more specific, the results of this article are divided into three theorems, which are stated at the end of Section 1. We make use of a result about Coxeter graphs; that is, graphs associated to Coxeter complexes. In Lemma 2.5 (proved in Section 2) we see that any such graph can be decomposed in terms of a lower rank Coxeter complex. By replacing the lower rank Coxeter complex by a building of the same type we are able to construct geometries which we see in Theorem A arise from non-thick buildings. In the geometries of Theorem A there are always some points only on thin lines (that is, lines of just two points). Theorem B shows that to any non-thick building we can always associate a geometry with some points on lines of just two points. Finally in Theorem C we discuss to what extent a general weak building of spherical type

-Geometriae-Dedicata 27 (1988), 15-47. © 1988 by Kluwer Academic Publishers.

1 6 S A R A H R E E S

must conform to the pattern of the buildings of Theorem A, and under what

conditions one of those buildings is forced.

1. B A C K G R O U N D S T A T E M E N T O F R E S U L T S

Almost all the notations and definitions related to the buildings of this article are those of [10]. The definition, however, of a building itself, in keeping with current accepted practice (for instance, in [13]), is that which in [10] is attached to weak buildings.

DEFINITION. Let A be a numbered simplicial complex. Suppose that there exists a set a of subcomplexes of A, each isomorphic to the Coxeter complex of some Coxeter group W = ( r l , r 2 . . . . . rn:(riry'iJ=l} with Coxeter diagram M. Then we call the elements of a apartments of A and call A a buildin# of type M or of type W (over {1, 2 . . . . . n}) if the following conditions hold:

(i) Any two simplices of A lie together in an apartment. (ii) If A and B are two simplices of A both contained in two apartments

and E' then there is an isomorphism from E to E' fixing both A and B. A is said to have rank n; maximal simplices have dimension n - 1.

This definition differs from that found in [10] only in that the condition of thickness is not required here. Notice that in the terms of this paper E(W) itself is a building of type M.

We say that A is a buildin# of spherical type or a spherical building if W is a finite Coxeter group. We are interested only in these buildings in this article. We shall limit ourselves sometimes (more often for notational ease than for any deeper reason) to the case of irreducible Coxeter groups (those which cannot be written as direct products of such groups of lower ranks). Since a building with apartments of type E(W 1 x W2) is just the join of two buildings with apartments E(W 1) and E(W2), this does not really restrict our results. The restrictions mean that we are concerned only with those Coxeter diagrams which are in the following list. Whenever these diagrams are referred to, they are assumed labelled as shown unless otherwise stated.

1 2 3 4 n--2 n--I n

An 0 0 0 0 ... 0 0 0

1 2 3 4 n - 2 n - 1 n

Cn 0 0 0 0 . . . 0 0 0

1 2 3 4 n--3 n - 1 n

Dn 0 0 0 0 ..- 0 0 0

I 0

n

W E A K B U I L D I N G S O F S P H E R I C A L T Y P E 17

1 2 3 5 6

E 6 O O O O O

0 4

1 2 3 5 6 7

E7 0 0 0 0 0 0

0 4

1 2 3 5 6 7

E8 0 0 0 0 0 0

0 4

8

O

1 2 3 4 1 2

F4 O O O O GE(m) O m O

1 2 3 1 2 3 4 5 5

H3 0 0 0 H4 0 0 0 0

NOTATION. For convenience we remind the reader of some of the most relevant notation from [101.

If W is a Coxeter group with Coxeter diagram M and generators r i, where i ~ I, then W {il is defined to be the subgroup of W generated by all the r; with ] # i. (Its cosets correspond to the simplices of type i in the Coxeter complex.) w o refers to the longest element of W where it exists.

If A is a building of type M and A is a simplex of type J in A, then we define the building st(A) (the star of A) on the set of simplices which contain A. There is an obvious correspondence between the star of A and the set of simplices disjoint from A but lying in a maximal simplex with A (we may say that such simplices are joined to A, or rather to every vertex of A), which we call (as in [13]) the residue of A, written Res(A) or ResA(A). Res(A) is then a building isomorphic to st(A).

We call maximal simplices of A chambers of A, and say that two chambers are adjacent if their intersection is a simplex of dimension n - 1, i-adjacent if further their intersection is a face of type I\{i}.

We say that two simplices of A are opposite if they are opposite in some (and therefore in every) apartment containing them both.

18 SARAH REES

We say that a rank n building A is thick if every (n - 2)-dimensional simplex of A is in at least three maximal (that is (n - 1)-dimensional) simplices, weak if some (but not every) ( n - 2)-dimensional simplex is in just two maximal simplices (here we differ slightly from [10]) and thin if every (n - 2)-dimensional simplex is in just two (n - 1)-dimensional simplices.

Examples exist of thick buildings, of thin buildings, and of (weak) buildings which are neither thick nor thin. The building of a group of Lie type is always thick while every Coxeter complex is always thin. In fact, every thin building is a Coxeter complex. (This is essentially immediate from the definition; there is only space in a thin building for one apartment, which must be a Coxeter complex.) By [10, (11.4)] every thick finite building of spherical type and rank 3 or more is the building of a group of Lie type.

EXAMPLE 1. Buildings of type A, correspond to n-dimensional projective spaces (see [10]; in the terms of this paper, introduced later in this section~the projective geometries are the 1-geometries of the building). We understand a projective space to be a linear space in which Pasch's axiom holds (as in [10]),but do not require lines to have more than two points.

A building of type A, is thick precisely when every line of the corresponding projective space contains at least three points. Thick An buildings of rank 3 or more are therefore all Desarguesian projective spaces, and each must arise as the system of subspaces of a vector space of dimension 4 or more.

On the other hand, a building of type A, is thin precisely when every line of the projective space contains exactly two points. The n-dimensional projective space with two points per line has as its points and lines the vertices and edges of a complete graph G on n + 1 vertices, and the asociated building (whose vertices of any one type are all the complete subgraphs of G of one particular size) is the A, Coxeter complex.

A weak building of type A, must have some lines (which we shall call thin lines) containing just two points, and some lines (which we shall call thick lines) containing more than two points. We can form these buildings as products of projective spaees of lower dimension. Given two projective spaces, of dimensions n 1 and n 2, respectively, we can form a projective space of dimension n 1 + n 2 + 1 whose points are all the points of either space and whose lines are all the lines of either space together with all pairs of points consisting of one point from each space.

EXAMPLE 2. Buildings of type C, and D, correspond to rank n polar spaces (see [10]). We understand a polar space to be a system of points, lines (viewed as sets of points) and singular subspaces satisfying the axioms of Buekenhout and Shult in [4].

WEAK B U I L D I N G S OF S P H E R I C A L TYPE 19

The thick buildings of type C, and D, are discussed by Tits in [10], and those

of rank 3 or more are covered by his theorem and so found as buildings of a group of Lie type. Any thick building of type C, or D. is associated with a polar space all of whose lines are thick; to any polar space with just thick lines we can associate either a thick building of type C. or a thick building of type D. (as well as a building of type C. which is not thick).

A polar space with two points on every line has as its points and lines the vertices and edges of a complete multipartite graph. The complete multipartite g r a p h K2, 2 ..... 2 with n parts gives the Coxeter complexes of types C. and D,; graphs with more than two vertices in each class give C, buildings which are neither thick nor thin.

Polar spaces with some thin lines and some thick lines are classified by Buekenhout and Sprague in [5]. There are two basic constructions, and all polar spaces of this kind can be constructed via a composite of these two constructions. A simple product of two polar spaces is constructed in a way exactly analogous to the product of two projective spaces. It has as its points all points from either polar space, and as its lines all lines from either together with all pairs of two points with one from each space. A second construction which we shall call the dualized projective space construction builds a rank n polar space with some thin lines from an ( n - 1)-dimensional projective space l-I. As points of the polar space we choose all points and all hyperplanes of II. As a line we may choose the set of all points on any line of I-I, the set of all hyperplanes through any ( n - 3)-dimensional subspace of l-I, or any pair consisting of a point of FI and a hyperplane through it. Both I-I and its dual feature as maximal singular subspaces of this polar space. If II is thick these are the only two singular subspaces of the polar spaces which are themselves thick as projective spaces; otherwise there are none.

In each of the above examples we have built weak buildings A of type M by somehow combining buildings of lower rank (and described by subdiagrams of the diagram for A). Now most of these constructions can be seen very clearly within the Coxeter complexes.

For example, given a complete graph G on n + 1 vertices (which models the A, Coxeter complex) we can find complete subgraphs G 1 and G 2 on, say, nl + 1 and 112 -1- 1 vertices which partition the vertex set. Then every edge of G either is an edge of G~ or an edge of G 2 or joins a vertex of G~ to a vertex of G 2. This gives us the Coxeter complex of type A, as a projective space product of Coxeter complexes of typesA,1 and A,2.

Again, we can find subgraphs G 1 and G 2 of a K2,2...,2 graph G with n parts, so that G1 is a K2,2...2 with n I parts and G 2 the same with n 2 parts, and so that every vertex of G is in exactly one of G~ and G 2, and every edge of G either is in G 1 or in G 2 or joins a vertex of G 1 to a vertex of G z. This gives us the Coxeter

20 SARAH REES

complex of type C, or D, as a polar space product of two Coxeter complexes of the same type but lower rank.

The construction that builds a C, or D. polar space out of a projective space and its dual (the dualized projective space construction) does not seem to be so obviously explainable via the Coxeter complex, but that is really because we are using a rather prejudiced view of the building, insisting on describing it in terms of the points and the lines of the polar space. It turns out to be easier to understand this particular building if we look at it rather from the point of view of a geometry whose points are the maximal singular subspaces of the polar space. We recall that buildings were defined as simplicial complexes, and so to any one building we can associate many different point-line geometries, choosing vertices (or indeed higher-dimensional simplices) of any kind as points, and lines in various ways.

DEFINITION. Suppose that A is a weak building of type M and rank n, and let I be the set {1,2 .... ,n}. For any i t I we define the (standard) i-geometry of A, A i, as a geometry of points and lines as follows. Ai has as its points all the vertices of A of type i. For any simplex V of A of type I\{i} we define a line of A~ as the set of all points of A~ joined to V. Notice that two points of A~ are collinear if they are vertices in adjacent chambers of the building.

More generally, for any subset J o f / w e say that two simplices of type J are j-adjacent (for somej in J) if they lie in j-adjacent chambers. We call a set of simplices of type J a j-line, forj in J, if it is the set of all simplices of type J joined to some simplex of type I\{j}; notice that any two simplices on a j-line are j-adjacent. We define a J-geometry to be any geometry whose points consist of all the simplices of type J, and whose lines are all possible j-lines forj in some subset of J. The J-geometry containing j-lines for every j in J is called the standard J-geometry and denoted by A s (though A(~} is abbreviated to A~). The J-geometry with no edges is called the trivial J-geometry.

The geometries described above are often (e.g. in [10]) referred to as the shadow geometries of the building. Clearly the standard 1-geometries of buildings of types A,, C, and D, are projective and polar spaces. The i-geometries of A, buildings for 1 < i < n are Grassmannian geometries.

Clearly if E = E(W) is a Coxeter complex then the lines of any J-geometry for Z lines must contain just two points. So any J-geometry for E is in fact a graph, and we shall call it a J-graph. Its edges will be called J-edges. A graph of this type will be called a Coxeter 9raph of type W (or M, where M is the relevant Coxeter digram). The (standard)/-geometry for Z will also be called the (standard) i-graph, and denoted Z~.

Our results in this article exploit the fact that any finite Coxeter graph can be


decomposed in terms of Coxeter graphs of a single lower rank Coxeter complex. More specifically, if p is a vertex of the/-graph E~ of a finite Coxeter complex E(W) we can find elements w of W, associated subsets J(w) of the indexing set I and subgraphs Ew(p ) of E i (which are in fact the orbits under the action of the stabilizer of the vertex p) so that each Ew(p) is isomorphic to the J(w)-graph of E(W~}). Edges between these subgraphs are determined by incidences between the corresponding simplices of ~(W {i~ ). (This result is given as Lemma 2.5 in Section 2.) Now given this decomposition of the Coxeter graph in terms of a Coxeter complex of lower rank, we can substitute any building of the same type for the smaller Coxeter complex. In this way we define a point-line geometry, usually with some thick and some thin lines, which will in fact be the/-geometry of a weak building.

CONSTRUCTION 1. Suppose that E is the Coxeter complex of a finite Coxeter group W with generators rl, r2,...,rn, and let I = { 1,2 ..... n}. For some i ~ I let F be a rank n - 1 building each of whose apartments is isomorphic to the smaller Coxeter complex E' = E(W/~). Then we can construct a point-line geometry Ag = Ai(Ei,F) as follows.

For each of the subgraphs E~(p) of E~ found as in Lemma 2.5 we form a corresponding J(w)-geometry, H~ = Hw(F), for F. As points of Hrr we use the simplices of F of type J(w), and for any k for which we have k-edges in E~(p) we have all possible k-lines in Hw. If ~w(p) consists of just a single vertex then Hw consists of a single point, and no lines.

Now the point set of A~ is formed as the disjoint union of the point sets of all the point-line geometries H~, as w runs through the set of elements of W which indexes the distinct subgraphs Zw(p) of Ei. As lines of Ai we have all the lines of each of the point-line geometries H~, which may be thick or thin. In addition we have some thin lines which arise in the following way:

(i) If the graph E~(p) consists of a single vertex which is joined to every vertex of the graph E w, (p) then we join the single point of the geometry Hw to every point of the geometry H~. by a thin line.

(ii) If E~(p) and Ew,(p ) are graphs of at least two vertices each and J(w) = J(w') then we join every point of Hw to the unique correspond ing point of H~, by a thin line.

(iii) If Ew(p) and Ew,(p ) are graphs of at least two vertices each and J(w) ~ J(w') then we join by a thin line any two points from H w and H w, which correspond to simpliees joined in F.

Any automorphism of the building F will induce a collinearity of the point-line geometry A~, and also an automorphsim of the underlying building (which will be recovered in Section 3).

22 S A R A H R E E S

EXAMPLE 1. I fE is of type E6, then for Z 6 (or El) we have three subgraphs {p} = Zid(p), E,l(p ) and Za(p), where a = r6rsr3r4rzr3rsr 6. Let E' be the Coxeter complex of type D 5 over { 1, 2, 3, ~, 5}. Z~l(p) consists of the neighbours of p, and corresponds to the 5-graph of E'. ~a(P), on the vertices at distance 2 from p in El , corresponds to the 1-graph of Y.'. In this case the opposite involution does not induce an automorphism of the graph.

Now i fF is any building of type D 5 over {1, 2, 3, 4, 5} as above we form three geometries Hia,H,I and H a. Now Hid is a single point, Po- Hrl is isomorphic to the 5-geometry F 5 of F, and Ha is isomorphic to Fx. Now A 6 is a point-line geometry whose points are all the points of Hid, H~ and H a. As lines we have the lines of H~ and H a together with all pairs {Po,q} where q is a point of lit, and all pairs consisting of a point of Hrl and a point o fH a which are joined in F.

Now A 6 is the 6-geometry of an E 6 building A. So points are vertices of type 6. Lines correspond to vertices of type 5. The vertices of types 4, 3 and 2 of A arise as the singular subspaces of A~ and the vertices of type 1, which should be rank 5 polar spaces, arise as the geodetic closures of pairs of points at distance 2 in A 6 .

EXAMPLE 2. IfX is of type E 7 then we have four subgraphs {p}, Er7(p ), Xa(p) (where a = worT) and {p'} = Zwo(P) for the graph E x. ErT(p ) and Ea(p), which are isomorphic, consist of the neighbours of p and of p' respectively. We identify Er7(P) as the 6-graph of a Coxeter complex of type E6, and Ea(p) as the 1-graph of the same Coxeter complex. Then the edges between Y,v(P) and Za(p) are, as in Lemma 2.5, joining pairs of vertices which are joined in the E 6 Coxeter complex. The opposite involution induces an automorphism of E 1 which swaps p with p', and ErT(p ) with Ea(p).

Now, given a building F of type E 6, we can build the 7-geometry A 7 of a weak building A of type E 7. Its points are the points Po and pl of trivial geometr ies Hid and Hwo together with the points of a geometry H,7 which is isomorphic to F 1 and a geometry H a isomorphic to F 6. As lines ofA 7 we have the lines of H,~ and Ha, all the pairs {Po, q} with q a point of H~7, all the pairs {p~,q} with q a point of H a, and all pairs of points from H,~ and H a which are joined as vertices of F. Lines correspond to vertices of type 6 of A. We can recover the remaining vertices of the building easily in terms of the points and lines joined to them.

In the cases where the opposite involution preserves the set of vertices of type i and induces an automorphism of the graph we can sometimes generalize the above construction.

CONSTRUCTION 2. Suppose that E is the Coxeter complex of a finite Coxeter group W with generators r 1 , r 2 ..... r,, let I = { 1, 2,...,n}, and suppose


that E i decomposes into subgraphs Ew(p) for various elements w of W as in Lemma 2.5. Let T be a set of elements of W representing the subgraphs Ew(p) (so each such subgraph is indexed by a unique element of T).

We suppose that we are in the following situation. First, for each w in T either Ewo~(p ) = Ew(p) (we define A to be the subset of such w) or WoW is also in T. Second we suppose that we can divide T \ A into two parts, B and w o B so that, for all w in B , J ( w ) = J(woW ) and if there are edges between Ew(p) and Z~,(p) with w in B and w' in Wo B we must have w' = w o w. In this case we say that Y~i has good opposite symmetry.

Now for some i e I let F be a rank n - 1 building each of whose apartments is isomorphic to the smaller Coxeter complex E(W~i}). Further, let A be an indexing set of at least two elements. Then we can construct a point-line geometry Ai = Ai(Ei, F,A) as follows:

The contruction is simply a more general version of Construction 1, and reduces to that one if A contains just two elements.

We form a single J(w)-geometry, H w = Hw(U) for each w in A, and IAI different ones, an Ht~, ~) for each 2 e A, for each w in B. As points of each H~ (or Ht~,~)) we use the simplices of F of type J(w),and for an y j for which we have j-edges in Ew(p) we have all possible j-lines in H w (or H~,z)).

Now the point set of Ai is formed as the disjoint union of the point sets of all the point-line geometries Hw, where w ~ A, and Htw,~ ~, where j e B and r ~ A.

As lines of A~ we have first all the lines of each of the point-line geometries H~, or Htw,~ }, which may be thick or thin. In addition, as before we have lines consisting of points from different geometries H w and HIw, z). If w and w' are both in A, and there are edges from Ew(p ) to E~,(p) we have thin lines of the same kind from Hw to H~,. Ifw and w' are both in B, and there are edges from Ew(p) to E~,(p) we have thin lines of the same kind from Hl~,a) to Hcw,,z), for each L. If there are edges from E~(p) to E~,(p) and w ~ A but w' ~ B or w 0 B then there are also edges from Zw(p) to Ewo~,(p ) in E i. In that case we have appropriate thin lines from H~ to each Htw,,a) as L runs through A. Finally,

when there are edges between E~(p) and E~ow(P) for w e B we insert lines of IAk points each joining corresponding points of all the Ht~,,a)s.

Again, automorphisms of F induce collinearities of A~ as do permutations of the set A. And these induce automorphisms of the underlying building which will be recovered in Section 3.

EXAMPLE. IfE is of type F 4 then E 4 has subgraphs {p}, E~4(p), E~(p), E~o,4(p ) and {p'} = E~o(P)- Z~4(P), the neighbourhood of p, and E~o~4(p), the neighbourhood of p', are both isomorphic to the graph of vertices and edges of a cube; that is, the 3-graph of a Coxeter complex of type C a. Eo(p) is isomorphic to the graph of vertices of an octahedron; that is, the trivial 1-graph of the same

24 SARAH REES

Coxeter complex. Corresponding vertices in E,,(p) and Ewor,(P) are joined, while any vertex in Ea(p) is joined to every vertex of E,,(p) or Ewo~,(P) which is joined to it in the C 3 complex. The opposite involution induces an automorphism of Za of order 2 which fixes ~ ( p ) as a subgraph, but swaps p with p' and ~,,(p) with ~0,,(P)"

Now Z# has good opposite symmetry so we can use the more general second construction. (We have A = {a}, B = {id, r4}. ) For any C 3 building F, and any set A of at least two elements we form ]A] copies Ht~l.~j of 1"3, the dual polar space, and one copy H a of F 1, the polar space associated with F. As points of the 1-geometry A a of an F4 weak building we have I A I points px together with all the points of the Ht, l,a)'s and Ha. We have all the lines of each Hrl.x and of Ha as lines of At. In addition each pa is joined by a thin line to every point of H~,~.a) (same 2), each point of H~ is joined by a thin line to every point of every H¢,~.x) that is joined to it as a vertex of F, and lines of IAI points each join corresponding points of the different H¢~l,x)'s.

Again lines of At correspond to vertices of type 3 of A. Type 2 vertices of A are singular planes and type 1 vertices the rank 3 polar spaces that arise as the geodetic closures of pairs of points at distance 2.

T H E O R E M A. Suppose that Z is the Coxeter complex of a finite Coxeter group

W with generators r~ r z . . . . ,r,, that F is a building with apartments each isomorphic to E(W{~) for some i ~ {1, 2 . . . . , n}, and that Z i is decomposed as in Lemma 2.5. Then the point line geometry Ai(~ ~, F), defined as above, is the

i-geometry of a weak building A(Ei, F) of the same type as E. I f E~ has good opposite symmetry then A~(Ei,F,A) is also the i-geometry of a weak building

A(Y~, F, A). In each of the point-line geometries A~(E~, F) and A~(£~, F, A) there is a point

which is only on thin lines. In fact this is generally true for appropriate point- line geometries associated with weak spherical buildings.

T H E O R E M B. Suppose that A is a we ~k building of spherical type. Then for some i there is a point p of the point line geometry A t with the property that all

lines of A i through p are thin.

EXAMPLES. IfA is of type A, then we can find a point of A~ only on thin lines when the projective space A~ can be found as a product involving an (i - 1)-dimensional space. For A of type C~ or D, and i < n, we have a point of A t on just thin lines when the associated rank n polar space is a product of polar spaces of ranks i and n - i. For A of type C~ and i = n we must be able to find the building also as a building of type D~, while any building of type D n which has a point of its n-geometry on just thin lines must arise from the dualized projective space construction.


The well-known constructions of weak buildings of types A,, C, and D, are really very straightforward to work with, and there seems little value in using the more general constructions of Theorem A to produce buildings of these types. It seems worthwhile, however, to observe the connection between these constructions and the constructions of Buekenhout and Sprague, and in doing so we recognize that even the second construction of this paper is not general enough to produce all examples. For in a building A(E i, F) the point p has a unique opposite (which is either the unique point of Hwo when that subgeometry contains just one type or otherwise the unique (as we shall see in Section 3) simplex of A incident with every vertex of Hwo). In the more general building A(E i, F, A) each p~ has many other opposites; that is, all the other Pu"

But any two points opposite to p~ are also opposite each other. This is not in general true in buildings, athough in Coxeter complexes each simplex has a unique opposite. Below we construct an example which we see from the behaviour of opposites cannot be a building of type A(ZI,F) or A(ZI,F,A).

EXAMPLE. Consider the weak building A of type C,~ which may be built as a product of two thick generalized quadrangles, Q1, with point set P1 and line set L 1, and Q2, with point set P2 and line set L 2. The rank 4 polar space has P1 u P2 as its point set and L 1 ~ L 2 ~d (P1 x P2) as its line set. Any line of L1 or L 2 corresponds to a 'point' ofA 2 which is just on thin lines ofA2, but we cannot find such a 'point' for any of the other geometries A i. Since both Q~ and Q2 are thick lines any element o fP 1 x P2 also has some thick 'lines' through it in A 2.

Now two lines of a polar space are opposite in the corresponding C, building precisely when no single third line is coplanar with both, so in this case we observe that two lines of the same quadrangle are opposite when they do not meet. But in general, in a generalized quadrangle it is easy to find two intersecting lines disjoint from any third line. So, in this case, we can certainly find lines opposite any line of L1 o r L 2 which are not opposite each other.

Where A and A~ are as in Theorem B, we know from the above example that we cannot prove in general that A~ is isomorphic to A~(E~, F) or A~(EI, F, A), but we can prove some general structure results, and in certain low rank cases (where E~ decomposes as a small number of subgraphs) we have a full description.

T H E O R E M C. Suppose that A is a weak building of spherical type M over

I = { 1,2 . . . . . n }, with apartments isomorphic to 32 = E(W). Now suppose that for

some i in I a point p of the point-line 9eometry A~ is on no thick lines. Let F be the buildin9 st(p). Then

(i) any line joining two or more points of A,,(p) lies within that set, and the 9eometry induced on Ar,(p ) is isomorphic to a J(i)-geometry for F, where

J(i) is the set of nodes joined to i in M; moreover, i f p' is any point of A i

26 SARAH REES

opposite p, then p' is on no thick lines and the geometries Ar,(p ) and Ar,(p') are isomorphic;

(ii) if any two points o f A t can be found in a single apartment with p, Ai is

isomorphic to At(E t, F); (iii) if M = E 6 and i = 1 or 6, or i f M = E 7 and i = 7, A t is isomorphic to

Ai(Z i, F).

I f M = E 7 and i = 1, or i f M = F 4 and i = 1 or 4, or i f M = E 8 and i = .8, A t is

isomorphic to Ai(Zt, F,A)for some indexing set A.

2 . T H E S T R U C T U R E OF CO X ETER G R A P H S OF S P H E R I C A L TYPE

In this section we suppose that W is a finite Coxeter group with generators r l , r 2 . . . . . r n and diagram M, and let I = {1,2 . . . . . n}. We investigate the /-graph E i (as defined in Section 1) of the Coxeter complex E = Y.(W).

From the standard description of the Coxeter complex E(W) in terms of the group W (see [10]) we find a similar description for the associated Coxeter graphs. Where for any subset J of I = {1,2 . . . . ,n}, W J is defined to be the subgroup of W generated by those rj with j q~J, the faces of E of type J correspond to the right cosets of W J in W. Two right cosets W J x and W S y

are j-adjacent, for somej E J, i f x y - 1 is an element of the double coset W J r~ W J. In particular, the vertices of the standard/-graph :E t correspond to the right

cosets of W {i} = ( r / j ~ i ) , and two vertices W{i}x and W(i}y are adjacent in the graph precisely when x y - 1 is an element of the double coset W {i/r i W ~}. By [10], W is the full group of automorphisms of Z, and provides a large group of automorphisms of the graph, W {~} stabilizes the vertex W ii} of Y~t; we are interested in the remaining orbits of W {g} on El. Throughout this section we shall write p for the vertex W ~g/of Zg.

We list below some properties of Coxeter groups which will be useful in our decomposition of the Coxeter graphs.

LEMMA 2.1. Let E t be the standard i-graph o f the Coxeter complex Y~(W) o f

a f lni te Coxeter complex W, and identify the vertices o f E~ with right cosets o f

W {q in W. Then

(i) the orbits o f W li} on ~i are in correspondence with the double cosets o f

W I~} in W; each such double coset can be written uniquely in the form

W {~} a W ~}, where a is the single element o f minimal length (as a word in the

generators r j ) in the double coset W {i} aW{g};

(ii) either W {g} is self-normalizing or its normalizer is W {q u W (~ w o W {~}, where w o is the longest word in W," in the second case r~ w o = w o r t and the

unique shortest element o f W li} w o W Ig} is w o w 1 , where w I is the longest

word in Wfi};


(iii) i f a is o f minimal length in the double coset W {i} a W ~1, W {~} n W {~}a = W 1ta~, where I \ l (a ) = {j e I \{i}; rj = r~,for some k ~ I\{i}}.

Proof. (i) Tha t the orbits of W/~l on Z i correspond to double costs of WIil in

W is s tandard pe rmuta t ion group theory. The rest is found in Bourbaki [1]. For future reference we note f rom there that if a is of shortest length in

W I~} a W Iil, a is called (I - (/})-reduced. Further , a is (I - (/})-reduced if and only if, for all elements x of W IiI, l(xa) = l(x) + l(a) = l(ax). (Here l(w) denotes the length of w as a word in the generators ri.)

(ii) (This p roof is due to D. Testerman.) Certainly the normal izer of W/~l

must be a union of double cosets of Wlel in W. We need to see what possible ( I - (/))-reduced elements can normalize W ~. So suppose that a is such an element. Then, since, for all j distinct from i, r~ e W ~) and ar t = r i a has length equal to bo th l(a) + l(r~) and l(r2) + l(a), we see that for al l j distinct f rom i,r~ is

a genera tor rk, where k =~ i. In the language of root systems this means that a takes the simple root ~j to + ~k" Since also l(arj) = 1 + l(a) > l(a)a(c~j) must be positive (e.g. by Car ter [6, p. 18]), in fact a(~tj) = ct k for some k ~ i. Also, since l(a) > O, a must take some positive roots to negative roots, so a(ct~) must be negative. Further , since the images under a of the simple roots must span the root system, a(c¢~) must be a negative linear combina t ion of a set of roots which includes c~.

N o w suppose that w E W {i}. Since l(wri) > l(w), w must take ~i to a positive root. So now, ifw 1 is the longest element of W l~l, w1(7~) must be a positive root. Since, as an element of Wlil, Wl must preserve (~j: j ~ i), while acting as a pe rmuta t ion of the root system spanned by all the %., we see further that

Wx(~) must be a positive linear combina t ion of a set of simple roots which includes ct~. By the previous pa ragraph we see that the coefficient of ¢t~ in awx(~) must be negative, so aw~(~i) is a negative root. On the other hand,

aw ~(~2) is a negative root for all j ~ / ( s i n c e l(w ~ r 2) = l(w 1) - l(r 2) < l(w Ow t (% ) is a negative combina t ion of those ~ with j ~ i, while for each j ~ i, a(c~) is

positive). So we see that aw~ takes every simple root to a negative root, and hence every root to a negative root. This forces aw~ to be the longest element

w o o f W . S o a w ~ = w o , a n d a = w o w 1. Finally observe that, since conjugat ion by w o permutes the generators rj, w o

normalizes W li} precisely when it commutes with r~. (iii) is a special case of L e m m a 2 of I-8]. It first appears in [10] and [12].

R E M A R K . The results of some useful calculations for Coxeter graphs of types E6, E 7 and E 8, in part icular some lists of (I -- { 1})-reduced elements for some i, are given in [2].

N O T A T I O N . We shall use the nota t ion Zw(p) to denote the set of vertices of Z~ which cor respond to cosets of W I~) contained in the double coset W ~} w W I~},

28 SARAH REES

and also the subgraph induced on this set of vertices. Then each Zw(p) is also the orbit of W (it on 2 i which contains the image of the vertex p under w. Of course, if W ~ w W ~i~ = W I~l w' W ~ then Ew(p) = Y,~,(p).

We introduced I(a) above, for ( 1 - {/})-reduced elements a of W as ~j ~ I:rj = r~, for some k e l}. We define in general I(w), for any element w of W, to be equal to I(a) when w is in the double coset W{~IaW ~}. We define also J(w) to be I(w)\{i}. For ease of notation we shall write I(0 rather that l(ri) and J(/) rather than J(ri). Notice that I(i) = {i} u {j:rj # r~' for any k} = {i} u {j:rj does not commute with ri} = {i} u {j joined to i by an edge of the Coxeter diagram}, so that J(i) is simply the set of nodes surrounding i in the Coxeter diagram.

It helps to give some further definitions at this stage.

DEFINITI ON (following [ 10, (2.30)]). If A and B are simplices of E we define a 9allery from A to B to be a sequence of chambers of E, the first containing A, the last containing B, and any two successive chambers being adjacent, and a minimal gallery from A to B to be such a gallery of shortest possible length. We define the projection of B onto A, which we write as projA B, to be the simplex containing A which is the intersection of all the first terms of minimal galleries from A to B. ProjA is a map from Y. to the star of A.

The following facts relating to projections will be useful.

LEMMA 2.2. (i) The projection of B onto A has its stabilizer the intersection of the stabilizers of A and B.

(ii) I f B is a chamber then so is projA B. (iii) I f B is a face of C, then projaB is a face of projaC. (iv) Where a is any ( I - {i})-reduced element of W and l ( a ) is defined as in

(2.1),for any x E W (i}, Wt~") x is the projection onto p of the vertex W {i} ax. For x and y both in W {~1, the vertices W {~} ax and W I~} ay have distinct projections onto p.

Proof (i),(ii) and (iii) are proved in [10, §§(12.5),(2.29) and (12.2)]. The first part of (iv) is almost a direct consequence of (i), since the projection of W ~1 ax onto p = W ~}, as a simplex containing the vertex W {~}, must be a coset in W {~} of some subgroup of W/~. For the second part of (iv) notice that for x and y in W{il, WI~}ax = W{~}ay when axy -a a - l e W{~}; that is, when xy -~ ~ W {~} c~ W ~}" = W I~). So we see that W t ~} ax = W {~} ay precisely when W ~ ) x = W t~")y.

LEMMA 2.3. (i)For any simplex V of Z the set of vertices of El joined to V and in E~(p) either is empty or maps under projp to the set of simplices of type I(a) joined to projp V.

(ii) Suppose that some simplex V of type J in E is joined to just two simplices


A and B of type K. Then K \ J = {j},for some j, and A and B must be j-adjacent.

Proof. (i) Certainly the stabilizer in W ~i~ of I/", which we shall call H, must preserve the set of vertices of E i in Ea(p) that are joined to V. Under the projection map these must go to a set of simplices of type l(a)joined to proj~ V, which is also preserved by the action of H. But now, by (2.2)(i) H is equal to the stabilizer in W of projp V, which acts transitively on the simplices of any one type in the star of projp V. In particular the set of all vertices of type I(a)joined

to projp V forms a single orbit of H. So no proper subset of these can be fixed as a set by H. The result follows.

(ii) As simplices in the Coxeter complex st(V), A and B have type K\J . The number of simplices of type K \ J in st(V) is equal to the index of W J~K in W J. Since this is two W JuK must be maximal in W ~, so [K\J[ = 1. Now the standard K \ J graph for st(V) must be connected, so A and B must be j-adjacent.

Now recall that when we defined our Coxeter graphs we defined each edge as the pair of vertices joined to a given (n - 2)-dimensional simplex of Y. So if two simplices A and B of type J are j-adjacent vertices of a Coxeter graph then they lie in j-adjacent chambers of E. For any pair of j-adjacent chambers C ~_ A and C' _ B the simplex C c~ C' would define the edge {A, B}. But since there may be many possible choices for C and many possible choices for C' many different (n - 2)-dimensional simplices may define the same edge of the graph. In fact a unique smallest face of E can always be found that defines the edge {A, B} as the set of simplices of type J joined to it. This follows from a result of Tits in [10, (12.11), (12.14)] which will be stated in more generality in Section 3. Any j-adjacent pair of simplices of type J can be found as the pair of simplices of type J joined to a unique simplex of type J \ { j } w J(j). We shall call this simplex (A, B). It is the intersection of all simplices C ~ C' as above.

LEMMA 2.4. Suppose that A in Ea(p) and B in Zb(p), where a and b are

(I - {i})-reduced and not necessarily distinct, are adjacent vertices of the graph Z i. Then projpA and projpB are the only simplices in Y of types I(a) or l(b)joined

to projp(A,B). I f I (a)# I(b) then projpA and projpB are joined in Z,. I f I(a) = I(b) but a v ~ b then projpA = projpB. I f a = b and projpA and projpB are distinct then they are j-adjacent for some j.

Proof. We use 2.3(i) to see that the set of simplices of type I(a)joined to the projeetitm 8f ~-q,~7~is~ih~:projection of the set of vertices in Za(p) joined to (q, r ) itself. Similarly for b. If I(a) ~ I(b), then since projp q and projpr are the unique simplices of their types joined to the projection of (q, r ) they must be faces of it. So certainly they are joined to each other. If I(a) = I(b) each of proj, q and projv r i's the unique simplex of type I(a)joined to the projection of

30 SARAH REES

(q, r ) , so the two are equal. Finally, if a = b, 2.3(ii) forces the projection of q and r to be adjacent.

L E M M A 2.5. Suppose that Y~ = E(W) is a finite Coxeter complex of type M over I = { 1,2 . . . . . n}. Choose i ~ I and let p be the vertex ofE i corresponding to (and so also fixed by) W ~il. For any w in W let Ew(p) denote the subgraph ofY. i induced on the orbit o f W Iil that contains pw. IfZw(p) is not a single vertex we can identify it with one of the Coxeter graphs, say a J(w)-graph, of the Coxeter complex Z(W ~il) of type I\{i}.

(i) I f Zw(p ) consists of a single vertex q then q is joined either to every vertex or to no vertex of any other graph Zw,(p).

(ii) I f E~(p) and Ew,(p ) are subgraphs each of at least two vertices, and J(w) = J(w'), then if there are any edges between Yw(p) and E~,(p) they join all pairs of vertices which correspond to the same simplices o fZ(W ~1).

(iii) I f E~(p) and Ew,(p) are subgraphs each of at least two vertices, and J(w) ~ J(w'), then if there any edges between Z,~(p) and E~,(p) they join all pairs of vertices which correspond to simplices which are joined in Z,( WI~).

(iv) The subgraph Zr/(p ) consists of the neighbours of p. J(ri) consists of those j in I joined to i by an edge of M.

(v) I f w o commutes with r i then Zwo(P) = {P'} is the second of two single vertex subgraphs and for all w, Yw(p' ) = E~ow(p ). Otherwise {p} is the only such graph, and the vertices of Ewo(p ) are precisely the vertices of ]E of type i joined to the opposite of the vertex W {~l.

Proof. Where I(a) and J(a) are as defined in this section, we see from 2.2(iv) that the projection map projp identifies the set of vertices of each Ea(p) with the set of simplices through p of type I(a). These correspond to simplices of type J(a) of the lower rank Coxeter complex E(W ~) = st(p). We can use (2.4) and the group action to see that edges on each Za(p) and between two such subgraphs (as in (i), (ii) and (iii)) are as given.

The first s tatement of(iv) is immediate from our earlier observat ion that two vertices W¢Jlx and W I~/y are adjacent precisely when x y - ~ ~ W ~1 r~ W ~. J(r~) = J(i) was described when the nota t ion J(w) was int roduced at the beginning of this section as the set of nodes of M joined to i by an edge.

For (v) notice that if E,(p) is a single vertex subgraph {p'} distinct from {p}, then W ~ a x = W ~ a for all x in W, so a normalizes W/~/. 2.1(ii) verifies that Zo(p) = Ewo (p) = { W I~ w o }, and that in this case w o commutes with r i. Now for any W, Ea(p' ) consists of the vertices of the form W~WoWX, for x in W (~/, or ra ther those in the double coset W ~ w o wW I~} = Z~o~(p).

The opposite involut ion maps p~= I~! ~ to the simplex W (~° w o . When Wo normalizes W ~ ffiis'is the Single vertex w ~i~ w o. Otherwise it°is not of tYlSe i, but

WEAK B U I L D I N G S OF SPHERICAL TYPE 31

is joined to those vertices W {i} x of type j where W li} x c~ W {~}w° w o = W {~} x c~ w o W {~} is non-empty. This happens when x e W {i} w o W {i}, or rather when W{~}x is a vertex of Zwo(P ).

3. P R O O F OF THEOREM A

In order to prove that the point-line geometries of Theorem A are indeed the /-geometries of buildings we need to see how to recover a building from one of its geometries. We need the results of [10,Ch. 12].

We suppose for the time being that A is any building of type I41, where W is a finite irreducible Coxeter group. It is only in this section that we make any real use of the irreducibility of W. Given the/-geometry of a building A we shall recognize each vertex of A of type distinct from i by the set of vertices of type /joined to it. (If W were not irreducible vertices of some types would be joined to every vertex of type i, and so could not be recognized in this way.) If V is any vertex, or indeed any simplex of A, we shall call the set of vertices of type / jo ined to V the i-shadow of V, and write this sh~ V. More generally, the J-shadow of V, written shj V, will denote the set of simplices of type J joined to V.

The following facts all appear in 1-10, Ch, 12].

LEMMA 3.1. (i) Any simplex V of A of type K contains a minimal face which has the same J-shadow as V. We call this face the 'J-reduction' of V; its type is the smallest subset K' of K that separates J from K in the Coxeter diagram. V is called 'J-reduced' if it is its own J-reduction.

(ii) Suppose that A and B are two smplices of A, and that A' is the J-reduction of A. Then shiB ~ shsA only if A' and B are joined in A and the type of A separates the type of B from J in the Coxeter diagram. The shadows coincide if and only if A and B have the same J-reduction.

REMARK. We see that all vertices of A are J-reduced for any J (here we need the fact that W is irreducible, otherwise some vertices might have empty /-reduction), also that, since the lines of A~ are simply the/-shadows of simplices of type I\{i}, each line of A~ arises as the/-shadow of a unique/-reduced flag of type J(/).

Now, following [10], we shall call a set of points ofA~ a subspace of A~ if it is equal to the/-shadow of some simplex of A. (Notice that by (3.1)(ii) above each subspace contains any line of A i of which it contains two or more points.) We shall define the type of a subspace X to be the type of the unique (/ ~i~)-17educed simplex of which it is the/-shadow. What (3.1) implies is that

~¢e know the set of subspaces of Ai, together with their types, then we can


reconstruct A. For we can certainly recover all the vertices from the list of subspaces. We have a complete description of the simplicial complex once we know which pairs of vertices are joined. (3.1)(ii) tells us that two vertices, x of typej and y of type k, are joined when either one of their/-shadows is a subset of the other or the two/-shadows intersect in a subspace of type {j,k}.

Now suppose that Z is the Coxeter complex of W, and p a chosen vertex of type i. We need to see how the/-shadow of a vertex ofl~ splits up across the various subgraphs of El.

LEMMA 3.2. Suppose that the 9raph Y.~ is decomposed as in Lemma 2.5, and each subgraph Ew(p) is identified with the appropriate J(w)-graph for Y(W {i}) = st(p). Then for any simplex V, the i-shadow of V intersects Zw(P) either trivially or in the J(w)-shadow of projp V.

Proof This is clear from the group action. Now we see that any simplex of E can be uniquely recognized from three

pieces of information, namely the simplex of st(p) onto which it projects via the map projp, the set of subgraphs Zw(p) which contain vertices in its /-shadow, and finally its type. The first two pieces of information allow us to recover its/-shadow, the last to determine which other simplices should be joined to it. So we may represent any simplex ofl~ by a triple (X,S,J), where X is a simplex of the Coxeter complex Z(W ~i/) = st(p), S is a set of elements of W which index the graphs Zw(p), and J is a subset of the type set I. Not every triple of this type will necessarily describe a simplex of Z, but, since the group action of W {~/must preserve the decomposition of the graph 22~, if one triple (X, S, J) does describe a simplex, then so must every triple (Y, S, J) with Y of the same type as X as a simplex of Y,(W/~} ). We say that a triple (K, S, J), where K is a subset of I\{i}, S a subset of { 1,2,. . . , m} and J a subset of I, is a feasible triple for the pair (E, Z i) if there are triples (X, S, J) which describe simplices of 22 as above. We have shown that we can recognize Z from the smaller complex E(W/~I), the graph E~, decomposed into its subgraphs Zw(p), and the set of feasible triples for Y,.

We are now in a position to reconstruct the full buildings A(E, F) and A(22, F, A) of the/-geometries A~(E, F) and A~(E, F, A) of Theorem A. We reconstruct the simplices of the buildings A(E,F) and A(E,F,A) in the same way as we reconstruct Z from Ei.

The Construction of Simplices

We define simplices of the buildings A(Ei, F) and A(Ei, F, A) as sets of points of the point-line geometries AI(22i,F) and Ai(EI,F,A). Again we can represent each simplex by a triple, (X, S, J). X will always denote a simplex of the building


F, J a subset of the type set I, and S a subset of the set which indexes the geometries Hw (or in the more general second case Htw,~/). The simplex described by the triple (X, S, J) is described as the set of points which intersects each Hw (or H~w,~)) which is indexed by an element of S, intersecting it (when it contains more than one point) in the set of its points which are joined to X in F.

Again we say that a triple (K,S,J), where S and J are as above and K is a subset of I, is a feasible triple for the geometry Ai(E~,F ) or A~(Ei,F,A) whenever we define a simplex of the associated building by the triple (X, S, J) for some simplex X of type K in F. Whenever (K, S, J) is feasible every possible triple (Y,S,J), where Y is of type K in F, defines a simplex of type J.

So all we have to do is decide which triples are feasible. For Ai(22~, F) it is easy. We just say that (K, S, J) is feasible for A~(E~, F) if it is feasible for (Z, Z~). For A~(Ei, F, A) we have to be a little more subtle since the subgeometries H w and Htw,x) are indexed by a slightly larger set than the set which indexes the subgraphs of Y~. Recall that we indexed the subgraphs of E~ by a set T, which we partitioned as A u B u w o B, and then the corresponding subgeometries of AI(Zi,F,A) by A u {(w,2):w~B,2EA}. In this situation each feasible triple (K, S, J) of (22, Z~) defines a whole set of feasible triples for Ai(Y~, F, A), one for each pair {/~, v} from A, namely the triples (K, Su, v,J ), where

Su,~ = (A n S ) u ( w , 2 : w ~ B n S n woS,2EA} w {(W,l~): we B n S} u {(w, v):WewoB n S}.

(The idea is that we intersect every H(w,Z ), for a given w ifB ~ S contains both w and w o w, but just H(w,u) if only w is in B n S, and just H(w,~) if only w o w is in B n S . )

As we run through all feasible triples for (Z, Ei), which correspond to all /-reduced simplices of Z, we define as above all feasible triples for Ai(Z ~, F) and A~(Z~,F,A) and hence all /-reduced simplices for the buildings A(Zi,F) and A(Zi, F,A). We complete the constructions of these buildings by first defining each simplex defined in this way to be joined to those points (vertices of type i) it contains. Then if X is a simplex of/-reduced type J and Y a simplex of /-reduced type J ' we say that X and Y are joined if either J (say) separates J ' from i and X is a subset of Y or J u J'is/-reduced and X n Y is a subset of that type. In general we define simplices as sets of pairwise joined vertices.

Notice that since each simplex of type J( i ) of Y~ is found as above as a pair of vertices joined by an edge of the graph Z~, each simplex of that type in A(Zi, F) or A(22~, F, A) corresponds to a line of the geometry Ai(Z~, F) or A~(E~, F, A). So its clear that each of A~(Y.~, F) and Ai(E~, F, A) is indeed the/-geometry of the relevant simplicial complex A(E~, F) or A(E~, F, A). What remains to be seen is that each of these is a building. To see this we simply have to recover

34 SARAH REES

apartments. We find each apartment via its /-graph, which we define as a subgeometry of the point-line geometry with just two points per line.

The Construction of Apartments

Suppose that E i is the/-graph ofa Coxeter complex Z(W), F a building of type W ~i/, and A an indexing set of at least two elements. Given any apartment E' of F we define an associated apartment graph G(E') for Ai(Zt, F). Its set of vertices is the subset of the set of points of A t which is the union of the singleton Hw's together with the intersection of each remaining Hw (viewed as a J(w) geometry for F) with the apartment E'. Two vertices of the graph are joined if they are collinear as points of A t. More generally, given an apartment E' of F and a subset T of size two of A we can define an apartment graph G(Z', T) for Ai(E~,F,A) exactly as above except that points of a subgeometry Htw,~) are included as vertices of the graph only when 2 is an element of T.

Clearly each of the apartment graphs as defined above is isomorphic to E t. Just as above we construct the remaining simplices of the Coxeter complex by defining/-reduced simplices (X,S ,J) for any X of type K in E' and (K,S,J) feasible for (Z, Z'). Because of the close connection between these apartments and the apartments of F, which is known to be a building, it is now routine to verify the building axioms for A(Zi, F) and A(Et, F, A) in terms of those for F. This completes the proof of Theorem A.

4. P R O O F OF T H E O REM B

This section is devoted to the proof of Theorem B, which we restate here for convenience.

T H E O R E M B. Suppose that, A is a weak building of spherical type. Then for some i there is a point p of the point line geometry A i with the property that all lines of A i through p are thin.

Note that we may rephrase the conclusion of the theorem to read as follows: ' . . . . for some i we can find a vertex p of A of type i so that whenever G is a simplex of type I \ { i } joined to p[Resa(G)l = 2.'

LEMMA 4.1. Suppose that A is a building of finite rank n, where n is at least 2. I f two disjoint simplices F and G exist, both with thick residue, so that F w G is a chamber, then A is thick.

Proof. The proof is by induction on n. We consider first the rank 2 case, where A is a generalized m-gon. In this case, as a rank 1 simplicial complex A is simply a graph. It is bipartite, with diameter m and girth 2m. The residue of


a vertex of A is simply its neighbourhood in the graph. Thus a vertex of A has

thick residue precisely when it has at least three neighbours. If x and y are at distance m in the graph they are opposite; then each neighbour of x is distance m-2 from a unique neighbour of y and opposite all others. Thus we have a correspondence between neighbours of x and neighbours of y. So x has thick residue precisely if y does.

So now suppose that F, G are as given above, where A is a generalized m-gon.

Then F and G correspond to a pair of adjacent vertices of the graph. Now let H be opposite G. Res(G) is thick, so G has at least three neighbours, and so does H. That means that we can find two neighbours J and K of H, both opposite F and between them opposite all other neighbours of G. Since J and K are opposite F and F has thick residue, both J and K have thick residue. Since every neighbour of G is opposite at least one of H and K, we see also that every neighbour of G must have thick residue.

Now finally, using connectedness, we see that every vertex of A has thick residue, that is A is thick.

We consider now the case where n is at least 3. We may assume in this case that the simplex G has rank at least 2.

Now if H is any non-trivial face of G, F and G k H are simplices with thick residues within Res(H), so Res(H) must be thick, by induction. The same is true i fH is a non-trivial face ofF. I f K is a simplex intersecting G in some non-trivial

simplex H, then since Res(H) is thick so is Res(K). Using connectedness we see that any simplex with a non-trivial face in Res(F) has thick residue in A.

Now suppose that F ' is a simplex of the same type as F and that for some simplex X in Res(F) ~ Res(F') both F w X and F ' w X are maximal. X has

rank at least 2, so we can write X as a union of two simplices Y and Z, both in Res(F) n Res(F'). Using the results of the previous paragraph we see that Y and Z both have thick residue in A and hence also within Res(F'). Thus Res(F') is thick.

Now we can use connectedness to show that every simplex in A has thick residue; that is, A is thick.

L E M M A 4.2. Suppose that A is a weak building o f finite rank n, and suppose that V is a vertex of A of type i, for some i, whose residue is thick. Then in A i all lines through V are thin.

Proof. Let G be a simplex in Res(V) such that G u V is maximal. Then Res(G) cannot be thick, since this would force A to be thick (by 4.2). So IRes(G)l = 2.

D E F I N I T I O N . Suppose that M is a Coxeter diagram over an indexing set I. Let J be a subset of I, which we view as a set of vertices of M. We call a vertex

36 SARAH REES

j in J a cut-node for J if M\{j} is disconnected, with every connected component of M\{j} either contained in or disjoint from J.

L E M M A 4.3. Let A be a building which is neither thick nor thin. Suppose that A contains a simplex F of type J whose residue is thick, that J contains a cut-node j, and that no simplex in A of type J \ {j} has thick residue. Then if V is the vertex in F of type j, in Aj all lines through V are thin.

Proof Let G be a simplex in Res(V) such that G u V is maximal. Then we

can write G as the disjoint union of a simplex G1 of type I \ J and a simplex G 2 of type J\{j}. Sincej is a cut-node the set of vertices of types in I \ J in Res(V) is

identical with both Res(F) and Res(V w G2). Now the residue of G is equal to the residue of Gxwithin the geometry

Res(G2). And since Res(F), equal to the union of Res(V) and Res(G2), is thick

within the residue of G2, though Res(G2) itself is not thick, by induction we see that the residue of G 1 within Res(G2) has size 2. So IRes(G)[ = 2.

D E F I N I T I O N . I fA is a building with diagram M over some set I we say that

we have thickness at nodej of the diagram if we can find some simplex F of type I \{ j} whose residue is thick.

L E M M A 4.4. Let A be a buildin9 with diagram M. I f we have thickness at node j of M then we have thickness at any node of M connected toj by a string of edges

of type© r, o , where m is odd.

Proof In a generalized m-gon, any opposite of any vertex with thick residue has thick residue. If m is odd, any opposite of a vertex is of the opposite type.

Now suppose that we have thickness at nodej of M and that F is a simplex of

type l \ { j } with thick residue. Let k be a node joined t o j by an edge of type

© m -©, where m is odd. We can write F = F x w V where V is of type k.

Then in the generalized m-gon Res(F 1 ) V is a vertex whose residue is thick. We choose W opposite to V in Res(F1). Then Res(F1 w W) is thick, and we have

thickness at node k.

Proof of the Theorem

Since A is not thin we must have thickness at some node of the diagram M. 4.4 forces thickness at node 1 except possibly in the cases where M = (7. or F 4. We consider these cases first.

I f M = C. or F 4 and we have no thickness at node 1, all thickness is to the right of the double bond in M. In the case of C. this means that some simplex of type { 1,2 . . . . . n - 1} has thick residue but none of type {1,2 . . . . , n - 2}. On the other hand, in the case of F 4 we have a simplex of type {1, 2, 3} with thick


residue, possibly also a simplex of type { 1, 2} with thick residue, but no vertex

of type 1 with thick residue. We can apply 4.3 with J = {1,2, . . . ,n - 1} and j = n - 1 when M = C n, and with J = {1,2,3} or J = {1,2} when M = F 4 to

find a vertex V only on thin lines. In all remaining cases we can assume that we have a simplex of type I \ { 1}

with thick residue. Hence we can find some i ~ I with the p roper ty that there is

some simplex F of type {i , i + 1 . . . . . n} with thick residue but none of type {i + 1, i + 2 . . . . . n}. Then~we can apply 4.3 with J = {i, i + 1 . . . . . n} to find a vertex V which is only on thin lines in all cases except those where a branch in the d iagram causes a problem. We have to consider these cases separately.

They occur when M = D n and i = n - 1, or when M = E6 , E 7 o r E 8 and i = 4.

The case M = Dn. Suppose that we have a simplex of type {n - 1,n} with thick residue but none of type n. If we have a simplex of type n - 1 with thick residue we can apply 4.2 to find V of type n - 1 and only on thin lines.

Otherwise A is a rank n polar space with no thick maximal singular subspaces,

but with a thick singular subspace of dimension n - 2, of codimension 1 in a maximal singular subspace. Using the classification of Buekenhout and Sprague in [5~ we see that we can find A as the direct p roduc t of a D 1 polar space and a D~_ 1 polar space in which some maximal singular subspaces are

thick. Any type 1 vertex which is a point within the D x polar space is a point only on thin lines.

We are left with the cases of M = E6, E 7 and E 8. We assume in each case that we have a simplex of type I \{1, 2, 3} with thick residue but none of type i \{1 , 2, 3,4}, and we show in each case that there must be a vertex of type

1 only on thin lines. In each of the three cases the residue of a simplex of type

I \ { i } is complete ly determined by the type 2 vertex in the simplex. So what we have to show is that, for some type 1 vertex V, whenever X is a type 2 vertex

incident with V, X is incident with precisely two type 1 vertices. We shall need in the following certain propert ies of polar spaces.

(a) First notice that if A' is a D s building which is not thick but contains a vertex of type 5 with thick residue; then A' must arise via the dualized

projective space construct ion of Buekenhout and Spague ([5]). In that case A' contains two 4-dimensional thick singular subspaces, the one already specified as the residue of a type 5 vertex and a second sharing no c o m m o n subspaces with the first. The second must arise as the thick residue of some type 4 vertex.

(b) N o w if A' is a D s building containing a simplex of type {4, 5} with thick residue, ibut none o f type 4 or 5, the max ima l singular subspaces o f A' are 3-dimensional and A' must arise as the produc t of a D1 polar space and a D 4 polar space which has some thick 3-dimensional singular subspace. I f V is


a type 1 vertex corresponding to a point of the polar space in the D 1 component then every type 2 vertex incident with V is incident with just two type 1 vertices; that is V is a point only on thin lines within A'. Also Res(V) is isomorphic to the 0 4 component of A', and so is a 0 4 polar space containing some thick maximal singular subspaces.

Notice also that if we know that A' contains no simplex of type 4 or 5 with thick residue but that for some type 1 vertex V Res(V) is a D 4 polar space containing some thick 3-dimensional singular subspaces, then we know that A' itself contains thick 3-dimensional singular subspaces and is a product as above, and that V is a point only on thin lines of A'~.

(c) Notice finally that if A' is a D, building and x, y are non-collinear points (type 1 vertices) of A', then x and y have disjoint residues which are isomorphic. For if U is a singular/-space through x a unique singular (i - 1)-subspace of U spans a singular/-space with y.

We now proceed to deal with the cases of E 6 , E 7 and E s.

The case M = E 6. We assume that A contains a simplex of type {4,5,6} with thick residue but none of type {5, 6}. The observation (a) above allows us also to assume that there is no simplex of type {4, 6} with thick residue.

We know that we have P of type 6 in A so that Res(P) contains thick 3-dimensional subspaces but no thick subspaces of higher dimension; Res(P) is a D 5 polar space. By (b) above we know that we can find V of type 1 in Res(P) such that if X is of type 2 in Res(V,P) then X is incident with just two type 1 vertices. Res(V, P) is a D 4 building with some thick 3-dimensional singular subspaces.

Now Res(V) can also be viewed as a D 5 polar space in which vertices of type 6 appear as points. Suppose that Q is a point of Res(V) not collinear with P. Then by (c) above, Res(V, Q) is isomorphic to Res(V, P) and so is a O 4 building with some thick 3-dimensional singular subspaces. By (c) we see that V is only on thin lines within Res(Q); that is, that i fX is of type 2 in Res(V,Q) then X is incident with just two type 1 vertices.

Now finally we suppose that X is any type 2 vertex in Res(V). Then we can find an apartment Z of A containing the simplex {V,P} and X. Within E there is a unique type 6 vertex Q in the D 5 Coxeter complex Res(V) which corresponds to a point of the D 5 polar space non-collinear witn P. P and Q share no type 2 vertices in their residues; each is incident with eight type 2 vertices in Res(V), within E, and in all of E there are just sixteen type 2 vertices incident with V. So X must be incident with one of P and Q, and therefore incident, by the above, with just two type 1 vertices in A.

The case M = E 7. We assume that A contains a simplex of type {4, 5, 6, 7}

WEAK B U I L D I N G S OF SPHERICAL T Y P E 39

with thick residue but none of type {5, 6, 7} or {4, 6, 7}. We can assume that we have a type 7 vertex R in A and a type 1 vertex V incident with R so that V is only on thin lines within Res(R), and that this is forced by the isomorphism type of Res(R). Basically Res(R) is of type E 6 a s above, and V is chosen within Res(R) as above.

Now if S is of type 7 in Res(V), non-collinear with R in the D 6 polar space Res(V), then, by (c) above, Res(V, R) and Res(V, S) are isomorphic, and so V is forced to be only on thin lines also within Res(S).

Finally, if X is of type 2 in Res(V) we notice that we can find an apartment 22 of A containing the simplex {V,R} and X. In Y we find thirty-two type 2 vertices incident with V. Sixteen are incident with R, the other sixteen with the unique type 7 vertex S of 22 that represents a point non-collinear with R in the D 6 Coxeter complex Res(V). So X is incident either with R or with S, and hence with just two type 1 vertices in A.

T h e case M = E s. We assume that A contains a simplex of type {4, 5, 6, 7, 8} with thick residue but none of type {4, 6, 7, 8}. We can assume that we have a type 8 vertex T in A and a type 1 vertex V in Res(T) so that V is only on thin lines of Res(T), and that this is forced by the isomorphism type of Res(V, T).

As above we show, by looking within an apartment, that every type 2 vertex X of Res(V) is incident either with T or with some type 8 vertex U in Res(V) which corresponds to a point of the polar space Res(V) non-collinear with T. Then Res(V, T) and Res(V, U) are isomorphic, so Vis only on thin lines both in Res(T) and in Res(U). Then X must be incident with just two type 1 vertices in A.

5. P R O O F OF THEOREM C

DEFINITION. Let A be a building of type W over I = {1,2 . . . . . n}. Let p be any point of A i. Now for any element a of W, define As(p) to be the set of points q of A i with the property that, for any apartment E of A containing both p and q,q is a vertex of the subgraph 22a(P) of Z i. We shall call the sets Aa(p) the p-orbits of A.

REMARK. That As(p) is always well defined follows from the second of the two building axioms that were given in Section 1. Every vertex of A of type i and distinct from p lies in precisely one set Aa(p).

Now remember that each line I of Ai was originally defined as the/-shadow of a simplex of type I \ { i } . But, by (3.1)(i),/could also be defined as the/-shadow of a unique simplex of type d(i). Now from the building axioms we have immediately

40 SARAH REES

LEMMA 5.1. Any apartment of A containino two points of a line I also contains the unique simplex of type J(i) whose i-shadows is I.

This justifies, among other things, using the notation I to describe not only the set of points of the line but also the simplex of type J(i) which determines it. Now when we say that an apartment Z contains l we shall mean that

contains the simplex l of type J(i), or, equivalently, that it contains two points of the line. (Certainly no apartment can contain more than two points of a line.)

LEMMA 5.2. No two lines ofAi have more than one common point. Proof Suppose that p and q are two points, both on two lines I and m. Then

we can find an apartment E, containing p, q, l and m. But in E both I and m are i-reduced with the same/-shadow. So l = m. Since this ensures that any line is defined by any two of its points, we may use the notation (p, q) to describe l, where p and q are any two of its points.

LEMMA 5.3. Any line l of A contains points from at most two p-orbits of A. I f I intersects distinct p-orbits A,(p) and Ab(p) , then there are edges between Ea(P) and Eb(p) in Ei, where E is any apartment through p. Every point ofl in A,(p) has the same projection X onto p, and every point ofl in Ab(p) has the same projection Y onto p. IfI(a) = l(b) then X and Y are equal. Otherwise they are joined in A.

I f every point ofl is in a single p-orbit A,(p), then the set of points ofl projects onto p as the set of simplices of type I(a) joined to projpl.

Proof This follows immediately from 2.4. For if q is any point of 1 we can find it in an apartment with p and 1. And any two apartments containing p and I are related by an isomorphism which fixes both p and l.

From now on we suppose that p is a point ofA i only on thin lines and that F is the building of type W ~i~ which is the star of p.

LEMMA 5.4. (i) IrA and A' are opposite simplices of A then the restriction of prOja, tO st(A) induces a (simplicial complex) isomorphism from st(A) to st(A'). In particular if p' is any point opposite p, then p'is only on thin lines ofAi.

(ii) I f p' and p" are both points opposite p, then they cannot be collinear. (iii) I f p' and p" are both points opposite p, then for every point q collinear with

p' there is a point r collinear with p" with the same projection as q onto p. (iv) I f p' is a point opposite p, then two points q and r collinear with p' have

distinct projections onto p.

Proof (i) The first statement is [10, (3.28)]. The second depends on this and the fact that i r a and A' are opposite and C is a chamber through A, then Some simplex C' through A is opposite C.

(ii) I fp ' and p" are collinear, then some apartment Z must contain p,p' and


the line (p', p "). Then Z contains a second point of (p ', p "), but cannot contain p" since it can only contain one point opposite p. So (p' ,p") contains at least three points. But p' is only on thin lines, so we have a contradiction.

(iii) If I is the line joining q to p' we let r be the second point on projp,, projfl. q and r are in the same p-orbit, namely Awo,,(p), so projpq and projpr are of the same type. We can see within the Coxeter complex that I and projp,, projfl both project onto p as projpl. 2.4 then shows us that projpr and projpq must be equal.

(iv) Since projp induces an isomorphism from st(p') to st(p), no two lines through p' have the same projection onto p. So any line through p' is determined by its projection onto p, and the apartment A containing the simplices {p} u projp(p',q) and {p'} u (if , r) must also contain (p' ,q). Since the lines ( i f ,q ) and (if, r ) are thin, E also contains q and r. So q and r, as two points of the same subgraph E . . . . (p) of Y i, have distinct projections.

LEMMA 5.5. (i) Any two points of Ar~(p ) have distinct projections onto p. (ii) Every line meeting Ar,(p ) in two or more points is contained in A,,(p). (iii) The projection map projp maps the point-line geometry induced on Ar,(p )

isomorphically onto a J(i)-geometry for F. Proof (i) Ifq and r are two points of A,,(p), then some apartment Y contains

p,(p,q) and (p,r), hence also q and r, since lines through p are thin. So q and r have distinct projections onto p, as distinct points of Z,,(p).

(ii) and (iii) follow immediately from (i) and 5.3.

LEMMA 5.6. I f no two points of Aa(p) have the same projection onto p, then any line with two points in Aa(p) is contained in Aa(P). Further, the map projp maps the point-line geometry induced on Aa(p) isomorphically to a J(a)-geometry for F.

Proof This is also a direct consequence of 5.3.

We can now deduce

LEMMA 5.7. Suppose that A is a weak building of spherical type W and p a point of the point-line geometry Ai which is only on thin lines. Let F be the building st(p), of type W {il. Then the following are equivalent:

(i) No two points in the same p-orbit have the same projection onto p. (ii) Any two points of A i lie in an apartment with p.

(iii) A i is isomorphic to AI(ZI,F) where Z is the Coxeter complex of W. Proof That (i) implies (iii) is immediate from the application of 5.6 and 5.3

to~ether. That (iii) implies (i) is clear from the construction of Ai(E~, F). For the p-orbits of A~ correspond to the subgeometries Hw, and the projection of a point in H w onto p is precisely the J(w)-simplex of F to which it corresponds. Now since no two points in an orbit of W {~ on Ei have the same projection onto the vertex W ~i}, (ii) must imply (i). Finally, ifq and r are any two points we

4 2 SARAH REES

can find an apartment E containing p, projpq, and projpr. E must contain points which project onto projpq and projpr, so (i) implies (ii).

It seems difficult to produce much more that is useful in the way of general arguments. We know that we cannot prove in general that a weak building of spherical type is isomorphic to a building of type A(Ei, F) or A(E i, F, A), for we have already seen weak buildings that do not conform to that pattern. And some of the Coxeter graphs are so huge and complicated (for instance the 3-graph of type E s is described in [2] as having 1437 distinct subgraphs Ew(p) ) that the idea of constructing more general examples than we might try to construct already by the methods of this article seems appalling. The point-line approach to weak buildings seems to be the wrong one when Coxeter graphs get to be as confusing as this. So we look now for more specific arguments which allow us to classify particular cases.

It is frequently true that for two vertices q,r at a distance 2 in a Coxeter graph E i the simplex projqr\{q} is such that its residue in E is described by a diagram M whose irreducible component containing the node i is of type D n or C n, with i in the position of the node labelled 1 in the Cn or D, diagram. The justification for this is a result of Cohen in [3]. If this is the case, then we call the pair {q,r} a symplectic pair. Then the vertices of type i in the residue of projqr\{q} consist of q,r and their common neighbours in the graph Z i. In many cases there are only two possibilities for a pair of vertices at distance 2 in E i, namely that they have a unique common neighbour or that they have several common neighbours and form a symplectic pair.

Now when A is a building and q and r are points of Ai we call {q,r} a symplectic pair if it is such in some (and so any) apartment containing both points. Then we have some simplex which we shall call (q, r ) of A, containing projqr\{q}, whose residue is a C, or D, building. The points of the polar space correspond to the type i varieties in the residue, and consist of q, r and all points collinear with both q and r, as well as any point s in the same q-orbit as r which has the same projection onto q as r. For any such s, any point collinear with both q and r is also collinear with s and ( q , r ) = (q , s ) . Further, (q ,r ) = (r,s).

If q is on no thick lines of Res(q, r ) we can use the work of Buekenhout and Sprague in [5] to see also that all lines through r within Res(q,r) must be thin. Often we can deduce more. For the diagram Res(q,r) must be a subdiagram for diagram M of A. In particular, if M has no double bonds, then Res(q,r) must be of type D,, and in that case q has only one point opposite it in Res(q,r) . So r is the unique point of its q-orbit whose projection onto q is projqr. This argument may be useful in quite general situations, usually when q and r are in distinct p-orbits and we know more about the p-orbit of q than


about the p-orbit of r, but it is especially useful when the special point p is one of the points in the symplectic pair. For we know then that all lines through p are thin, so certainly all lines of Res(p,q) through p are thin.

In particular we have

LEMMA 5.8. Suppose that A is a weak building with diagram M. I f q is any point forming a symplectic pair with a point p that is only on thin lines, then

(i) all lines joining q to Ari(p) are thin; (ii) I f M contains no double bonds, then no two points in the same p-orbit as

q have the same projection onto p.

We can apply the above results to classify completely certain situations. We look at the low-rank cases (that is, the cases where there are few p-orbits) where

A is of type E6, E v, E s or F 4 with i = 1 or n (excluding the E 8 case with i = 1 where there are p-orbits).

(i) Suppose that A is a weak building of type E 6 and that a point p of A6 is on no thick lines. We have already described the 6-graph of type E 6 as an example in Section 1, so we shall not repeat all the details here. We need the following information.

The 6-graph E 1 for E 6 splits into subgraphs {p}, Y~6(P) and Ea(p ) (where a = r 6 r 5 r 3 r 4 r 2 r 3 r 5 r6). J(r'6) = 5 and J(a) = 1. El(p) consists of the neighbours of p, while every point of Ea(p) is in a symplectic pair with p.

Given p as above in the weak building A we let F be the D 5 building st(p). Now 5.5 tells us that no two elements of Ar6(p ) project onto the same simplex through p, while 5.8(ii) tells us that no two elements of A~(p) have the same projection. Now 5.7 shows that AI is isomorphic to A6(]E6,F ).

(ii) Suppose that A is a weak building of type E 7 and that a point p of A 7 is on no thick lines. The 7-graph Y~7 for E 7 for was also described in Section 1. We need the following information.

The 7-graph of type E 7 splits into subgraphs {p}, E,7(p),Za(p) and

Ewo(p) = {P'}. J(rT) = 6, and J(a) = 1. EfT(p) consists of the neighbours of p, and Y,~(p) of the neighbours of p'. If q is in E,(p) then {p, q } is a symplectic pair. In fact any two vertices at distance 2 with more than one common neighbour form a symplectic pair.

We let p and A be as above and let F be the E 6 building st(p). Again 5.5 tells us that no two elements of ArT(p ) have the same projection onto p, and 5.8 that no two elements of A,(p) have the same projection onto p. To finish we have to show simply that Awo(P ) consists of a single point.

Suppose that p' is_a point in Awo(p ). If q is any point of Aa(p) we can find an apartment contaning p,p' and projpq, and also (since no two points of A~(p) have the same projection onto p) q. So p' is collinear with every point of Aa(P). If

44 SARAH REES

p' and p" are both points of Awo(p), lots of points are collinear with both, so {if, p"} forms a symplectic pair. But now if r is any point of ArT(p ), {p',r} is a symplectic pair, and (p' ,r) = (if, p"). In particular, ( i f ,p") contains two points, p" and r, which are not collinear with p'. But (p',p") must be of D, type since the E 7 diagram contains no double bonds, so this cannot happen, and Awo(p) must consist of a single point. Now we can apply 5.7 to see that A 7 is isomorphic to A7(l~7,F).

(iii) The 1-graph of type ET, the 1-graph of type F , (and hence also the 4-graph since the two are isomorphic) and the 8-graph of type E 8 are all of the same basic shape. For each we have five subgraphs ofEi, {p}, E~,(p), Ea(p), Eb(p)

(where b = Worl) and Ewo(P) = {P'}. J(rl) = J(b)is a single node distinct from the single node J(a). There are edges joining {p} to E,l(p), {p'} to Eb(p), Ea(p) to E,l(p ) and ]gb(p) , and E~l(p ) to Y~b(P)" The opposite involution swaps p with p' and E~,(p) with Eb(p).

Any two vertices at distance 2 with more than one common neighbour form a symplectic pair.

We suppose that A is a weak building of type F 4, E 7 or E 8 , that i = 1 or 4 in the F 4 case, 1 in the E 7 case and 8 in the E s case, and that some point p of A~ is on no thick lines. Finally, we suppose that F is the building st(p). Then again 5.5 shows that no two points of A~l(p ) have the same projection, while 5.8 says the same about points of A~(p). 5.5 and 5.6 tell us that each of these is isomorphic to the appropriate geometry for F, and 5.8 that lines between A~l(p) and A~(p) are thin.

Ifp ' is opposite p, then A~o,,(p ) _ Ar~(p'). 5.4(iv) tells us that no two points of A . . . . (p) joined to p' have the same projection. Moreover, since every line through p can be found in an apartment with p', for any point q in A~l(p ) there is a point q' joined to p' in A~o~,(p ) with the same projection onto p. So we have a correspondence between the points of A~,(p) and the points of A~,(p').

By looking in apartments containing p and p' we see that if q' in A,,(p') has the same projection onto p as q in A~,(p),q' is collinear with q and also with every point of A~(p) that is collinear with q. This implies that Aa(p) = Aa(p'), and, since points in Aa(P' ) can be collinear only with points of A,,(p') and A . . . . (p'), that A~,(p)uA .. . . (P) = Ar,(P')uA .. . . (p'). Notice that since the projection of a point of A~,(p) u Awor,(p ) onto p is determined by the set of points coUinear with it in A~(p), whenever two points of this set have the same projection onto p they have the same projection onto p'.

It follows also, since all the rest of the points of the building have been accounted for, that we must have {p} u Awo(P ) = {p'} w Awo(p' ). So ifp' and p" are both opposite p, then p" is also opposite p'. Then p' and p" can be collinear with no common points, so we can partition the points of A . . . . (p) into the sets of points collinear with the various opposites of p.

WEAK B U I L D I N G S OF S P H E R I C A L TYPE 45

Suppose we index p and its opposites by the elements of some indexing set A, so that p is Po, say, for some 0 in A. Any two points p~ and pu for 2 and ~ distinct in A are opposite, and the sets {p~ }, Ar,(pz), as 2 runs through A, together with Aa(po) (= Aa(p~) for all 2) partition the points of A~. If2 :~ kt, Ar,(p~) is contained in Awor,(P~). An apartment containing p~ contains points of A,~(p~) precisely when it contains p~. For each pa,A,,(p~) is isomorphic to the J(i)-geometry for F. Lines from pa to A,,(pz) and from A,,(pa) t o Aa(pz ) --- Aa(p) are thin. By 5.4(i) F is isomorphic to st(pz) for any 2 in A.

It remains to consider lines between the various A,,(pz)'s. Looking in apartments we see that points from distinct Ar,(pz)'s are collinear precisely if they have the same projection onto p (or any other pu). Given three points q, r

and s all with the same projection onto p, we see, by looking in an apartment contaning q , ( q , r ) and s, that s is on the line (q,r). So the lines across the distinct A,i(p~)'s join all the points with the same projections onto p.

This verifies that A i is isomorphic to A~(Z~,F,A).

We have now proved everything that is necessary for Theorem C as well as various other technical results (such as, for instance, 5.4 which was never completely used in the special cases we considered) which would be of use in considering other more complicated special cases.

6. M O R E G E N E R A L S I T U A T I O N S

It was convenient in this article to put certain restrictions on the buildings under consideration. We dealt, for instance, only with weak buildings of spherical type. It is interesting to see how much we can do when we drop these conditions.

When W and E(W) are no longer finite, most of Lemma 2.5 remains true. The Coxeter graph Y.~ will usually then be infinite. The subgraphs Ew(p) may still be finite; they are, for instance, if W is an affine Coxeter group, making E(W) affine type. But there will usually be infinitely many such subgraphs. In each double coset of W ~i~ in Wthere will still be a unique shortest ((I - {i})- reduced) element, and the projection map will still associate the vertices and edges of Ea(P) (where a is (I - {/})-reduced) with the vertices and edges of a J(a)-graph for W/i~. All that really fails is statement (v) of the result, for we can no longer define a longest element of W. Lemma 2.5 does not in any way depend on the irreducibility of W. Provided we have some kind of description of the decomposition of the Coxeter graph Z i we can construct geometries of the type A~(Ei, F). Of course, if W is not finite we cannot look for opposite symmetry, so we.would not build examples of the more general type A~(E~, F, A). If W were reducible we could not reconstruct a complete building A(Z~, F) in terms simply of the shadows of its simplices, since many simplices

46 SARAH REES

would have the same shadows; many distinct buildings of type W would admit

Ai(E i, F) as their/-geometry. We see that the essence of Theorem A holds in

more general situations. The proof of Theorem B relied to a certain extent on induction on n, so it

would be inconvenient to work without finite rank. However, none of the

lemmas leading up to the proof made any use of the finiteness of W or even of

the fact that A is a building (as opposed to merely a Tits geometry of type M-see [13] for a definition). The result would extend very easily to any Tits

geometry, building or otherwise described by many string diagrams (so, for

instance, the result of Theorem B also holds for Tits geometries with C3 and F 4

diagrams; it was mostly diagrams with branches in them that caused more

problems and forced us to exploit, for instance, the existence of apartments

(which do not necessarily occur in more.general Tits geometries) in buildings.

The arguments we used to deal with the buildings of type E6, E 7 and E a would

certainly extend to give us the same result for infinite buildings of non-spheri-

cal type Eg ,E lo . . . . .

The general part of Theorem C does not depend on the irreducibility of W at all; the only thing that finiteness of W ensures is the existence of opposites. But

even in the finite case more could certainly be proved in the direction of this

result. We could certainly investigate more particular cases, and there is much

technical machinery set up in Section 5 that is not thoroughly exploited in the

cases that are considered there. We know, however, from the example in

Section 1 that the constructions of Theorem A are not general enough to give

us all weak buildings and it seems that most useful more general results would

come from an approach different from the point-line approach adopted in this

paper.

REFERENCES

1. Bourbaki, N., Groupes et Alg~bres de Lie, Chaps 4-6, Hermann, Paris, 1968. 2. Brouwer, A. E. and Cohen, A. M., 'Computation of Some Parameters of Lie Geometries' in

Algorithms in Combinatorial Design Theory (eds C. J. and M. J. Colbourn), Ann. Discrete Math. 26 (1945), 1-48.

3. Brouwer, A. E., Cohen, A. M. and Neumaier, A., Distance Regular Graphs, Chap. 5 (preprint). 4. Buekenhout, F. and Shult, E. E., 'On the Foundations of Polar Geometry', Geom. Dedicata

(1974), 155-170. 5. Buekenhout, F. and Spragne, A., 'Polar Spaces Having Some Lines of Cardinality Two', J.

Comb. Theory (Series A) 33 (1982), Note, 223-228. 6. Carter, R. W., Simple Groups of Lie Type, Wiley, London, 1972. 7. Scharlau, R., 'A Structure Theorem for Weak Buildings of Spherical Type' (preprint). 8. Solomon, L., 'A Mackey Formula in the Group Ring of a Coxeter Group', J. Algebra 41

(1976), 255-268. 9. Tits, J., 'Two Properties of Coxeter Complexes', Appendix to the above.

WEAK BUILDINGS OF SPHERICAL TYPE 47

10. Tits, J., Buildings of Spherical Type and Finite BN-Pairs, Springer Lecture Notes No. 386, Berlin, Heidelberg, New York, 1974.

11. Tits, J., 'Endliche Spiegelungsgruppen, die als Weylgruppen auftreten', Invent. Math. 43 (1977), 283-295.

12. Tits, J., 'Groupes et G6om&ries de Coxeter', mimeographed notes, Institut des Hautes Etudes Scientifiques, 1961.

13. Tits, J., 'A Local Approach to Buildings' in The Geometric Vein (The Coxeter Festschrift), (ed. Ch. Davis et al.), Springer, Berlin, 1982, pp. 519-547.

Author's address:

S a r a h Rees,

D e p a r t m e n t of M a t h e m a t i c s ,

U n i v e r s i t y o f B i r m i n g h a m ,

P .O. Box 363,

Birmingham, B15 2 T T ,

U.K.

(Received, September 1, 1986)

Weak buildings of spherical type

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