Journal of Geometry 0047-2468/85/010077-1151.50+0.20/0 Voi.25 (1985) �9 1985 Birkh~user Verlag, Basel

PRODUCTS OF TITS GEOMETRIES: A CONSTRUCTION OF C N GEOMETRIES

WITH NON-CLASSICAL RESIDUES.

Sarah Rees

A method of Buekenhout and Sprague's is generalised to produce non-classical Tits geometries of type C n for any n ~ 4.

i. INTRODUCTION

This paper is motivated by a paper of Buekenhout and Sprague

([5]) which, in classifying polar spaces with some lines of car-

dinality two, exhibits a method of combining polar spaces to

form polar spaces of higher rank. In other words, high rank

polar spaces are constructed as products of polar spaces of

lower rank.

But the polar spaces of rank n are merely the classical exam-

ples of Tits geometries of type C n. I show that Buekenhout and

Sprague's construction has a wider application. Formalising the

construction in order to define, more generally, the product of

a collection of Tits geometries, I prove the following result.

THEOREM. Suppose that SI,S 2 .... ,S k are Tits geometries of

Cn 2 " ' ' , C n k �9 types Cnl , , , each n i ~ I (By a geometry of type

C 1 we understand a set of at least two unrelated points.) Then

prod(S1,S 2 ..... Sk) is a Tits geometry of type C N, where

N = n l+n 2+..o+n k.

Thus we may construct high rank C n geometries as products of

C n geometries of lower rank. Such geometries are not thick;

neither are they thin, unless all constituents are thin. Each

constituent geometry appears as a residue.

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Various geometries are known of type C 3 which are not build-

Ings; a few are known which are not even covered by buildings.

By including such geometries as constituents we can build pro-

duct geometries of type C n, for arbitrarily large n ~ 5~ which

are not covered by buildings. Such geometries may be finite or

otherwise, by appropriate choice of constituents.

Note. I have chosen here to use the term product where other

authors, ~ncludlng Buekenhout and Sprague, have used the term.

s_~_~o My motive is to avoid confusion with another use of the

term sum in diagram geometry; it has now become fairly standard

to call a geometry a direct sum if its diagram splits into two

or more disjoint components (see Valette, [i0]).

2. BACKGROUND TO TITS GEOMETRIES.

I shall give here just a brief resume of what I mean by a Tl~

geometry. Basically I assume that the reader is already fami-

liar with the concepts; if not he may refer to [2], [7] or t9]

for details. (He should observe, however, that I work without

axiom 5 of [2].)

By a Tits geometry S I mean a collection of objects (varietles)

of different types indexed by a (frequently finite) set ~o The

size of ~, IAI, is called the !ank of S. A symmetric and re-

flexive binary relation I relates varieties of S, two dis-

tinct varieties of the same type never being related~

A set of palrwlse incident varieties is called a fl__~. The rank

of a flag is defined to be the size of the subset of ~ which

indexes its varieties~ Maximal flags are required to be of rank

IAI. Further, every non-maxlmal flag is contained in at least

two maximal flags.

The set of varieties not in a flag F but incident with every

variety of F forms a subgeometry of S known as the residue

S and written Re____ss~(F)._ The rank of ReSs(F) is of F in

defined to be the difference between the size of A and the

rank of F.

Rees 79

The incidence graph of the geometry (that is, the graph whose

vertex set is the set of varieties of S, two varieties being

joined by an edge if related by I) is required to be connected;

the same is required to be true for the incidence graph of any

residue geometry of rank two or more.

The structure of the geometry is defined by a diagram which con-

sists of a set of nodes (indexed by A) Joined by labelled edges.

The construction of the diagram is inductive in the sense that

subdiagrams spanned by subsets of size at least two describe re-

sidue geometries of rank at least two. (For further details, see

[2], [7], or [9].)

The classical examples of Tits geometries are Tits buildings,

geometries often associated with groups of Lie type and described

by Coxeter diagrams. As examples we see for the buildings of

type A n the n-dimensional projective spaces, for the buildings

of type C n the rank n polar spaces (see [6] for details).

Not all Tits geometries described by Coxeter diagrams, however,

are buildings. Every Tits geometry of type A n is a building

of type A n and hence an n-dimensional projective space. On the

other hand, geometries of type C 3 exist, both infinite (see

Tits [8]) and finite (for example, Neumaier's ~ geometry, see

[5]) which are not even covered by buildings. (We say that a

geometry is covered by a building if it is a homomorphic image

of a building by a homomorphism whose restriction to any residue

is an isomorphism.) As a consequence of the Theorem there exist

also geometries of type C n which are not covered by buildings

for every integer n greater than 3.

3. PRODUCTS OF PROJECTIVE SPACES.

Although as a Tits geometry a projective space is defined to be

a collection of points, lines and (necessarily singular) sub-

spaces of various dimensions, the structure of the space is

completely determined by its sets of points and lines. Thus in

determining here the projective space which is the product of a

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collection of projective spaces we need to do no more than de-

fine the sets of points and lines of the new space.

Suppose that P1,P2 ..... Pk are projective spaces of (projective)

dimensions nl,n2,...,n k, (We define a projective space of di-

mension 0 to be a single point, a projective space of dimension

1 to be a set of points and a single line, incident with all

points.) We define the product of P1,P2 ..... Pk to be that pro-

jective space whose point set is the disjoint union of the point

sets of PI,P2 ..... Pk and whose line set is the disjoint union

of the line sets of P1,P2 .... 'Pk together with all pairs

[xi,xjJ of points with xi,x j in distinct constituents Pi.Pjo

The new space has projective dimension n l+n 2+...+n k+k-1.

All subspaces of each Pi and each Pi itself are subspaces of

the product space.

This construction is well known, and appears, for example in [4 7 .

4. PRODUCTS OF POLAR SPACES.

A polar space, described as a Tits geometry as a geometry of

points, lines and singular subspaces, may also be reeognised

completely by its sets of points and lines. Thus again, to de-

fine the polar space which is the product of a collection of

polar spaces we need only define the point and line sets of the

new space.

Suppose that ~I,N2,...,Uk are polar spaces of ranks

nl,n2,...,n k. We define the polar space which is the product of

these spaces to have as its set of points the disjoint union of

the point sets of W1,U2,...,Uk, and as Its set of lines the dis-

Joint union of the line sets of N1,W2 ..... U k together with all

pairs [xi,xjJ of points with xi,x j in different constituents

~i,Nj. The new polar space has rank n l+n 2+...+nk~ For any

~i' every singular subspace of Ui appears as a singular sub-

space of the new space. Ui itself, of course, is not singular

(that is, within ~i there exist pairs of non-collinear points)

so is not a singular subspace of the new space.

Rees 81

This construction is described in [3].

5. PRODUCTS OF TITS GEOMETRIES.

The obvious parallels between the two product constructions de-

scribed above cannot have escaped the reader. It does not seem

unreasonable to suppose that we might somehow generalise this

construction to define products of other types of geometries.

The constructions given above are defined in terms of points and

lines only. This sort of description is fine for projective

spaces but not for Tits geometries in general, which are not

necessarily completely defined by their point and line sets. We

prefer therefore to formalise our construction in such a way that

it describes all the varieties of the geometry.

DEFINITION 1. Suppose that S1,S 2 ..... S k are Tits geometries.

Suppose that, for each. i, S i is of rank n i. indexed by the set

[1,2 ..... ni). Let S~ denote its set of varieties of type J, for each j = 1,2,...,n i. Define S, the product of

S1,S 2 ..... S k, written prod(SlLSS2~Skl, as follows.

S is a disjoint union of sets S1,S 2, S N ..., , where N = ~n i

and where

SI = ~k S I $2 = U k 2 k ~ S I I 1 i' i = I Si W U S x i,j=! J'

ni~2 i < j

and in general

sJ U k Jl J2 Jr = X x...X S i

il,i2,...,i r = 1 Si! Si2 r

il<i 2<...< i r

Jl +" " "+ Jr = j

each Jm ! n i m

(Note that, as expected, where we call type 1 varieties of each

geometry points and type 2 varieties of each geometry lines, we

see the point set of S as the union of the point sets of its

constituents and the line set of S as the union of the line

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sets of its constituents together with all pairs of points from

distinct constituents. We see other varieties of S simply as

sequences of varieties of the constituent geometries, of varying

lengths.)

Define an incidence relation on S as follows.

If v = (Vl,V 2 ..... v r) ES i, w = (Wl,W 2'...,w s) ES j and i _< jo

say that vIw in S precisely if for each m= 1,2 ..... r0 there

exists m' in the set [I,2 ..... s] such that v m and Wm~ are

varieties in the same constituent geometry S i , and such that in m

S i v m and Wm~ are incident, with v m of no higher type than m

w m

Although I shall not prove this here (the Theorem proves a more

general result), the reader can verify for himself that this con-

struction formallses the product for polar spaces described ~n

section 4 if each S i is taken to be a building of type Cn.o 1

~owever, this is not quite the right formallsatlon for a product

of projective spaces. For a start, the product of a collection

PI~P2 .... 'Pk of projective spaces of dimensions nl~n 2 ..... n k

(Tits geometries of types A An2, A ) is a projective n I ..... n k

space of dimension n l+n 2+o..+n k+k- 1 (a Tits geometry of

An l+n2+...+n k+ k- 1 ) and not of dimension n l+n 2+..,+n k.

which we should expect if the product in Definition 1 were the

correct formalisatlon.

Note also (and this is the explanation for the rise in rank)

that each constituent space itself is a subspace of the product

geometry, although it is not a variety of itself as a Tits geo-

metry.

We want, too, to admit the possibility that one of the consti-

tuent spaces might be a single point, and thus not a Tits geo-

metry in the sense of section 2.

Products of projective spaces are covered by our second defini-

tion.

Rees 83

DEFINITION 2. Suppose that Sl,S 2 .... ,S k is a collection such

that each S i is either a Tits geometry of rank n i ~ i, indexed

by the set [1,2 ..... ni], or a single point, known here as a

geometry of rank n. = 0. i n.

i S 2 ,S i denote If S i is a geometry of rank n i ~ I, let Si' i .... i

the sets of varieties of S. of types 1 2 . n i and let ni+l 1 ' '" "'

S i denote the singleton set [Si}. Extend the incidence rela-

tion of S i so that also S i is defined to be incident with

itself and with each of its varieties.

If S i is a single point, call the set [Si} by the name S~.

Define S i to be incident with itself.

Define S, the extended product of SI,S 2 ..... Sk, written

ext.prod(Sl,...,Sk~ , as follows.

S is a disjoint union of sets SI,s2,...,S N, where

N = Eni+k-l, and where

s I = uk 1 i=l Si'

i 1 Si U U S i • S i,j=l

each ni~l i< j

and in general

S j = Uk Jl J2 Jr

Sil ~ Si2 x...• S i il,12,...~ r

i I < i 2 <...< i r

Jl + J2+'" "+ Jr= j

each Jm -~ njm + i

(Note that the sequence (S1,S 2 ..... S k) is not an element of S~

although, for example, (S1,S 2 ..... Sk_l.V) is for each variety

v of Sk, as is also the sequence (SI,S 2, ..;..Sk_l).

Define an incidence relation on S as follows.

84 Rees

If v = (Vl,V 2 ..... Vr) ~S i, w = (wi,w 2'.o-,w s) gS j and i~j,

say that v I w in S precisely if there exists, for eecn

m = 1,2,...~r, some m' E [i,2 ..... sJ such that v m and Wm,

are varieties in the same geometry Sim (or equal to Si )~ and

such that in S i v m and Wm~ are incident, with v m of no m

higher type than Wm,.

Now suppose that SI,S2,.o~ k are Tits geometries of types

. (where by a Tits geometry of type A O we under- Anl,An2~ --,Ank

stand a single point). Such geometries are necessarily the geo-

metries associated with projective spaces of dimensions

nl,n 2 .... ,nk. Ext.prod(Si,S 2 ..... Sk) is the Tits geometry of

which is associated with the product of these type Azni + k-!

projective spaces as defined in section 5.

6. THE CONSTRUCTION OF PRODUCT GEOMETRIES OF TYPE CN~

In this section we prove the Theorem stated in the introduction.

First observe that since clearly prod(Sl,S 2 ..... S k) =

prod(prod(S I .... ,Sk_!),S k)

We proceed by induction on

prod(Sl,S2). If N = 2, then n l=n 2=!.

we may assume that k = 2.

N = n l+n2o Write S for

S I and S 2 are two sets of unre-

lated points. S is simply the geometry of vertices and edges

of a complete bipartite graph. This is a well known rank two

polar space, the dual of a "grid" quadrangle. Thus S is a

building of type C 2.

More generally, suppose that S I is of type Cnl and S 2 of

type Cn2 , where N=n l+n 2>2.

All that we need to check is that S is described by the appro-

priate diagram CN~ the other geometry axioms are clearly satis-

fied. It is sufficient to check that the residue of each

variety of S is described by the appropriate subdiagram of CN~

Rees 85

Suppose first that x is a variety in S of type i. Then x

is either a variety of type 1 in S 1 or a variety of type 1 in

S 2. We may assume, without loss of generality that x is a

variety of S 1.

Then ResS(x) consists of three sorts of varieties

(a) those varieties v in ResSl(X )

(b) those pairs of varieties (x,w) where w {S 2

(c) those pairs of varieties (v,w) where vEResSl(X ) and

w E S 2 .

If n I = i, then ResSl(X ) is empty. In this case, clearly

ReSs(X) is isomorphic to S 2 and thus is a geometry of type

Cn2 = CN_I, as required. If n I is greater than i, we see

that ResS(x ) is the product of the two geometries ReSSl(X )

and S2, of types Cnl_l and Cn2 respectively. Thus, by in-

duction, ResS(x) is a geometry of type CN_ I.

Suppose next that x is a variety in S of type N. _~nen x

is a pair (v.w), where v is a variety in S 1 of type n 1

and w is a variety in S 2 of type n 2.

In ResS(x) we see the following kinds of varieties

(a) all varieties of ReSsl(V ) and v itself

(b) all varieties of ResS2(W ) and w itself

(c) all pairs (t,u), where t ~ReSsl(V ) or t =v and

u ERess (w) or u=w, excluding the pair (v,w).

Thus we see that ResS(x ) is isomorphic to ext.prod(ResSl(V),

ResS2(W)). Now ResSl(V ) and ResS2(W ) are geometries of

types Anl_l and An2_l respectively. Thus we see that

ResS(x) is a geometry of type AN_ I.

Finally suppose that x is a variety in S of type i, where

l<i<N.

86 Rees

First observe that if y is a variety in ReSs(X ) of type less

than i and z is a variety in ResS(x ) of type greater than

i then y I z. This is clear from the definStion of ~ncidence

in S and from the fact that both S I and S 2 are defined by

string diagrams.

Thus we see that the diagram for Ress(X ) splits into two dis- ~

joint portions, one on the nodes i+ i .... aN, the other on the

nodes 1,2,...,i-i. These two portions must be the appropriate

subdiagrams for the diagrams for residues of varieties of S of

types ! and N respectively. Thus ResS(x ) is described by a

diagram which is a disjoint union of an Ai_ I and a Cn, i o

Hence we see that prod(SI,S2) is a geometry of type CNo

7. OTHER PRODUCT GEOMETRIES.

t b Since every geometry of type D n is a building of type D n [ y

[i]), all geometries of type D n are actually polar spaces.

Thus, naturally, we can define products of geometries of type

D n. The fact that the D n diagram is not a string diagram

means that we have to be a little careful, but there is no real

problem. Still, there Is little interest in this construction,

since it only yields buildings and will not give rise to any un-

usual geometries.

It seems unlikely that we can use constructions similar to those

of section 5 to build geometries described by any further

Coxeter diagrams.

For a start, it seems reasonable to suppose that under such a

construction a product of buildings would be a building. But in

all product geometries constructed as in section 5 two points

are either collinear with each other or collinear with the same

third point. The only buildings in which this is true are the

An, C n and D n buildings.

Again, if some such construction were to work to give products

of geometries described by other classes of spherical diagrams

Rees 87

(that is, Coxeter diagrams describing finite reflection groups)

surely we should expect such a class of diagrams to be infinite;

for we should presumably be able to build examples of arbitrarily

high rank? However, the only infinite classes of spherical dia-

grams are the An, C n and D n diagrams.

REFERENCES

[i] BROUWER, A.E. and COHEN, A.M.: Some remarks on Tits geome- tries, Indagationes Mathematicae 4~ (1983), 393-400.

[2] BUEKENHOUT, F.: Diagrams for geometries and ~roups, Journal of Combinatorial Theory, Series A 2_~7 (1979), 121-151.

[3] BUEKENHOUT, F. and SPRAGUE, A.: Polar spaces having some lines of cardinallty two, Journal of Combinatorial Theory, Series A 33 (1982), Note, 223-228.

[4] MAEDA, F. and MAEDA, S.: Theory of Symmetric Lattices, Springer-Verlag, Berlin, Heidelberg, New York, 1970, 71.

[5] RONAN, M. and STROTH, G.: Minimal parabolic subgroups for the sporadic groups, preprint.

[6] TITS, J.: Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics no. 386~ Springer-Verlag, Berlin, Heidelberg, New York, 1974.

[7] TITS, J.: Buildings and Buekenhout Geometries, in Finite Simple Groups, ed. M. Collins, Academic Press, New York, 1981, 309-320.

[8] TITS, J.: A local characterisation of buildings, preprint and early version of [9].

[9] TITS, J.: A local approach to buildings, in the Geometric Vein (The Coxeter Festschrift), ed. Davis, Grunbaum, Sherk, Springer-Verlag, Berlin, Heidelberg, New York, 1981, 519-547.

[10] VALETTE, A.: Direct sums of Tits geometries, Simon Stevin i{ (1982), 167-18o.

Sarah Rees, J �9 Departement de Mathematique,

Service de G@om4trie, C.P.216, Universite Libre de Bruxelles, Campus Plaine, Boulevard du Triomphe, B-1050 Bruxelles, BELGIUM.

(Eingegangen am q0. Mai q98~)