On a problem on chamber systems

Geometriae Dedicata 60: 225-236, 1996. © 1996 KluwerAcademic Publishers. Printed in the Netherlands.

225

On a Problem on Chamber Systems ANTONIO PASINI Department of Mathematics, University of Siena, Via del Capitano 15, Siena, 1-53100 Italy e-mail: [email protected]

(Received: 22 November 1993) Abstract. It is well known that a geometry belonging to a disconnected diagram is the direct sum of geometries corresponding to the connected components of the diagram. On the other hand, chamber systems with a disconnected diagram exist which do not split as direct products of components of smaller rank. Many finite examples of this kind are discussed in Groups of Lie Type and their Geometries (CUP, 1995, pp. 185-214), but none of them is simply connected. In this article, we construct a simply connected finite example.

Mathematics Subject Classification (1991): 51E24. Key words: diagram geometry, chamber systems, direct sum theorem cell-geometries, reducibility problem

1. I n t r o d u c t i o n

The Direct Sum Theorem for geometries states that every geometry belonging to a disconnected diagram is the direct sum of subgeometries corresponding to the connected components of the diagram. This theorem is one of the most powerful tools in diagram geometry. Unfortunately, the analogous statement does not hold for chamber systems in general. Some counterexamples are described in [6]. However, none of them is 2-simply connected. Some 2-simply connected counterexamples can be obtained by some kind of free construction ([10, 6.1.6.b]), but examples constructed in this way are not finite (nor even locally finite). Thus, we might still hope that, if we restrict our interest to (locally) finite 2-simply connected chamber systems, then the analogue of the Direct Sum Theorem is valid. I am going thwart that hope here, constructing a finite 2-simply connected counterexample. This paper is organized as follows. The problem we deal with is discussed in Section 2. In Section 3, I define what I call the cell-geometry of a chamber system. The following observation made in [2] turns out to be useful for recognizing that some geometries are in fact cell-geometries of chamber systems: a chamber system is nothing but a partial plane with parallelism. In Section 4 1 prove that thin-lined C~-geometries are precisely the cell geometries of chamber systems with trivial diagram (that is, belonging to diagrams with no strokes). Thus, a classification of chamber systems with trivial diagrams is equivalent to a classification of all thin-lined C,~-geometries. Such a classification

226

ANTONIO PASINI

is likely to be hopeless. Nevertheless, a great deal of information is available on finite thin-lined C3- geometries, hence on finite chamber systems of rank 3 with trivial diagram. I exploit that information in Section 5 to construct the above-mentioned counterexample. Finally, again using some information on finite thin-lined C3- geometries, I prove that the automorphism group of that example is not transitive on the set of chambers (Section 6). The reader is referred to [4] and [6] for basic notation concerning geometries and chamber systems.

2. The Reducibility Problem A chamber systems is said to be reducible [4, §12.5.2] if it splits as a direct product of some of its truncations. Otherwise, it is called irreducible. Clearly, every reducible chamber systems C splits as the direct product of a finite number of irreducible chamber systems and that splitting is unique, modulo permutations of the factors. The factors of that splitting are called the irreducible components of C. We say that a chamber system C with disconnected diagram graph D is completely reducible if C admits truncations over every connected component of D and those truncations are the irreducible components of C. The following is well known [4, Cor. 4.8]: THEOREM 2.1 (Direct Sum Theorem). Chamber systems of geometries are completely reducible. Unfortunately, this statement fails to hold for non-geometric chamber systems (see [6, §4]). This has some unconfortable consequences in classification problems. For instance, assume we are dealing with a chamber system C belonging to a given Coxeter diagram D of rank _> 3 and that we want to know ifC can be obtained as a 2-quotient of a building. According to a well-known theorem of Tits [10, §5.3] we should check if all rank 3 residues of C belonging to spherical Coxeter diagrams are 2-quotients of buildings. The connected spherical Coxeter diagram of rank 3 are A3, C3 and H3. The disconnected spherical Coxeter diagrams of rank 3 are the diagrams A1 + AI + A1 and A1 + I2(ra)(3 < m < oc) depicted below. (AI+A1 +A1)

*

*

(At +

•

•

* •

Since buildings are geometries and the Direct Sum Theorem holds for geometries, a chamber system belonging to a disconnected Coxeter diagram of rank 3 is a building if and only if it is reducible. (Note that, since all chamber systems of rank 2 are geometries, reducibility and complete reducibility are the same for chamber systems belonging to A1 + A1 + A1.)

227

ON A PROBLEM ON CHAMBER SYSTEMS

Thus, a chamber system belonging to a Coxeter diagram D is 2-covered by a building if and only if, for every triple of types J such that D induces A 1 + A1 + A1 or A1 + I2(ra) on J, all residues of C of type d have reducible universal 2-covers and, for every triple of types K such that D induces A3, C3 or tt3 on it, all residues of C of type K are 2-covered by buildings. Actually, irreducible chamber systems exist belonging to a disconnected diagram A1 + Aa + A1 and A1 + I2(ra) (see [6, §4]). Some of them are simply connected (Tits [10, 6.1.6.b]). However, the examples mentioned by Tits in [10] are neither finite nor transitive. Thus, one might still hope to prove that all simply connected finite and transitive chamber systems of type A1 + A1 + A1 and Aa + I2(m) are reducibile. I am going to partially deceive this hope. In Section 5 I will construct a finite 2-simply connected but irreducible chamber system of rank 3 with diagram A1 + A1 + A1 and order 9 at every type. However, that chamber system is not transitive (Section 6). Thus, we might still hope that the above can be proved at least in the transitive case.

3. Cell-Geometries 3. ].

THE CELL-GEOMETRY OF A CHAMBER SYSTEM

Let C be a chamber system of rank n > 1. We can construct a geometry of rank n over the set of types {0, 1 , 2 , . . . , n - 1} by taking as elements of type i the cells of C of rank i, and defining the incidence relation as follows: given two cells X, Y of type J and If, respectively, we declare X and Y to be incident if either X C_ Y andJCKorYCXandKC J. It is easy to check that this is indeed a geometry. We call it the cell-geometry of C and we denote it by F(C) (if C is geometric, then F(C) is just the dual of the flag-geometry of F(C) as defined in [4]). It is not difficult to prove that F(C) belongs to the following diagram, where the label X on the first stroke denotes a class of rank 2 geometries containing the cell-geometries of the rank 2 residues of C and the integers 0, 1 , . . . , n - 1 are the types: 0 --

X

1

2

:

•

......

n--2

n--1

.

.

Furthermore, F(C) is thin at all types i > 0. For instance, if C belongs to a Coxeter diagram with all strokes of weight ra, then F(G) belongs to the following Coxeter diagram: (2m) In particular, if the diagram of C is trivial (i.e. if ra = 2) then following diagram: (Cn)

.

.

4__

......

-_

_-

F(C) belongs ot the

228 3.2.

ANTONIO PASINI C H A M B E R S Y S T E M S A N D PARTIAL L I N E A R SPACES

Since a geometry is a graph, we can consider its girth, which is an even integer ___4 if the geometry has rank 2. Given a geometry F of rank 2, its gonality is half of its girth [4]. Geometries of rank 2 and gonality >__ 3 are called partial linear

spaces. Let F be a partial linear space. We can take the integers 0 and 1 as types for F calling the elements of F of type 0 (respectively, 1) points (lines). According to [2], a parallelism of F is a partition II of the set of lines of I' such that every point is incident to precisely one line in each of the classes of II. Every chamber system C can be viewed as a partial linear space with parallelism. The chambers and the panels of C (which are the elements of type 0 and 1 of the cell-geometry F(C)) are the points and the lines of that partial linear space. Two lines are declared to be parallel if they have the same type as panels of C. We denote by H(C) this partial linear space with parallelism. (Actually, we have more than just a partial linear space with parallelism here. Indeed the relation 'being of the same type' between cells of C is a parallelism of F(C) in the meaning of [2] and it induces II on II(C).) Conversely, let II = (F, II) be a partial linear space with parallelism and assume that l] has a finite number n of classes (that is, every point of F is incident to a finite number of lines). If we take the points of F as chambers and the equivalence relations defined by the classes of II as adjacency relations, then we get a chamber system C(II) of rank n. Clearly, H(C(H)) TM H. On the other hand, given any chamber system C, we have C -~ C(II(C)). The correspondence between chamber systems of rank n and partial linear spaces with parallelism with n lines on every point will be exploited in the next subsection to characterize cell-geometries of chambers systems. 3.3.

A C H A R A C T E R I Z A T I O N OF CELL=GEOMETRIES

In this subsection I? is a geometry of rank n with diagram as follows, with X denoting some class of partial linear spaces: 0

(X.an_l)

X

1

2

.

•

(~

points

lines

n--2

• dual lines

n--1

• dual points

We call the elements of type 0, 1, n - 2 and n - 1 points, lines, dualpoints and dual lines respectively, as in the above picture. We say that two dual points of F are dually collinear if they are incident to the same dual line. The dual colIinearity graph G(F) of F is defined by taking the dual points as vertices and the dual collinearity relation as adjacency relation. If F = F(C) for some chamber system C, then 6(F) is the incidence graph F(C) of C. The parallelism relation I] of F(C) (§3.2) induces an n-partition on G(F) -- F(C), which is in fact the type-partition of the geometry F(C).

229

ON A PROBLEM ON CHAMBER SYSTEMS

THEOREM 3.1. The following are equivalent: (i) F is the cell-geometry of a chamber system. (ii) F is thin at the types 1, 2 , . . . , n - 1 and its dual collinearity graph is n-partite.

Proof It is clear from the previous remarks that (i) implies (ii). Let us prove that (ii) implies (i). Let r be thin at the types 1 , 2 , . . n - 1, let G(r) be n partite and l e t / = { i)i=1 be an n-partition of G(F). The diagram forces the point-line system of r to be a partial linear space. Given an element x of type F of type k, let a ( x ) be the set of dual points incident to F. Then a(x) is a k-clique of G(F), hence it meets just k of the classes [1, I2, . . . , In of 1. Let J be the set of the classes of I that contain an element of ~r(x). Then we say that x has I-type I - J. In particular, if x is a line of F, then its •-type is an element of I. Given two lines x, y, we write x ]]y to mean that x and y have the same •-type. The residue of a point of r is a thin geometry of type An- 1, hence and (n - 1)dimensional simplex [10, Prop. 6]. Therefore every point is incident to precisely one line of every/-type. That is, 11is a parallelism on the point-line system of F. Hence it defines a chamber system C and r = F(C). [] COROLLARY 3.2. Let r be 2-simply connected and let X denote a class of cell-

geometries of chamber systems of rank 2. Then F is the cell-geometry of a (uniquely determined) 2 simply connected chamber system. Proof The hypothesis made on X forces F to be thin at all types but possibly 0. As F is 2-simply connected, its dual collinearity graph admits a (unique) n-partition, by Theorem 7 of [5]. By Theorem 3.1, r = F(C) for a uniquely determined chamber system C. Every 2-covering f • C -+ C of C gives us a 2-covering F ( f ) " F(C) --+ F(C) = F. As 17 is 2-simply connected, F ( f ) is an isomorphism. Hence f is an isomorphism, too. Thus, C is 2-simply connected. []

4. Thin-Lined Cn Geometries Let F belong to the diagram Cn. The elements of F of type 0, 1 and 2 are called

points, lines and planes respectively. We call those of type n - 2 and n - 1 dual lines and dual points respectively. 0

points

1

lines

2

planes

n -

2

dual lines

~ -

1

dual l)oin(,s

We say that 1~ is thin-lined if it is thin at the type 1 (hence it is thin at every type

i n. We must show that ?Tb ~ n .

We will prove this by induction on n. When n = 2 the statement is obvious. Let n > 2 and let m > n, if possible. Then, for every dual point u, there is a point a ~ c~(u) but such that a ± 2 or(u). By a well-known property of Cn-geometries there are a dual point v incident with a and a dual line w is incident with u such that v and w are incident (see [4, 7.4.1]). Let b0, b l , . . . , b~-i be the points of a(u). We can assume to have ordered them in such a way that the first n - 1 of them form a ( w ) . For i, j = 1 , 2 , . . . , n - 1, i < j let xi,j be the line joining bi with bj in F~. Note that xi, j is also a line of F~o for 0 1 if n = 1. Let X1, X2, X3 be pairwise disjoint sets of size ral, m2 and ra3 respectively and let I / xyz be a set of size n. For every triple (x, y, z) E X1 × X2 X X3, let Lzyz = [mi, j ) i , j E I be and (n x n)-latin square with entries in I. Let us take U3~=IX~ as set of points. As lines we take the pairs ({x, y}, i) with x , y points not in the same set X~ for any r = 1,2,3. As planes we take the triples ( ( { x , y } , i ) , ( { y , z } , j ) , ( { z , x } , k ) ) with (x,y,z) C X1 x X2 x X3 and xyz k = mi, j .

The points of a line ({x, y}, i) are x and y. The points and the lines of a plane ( ( { x , y } , i ) , ( { y , z } , j ) , ( { z , x } , k ) ) are x , y , z and ( ( { x , y } , i ) , ( { y , z } , j ) , ({z, x), k)) respectively. Let r be the geometry defined as above. Then r is a thin-lined C3-geometry. {X~}3~=1 is the unique 3-parition of the collinearity graph G(F) of F (see Lemma4.1). r is a p o l a r space if and only if n = 1. I f r a l = ra2 = ra3 = m, then I' admits order ran - 1 at the third node of the diagram. Note that ral = ra2 = ra3 = 1 if and only if every point of I' is incident to all planes; that is if and only if I' has just three points. If this happens, then I' is said to be fiat. Clearly, if r is flat, then it arises from a unique latin square. The following has been remarked by S. Rees in [7, (5(i))]:

ON A PROBLEM ON CHAMBER SYSTEMS

233

L E M M A 5.1 (Rees). Let F be a flat thin-lined finite C3-geometry and let L = (mi,j)i,jE1 be its latin square. For i E Jr, let pi be the ith row of L, viewed as a permutation of I, Then F is a 2-quotient of a polar space if and only if the set {Pi } i~ I is a coset of a group of permutations of I. We can now construct our example. Let us take ml = m2 = m3 = 2, n = 5 and all latin squares Lxyz equal to the following one: 01234 10342 24013 32401 43120 Let I' be the resulting C3 geometry. L E M M A 5.2. F is 2-simply connected but it is not a polar space. Proof. F admits a flat 2-quotient F described by the following latin square: 0123456789 1034265897 2401379568 3240187956 4312098675 5678901234 6589710342 7956824013 8795632401 9867543120 Such a quotient is obtained as follows. Let {X~}~=I 3 be the (unique) 3-partition of ~(F) (Lemma 4.1) and let g be the permutation of the set of points of interchanging the two elements of X~ for every r = 1,2, 3. It is not difficult to see that g can be extended to an automorphism 7 of F defining a 2-quotient F / ( 7 ) of and that F / ( 7 ) is isomorphic with the fiat geometry F described by the above (10 × 10)-latin square.

234

ANTONIOPASINI

By Lemma 5.1, the universal 2-cover F of I? is not a polar space. On the other hand, F is a 2-quotient of F, since it is a 2-cover of F. If F is a proper 2-quotient of F, then I', is a 5-fold cover of F. This forces F to have 30 points. Hence it is a polar space, since it has order 9 at the third node of the C3-diagram. However, F is not a polar space, as we have remarked. Hence F ~ F. That is, F is 2-simply connected but it is not a polar space. [] By Theorem 4.6 and Lemma 5.2, the chamber system j = C(F) is 2-sirnply connected but irreducible. Therefore: THEOREM 5.3. There are irreducible 2-simply connected finite chamber systems with trivial diagram.

6. On Transitivity 6.1. SOMENOTATION Given a chamber system C of rank n, let II be the parallelism of F(C) as defined in Section 4.2 and let G be the subgroup of Aut (F(C)) preserving the partition defined by the equivalence relation [I. Then G = K °° .G ~ where K °° is the stabilizer in G of all classes of II and G = G / K ~ is the group induced by G on the set of those classes. (The classes of 11can be viewed as 'elements at infinity' of the elements of F(C).) Given a point a of F(C), let Ga be the stabilizer of a in G. Then Ka = K °° VlGa is the elementwise stabilizer of the residue of a and G a / K a is naturally embedded in G ~ . Clearly, Aut(C) = t ( ~. Hence C is transitive if and only if K ~ is transitive on the set of points of F(C). If this happens, then G °° ~ G a / K a for every point a

ofF(c). 6.2. AGAIN ON FINITE CHAMBERSYSTEMS OF RANK 3 WITH TRIVIAL DIAGRAM In this subsection C is a finite chamber system of rank 3 with trivial diagram and F is the dual of F(C). As we have remarked in Section 5, F arises from three disjoint sets X1, X2, X3 and a set E = {Lxvz}(x,v,z)en~=lx r (

xyz

\

of latin squares Lxyz = [mi, j )i,jeI with entries from the same set I. We can assume that I = { 1 , 2 , . . . , n}. It is not difficult to see that the automorphisms of F belonging to K ~ are 4-tuples (Pl, P2, P3, ( Ofyz, /3xyz ,7 xyz\)(x,~,z)en~=lX,,)x

ON A PROBLEM ON CHAMBER SYSTEMS

235

with p~. a permutation of X~ (r = 1, 2, 3) and ct~v~ ,/3 ~u~, 7 xy~ permutations of [ such that 7xyz[~Xyz'~ ~npI(x)P2(y)P3(Z) k "~i,j ] . . . . e~xy~(i),3.~y~(j ) 3 for every (x, y, z) E II~=lXr and all i , j E I. 3 Given (a, b, c) E II~=~X~, let K ~ c be the stabilizer of the points a, b, c in K °°. Given an automorphism f -- (Pl, P2, P3, ( v~xyz, flxyz, ,~xyZ)(x,y,z)GH3=lXr ) in I ( ~ , the action of f on Labc only depends on the permutations a ~bc,/3 ~b~ and 7abe. We denote that action by f~z.y, writing u,/3 and 7 instead of ~bc, fl~b~ and 7~bc, for short. Clearly. mc~( i),3(j ) = ~/ ( mi,j )

(1)

for all i, j E I, where we write mi, j instead of mi,abc j , for short. We set: K~

= {f~z'ylf E I ( ~ } .

Permuting the rows and the columns of all latin squares of Z; if necessary, we can always assume that ml,i = mi.1 = i for every i E I. Thus, given f~3"y E I ( ~ , we have: m~(1),3(i ) -: m~(i),3(1 ) = 7 ( i )

(2)

for every i E I. The ith row (column) of L~uz can be viewed as a permutation Pi (respectively ql) of I. By (2) we get Pa(1) = ")'/3-1,

q3(1) = 7 a - l -

(3)

Thus, the products 7f1-1 and 7 a -1 only depend on the values a(1) and fl(1), respectively. In particular since both Pl and ql are the identity permutation, 7=/3

if

a(1)=l

and

7=a

if

3(1)=1.

(4)

We can n o w prove the following, PROPOSITION 6.1. Let C, F and Lab¢ = (mi,j)i,j6i be as above, with ml,i = ral,~ = i f o r every i E I = { 1 , 2 , . . . , n} and let pi( i E I) be the rows of the latin square L~b~, viewed as permutations of I. If C is trasitive, then for every i E I there is a permutation/3~ of I such that/3i(1) = 1 and pp~(j) = pi/3ipjfl~ 1 for every j E I.

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ANTONIO PASINI

Proof. As C is transitive, K ~ ( = Aut(C)) is transitive on the set of planes of F (which are the points of F(C), that is the chambers of C). In particular, the planes (({ a, b }, i), ({ b, c} , 1), ({ c, a} , mi,1) ) are contained in one orbit of I ( ~ . Therefore, for every i E I there is an element f ~ . ~ E/-(~c with ~i(1) = i and /~i(1) = 1. For such a choice of f~P~-Yi we have 7i = ai and pi = ~i/3~ -1

(5)

by (3) and (4). By (1) we obtain ~-o~ ./q-1 f4-1

Comparing (6) and (5) we get Pp~fli(j) = PifliPj/3[ -1, as we wanted.

(6) []

COROLLARY 6.2. The chamber system C constructed is Section 5 is not transitive. (Staightforward, by Proposition 6.1.) References 1. Buekenhout, F.: The basic diagram of a geometry, in Geometries and Groups, Lecture Notes in Maths 893, Springer, 1981, pp. 1-29. 2. Buekenhout, E Huybrechts, C. and Pasini, A.: Parallelism in diagram geometry, Bull Soc. Math. Belgique-Simon Stevin 1 (1994), 355-397. 3. Kantor. W.: Generalized polygons, SCABs and GABs, in Buildings and the Geometry of Diagrams, Lecture Notes in Maths. 1181, Springer, 1986, pp. 79-158. 4. Pasini, A.: Diagram Geometries, Oxford University Press, 1994. 5. Pasini, A.: Shadow geometries and simple connectedness, European J. Combin. 15 (1994), 17-34. 6. Pasini. A.: The direct sum problem for chamber systems, in W. Kantor and L. Di Martino (eds.), Groups of Lie Type and their Geometries, Cambridge University Press, 1995, 185-214. 7. Rees, S.: Finite C3 geometries in which all lines are thin, Math. Z. 189 (1985), 263-271. 8. Ronan, M.: Coverings and automorphisms of chamber systems, European J. Combin. 1 (1980), 259-269. 9. Schaflau, R.: Geometrical realizations of shadow geometries, Proc. LondonMath. Soc. 61 (1990), 615-656. 10. Tits, J.: A local approach to buildings, in The Geometric Vein, Springer, 1981, pp. 519-547.