J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 382, Springer--Verlag, Berlin, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....#) hermitian form or pseudo quadratic form vanishes. To state the Main Theorem we introduce some notation. Let L be a division ring and W be a (left )vector space over L endowedwitha(#, #) hermitian form or a pseudo quadratic form q (with associated (#, #) hermitian form f) in the sense of [Ti, 8]. We may assume that # = 1and# 2 = id.Welet Rad(W, f) w f(w,x) 0 for all x W , x # = w # f(w,x) 0 for x W, # min = c L , #max = c # L c . If Rad(W, f) 0,thenf is said to be non degenerate. Further f is trace valued, if f(w,w) # c #c L for ....

Tits, J.: Buildings of spherical type and finite BN-pairs. Lecture Notes in Math., 386. Springer--Verlag, Berlin Heidelberg New York, 1974.

....three geometries associated with PSL(3, 4) Harald Gottschalk # 1 Introduction In [Bu86] F. Buekenhout started a research program in order to find a unifying combinatorial approach to all finite simple groups, inspired by J. Tits famous theory of buildings (see e.g. [Ti74]) that is to classify diagram geometries fulfilling various conditions for all of these groups. This attempt has recently led to several collections of geometries (see [BCD95a] BCD95b] BDL94] and [BDL95] using Cayley (see [De94] The present paper should be regarded as a part of this ....

....n is said to be firm if s i 1, thin if s i =1and thick if s i 2 for all i I . It is called residually connected if its incidence graphs are connected for every residue # J with J # I 2. This is also called condition RC. To check whether # is connected, we have the following lemma [Ti74]. 2.1 Lemma. Let # be a geometry over I and G be a flag transitive automorphism group of #. For every i I , G i denotes the stabilizer of an element of type i. Then # is connected if and only if G i : i I = G. Let # be a flag transitive geometry. Then we say that # fulfills the ....

Tits, J. Buildings of spherical type and finite BN-pairs. Lect. Notes in Math. 386, Springer-Verlag, Berlin-Heidelberg-New York (1974).

....J) is isomorphic to the coset geometry (G J , G J# k : k J) The Borel subgroup of the geometry is the subgroup B = G I = i#I G i . See [1] for more explanation of these terms. Condition (G3) was re phrased in terms of the subgroups G i by Buekenhout and Hermand [4] following Tits [7], as follows: For any J I with J # 3 and any j J, we have G j G k G j G k . Moreover, if this holds for one j J, then it holds for all. We refer to this as condition (BH) The coset geometry is residually weakly primitive, or RWPRI, if the following condition holds: ....

J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 382, Springer--Verlag, Berlin, 1974.

....explosion of literature on singular spaces, see the bibliography of [Bal95] Examples of spaces with curvature bounded above (henceforth CBA spaces) include: ffl Complete Riemannian manifolds with sectional curvature bounded above. ffl Euclidean, spherical, and hyperbolic Tits buildings, see [Tit74, Ron89, KL97] ffl Complexes with piecewise constant curvature. DJ91, CD95, Ben91, BB94, S93] construct examples with interesting geometric and topological properties. ffl Limits of Hadamard spaces 1 , such as Tits boundaries and asymptotic cones. These have a number of applications, see ....

J. Tits. Buildings of spherical type and finite BN-pairs, volume 386. Springer, 1974. 42

....if c j = d j for all j #= i. Of course, the lattice of subspaces of a finite dimensional vector space over a division ring is semimodular (even modular) and has finite height. Buildings of type A n = Sym n 1 , n # 3 are flag complexes of finite dimensional vector spaces over division rings [39]. Therefore buildings of type A n constitute a special case of flag complexes of semimodular lattices of finite length. From now on we assume that we have a chamber system C with a Coxeter metric # : C C # W. 2.3 Apartments A # isometric image of the Coxeter complex W in C is called an ....

J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math. 386 Springer-Verlag, 1974.

....some arguments in [11] but it is very di#erent from [9] in the technique used. Besides its uniform approach, it has the advantage of avoiding references to Tits s classification of buildings, a very di#cult result with a long proof which still exists only in Tits s original written account [24]. A group of finite Morley rank is said to be of p # type, if it contains no infinite abelian subgroup of exponent p. Notice that a simple algebraic group over an algebraically closed field K is of p # type if and only if char K #= p. Other definitions can be found in the next section. The aim ....

J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathematics 386 (Springer-Verlag, New York, 1974).

....Galois and continuing through Cooperstein [12] and Kantor [16] Cameron [6] has some geometric speculations on Aschbacher s Theorem. Carter [8] discusses groups of Lie type (identifying many of these with classical groups) The natural geometries for the groups of Lie type are buildings: see Tits [23] for the classification of spherical buildings, and Scharlau [21] for a modern account. The other papers in the bibliography discuss aspects of the generation, subgroups, or representations of the classical groups. The list is not exhaustive ....

J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 382, Springer--Verlag, Berlin, 1974. 96

....be joined by a gallery of type i 1 i k . The Coxeter complexes are buildings. In the special case when W = Sym n is the symmetric group on n # 4 letters, buildings of type W , or type A n as they are usually called, are flag complexes of finite dimensional vector spaces over division rings [18]. Therefore buildings of type A n are a special case of flag complexes of semimodular lattices of finite length. 8 5 Abels complex for a semimodular lattice. Herbert Abels [2] had shown that the flag complex of a semimodular lattice of finite height n has a natural structure of a chamber ....

....1 Actually we use the inverse of the permutation defined in [2] 9 . When we consider topological realisations of C and A, the map # a gives rise to a retraction, in the topological sense, of the topological space of C onto that of A. Retractions of buildings were introduced by Tits [18]; they play an important role in the structural theory of buildings. Coxeter matroids. Now let us identify the apartment A with W and, given a chamber c # C, define the map c : W # W by the rule c (w) #w (c) The main result of [5] can be stated in the following form. A map : W # ....

J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math. 386 Springer-Verlag, 1974.

....Thus H arises from an embedding in this case. In the second case H is a non degenerate polar subspace. But then, applying the Buekenhout Lefevre Dienst theory to H, this time, we see that e H : H # P#e(H)# is again a natural embedding of H which is a dominated embedding in the sense of Tits [10]. Lemma 8.6 of Tits [10] then shows P#e(H)# is a projective hyperplane of P(V )ande(H) exhausts all singular points in this hyperplane. Thus in this case H also arises from the embedding. This proves the result for n = 4, so we may now assume n 4(3) Geometric hyperplanes of the half spin ....

....embedding in this case. In the second case H is a non degenerate polar subspace. But then, applying the Buekenhout Lefevre Dienst theory to H, this time, we see that e H : H # P#e(H)# is again a natural embedding of H which is a dominated embedding in the sense of Tits [10] Lemma 8. 6 of Tits [10] then shows P#e(H)# is a projective hyperplane of P(V )ande(H) exhausts all singular points in this hyperplane. Thus in this case H also arises from the embedding. This proves the result for n = 4, so we may now assume n 4(3) Geometric hyperplanes of the half spin geometries arise from ....

J. Tits. Buildings of Spherical Type and Finite BN-pairs, volume 386. Springer-Verlag, Berlin, 1974. Lecture Notes in Math.

....3) We will freely use them in this paper. Since a diagram can be viewed as a graph, we can speak of paths in it, of its connected components, and so on. Thus, we can state the following definitions. Let # be a diagram over a finite set of types I,let0# I and let X,Y # I . Following Tits [39], we shall say that X separates 0 from Y in # if there is no path in # X joining 0 to some element of Y . Let # # be a diagram over J # 0 ,letK # J and let # be the diagram over J obtained by removing 0 from # # .Wesaythat# # is a (0,K,#) diagram if K separates 0 from J K in # # . We freely ....

J. Tits. Buildings of Spherical Type and Finite BN-pairs, volume 386 of Lect. Notes in Math. Springer Verlag, 1974.

....of Mathematics Rutgers University New Brunswick NJ 08903 USA Neil White Department of Mathematics University of Florida Gainesville, FL 32611 USA 12 June 1997 De nitions and notation used in the paper are mostly standard. Those related to Coxeter groups and buildings can be found in [11, 13, 15], to lattices and matroids in [18] See also the forthcoming book [4] Introduction This paper is devoted mostly to the explanation of the equivalence of two de nitions of representability of matroids: classical, in terms of vector 1 con gurations, and a more general de nition, in terms of ....

....of type i 1 i k . The Coxeter complexes are buildings, with a W metric (x; y) x 1 y. In the special case when W = Sym n is the symmetric group on n 4 letters, buildings of type W , or type A n as they are usually called, are ag complexes of projective spaces over division rings [15]. Therefore buildings of type A n are a special case of chain complexes of semimodular lattices of nite height. 5 Abels complex for a semimodular lattice. Herbert Abels [2] has shown that the chain complex of a semimodular lattice of nite height n has a natural structure of a chamber complex ....

J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math. 386 Springer-Verlag, 1974.

....by the editors November 1993 Communicated by J.Doyen AMS Mathematics Subject Classification : 51A. Keywords : incidence geometry, building, polar space and line space. Bull. Belg. Math. Soc. 2 (1995) 87 97 88 S. Lehman Each of these diagrams corresponds to a class of buildings (Tits [16]) If the node labelled i is singled out, we get a diagram called A n,i or C n,i respectively. Geometrically, this amounts to the construction of a line space (the definition is given below) with one type (namely i) of vertices of a building # of type X n ,whereX n is a Coxeter diagram of ....

....if there exists a flag F of cotype i such that l is the set of all i elements incident with F . We call this line space an X n,i building space and we denote it by S(#,i) We can make a similar construction from a geometry. The A n,1 building spaces correspond exactly to projective spaces (Tits [16]) and the classical work of Veblen and Young [17] characterizes the latter in terms of points and lines. Buekenhout Shult s characterization of polar spaces [4] gives an analogous result for C n,1 building spaces. It seems reasonable to try to find a similar characterization for all building ....

J. Tits, Buildings of spherical type and finite BN-pairs , Lecture Notes in Math., 386, Springer-Verlag, Berlin, 1974.

....3 Then either Gamma contains a free nonabelian subgroup or X is a Euclidean space and Gamma is a Bieberbach group whose rank is dimX. For thick Euclidean buildings of dimension 3, this follows from Tits theorem quoted above [Ti1] and his classification of spherical buildings of rank 3 (see [Ti2]) The proof of our main result, Theorem C, consists of two parts. In the first part we consider the case when all faces of X are flat Euclidean triangles and all links have diameter and show by an elementary argument that X is either a product of two trees or a thick Euclidean building of type ....

J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Math. 386, SpringerVerlag, 1974.

....at distance 6 from each other are called opposite. If two elements x; y are not opposite, then there exists a unique element incident with x and at minimal distance from y, and we denote that element by proj x y (it is directly related to the usual projection mapping in buildings, see Tits [11], Subsection 3.19) If two elements x; y are opposite in Gamma, then the set Gamma i (x) Gamma 6 Gammai (y) i = 2; 3, is non empty (it has the same cardinality as Gamma 1 (x) and Gamma 1 (y) and is denoted for short by x y [i] For i = 2, we sometimes write x y [2] x y , see ....

J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math. 386, Springer, Berlin--Heidelberg--New York, 1974.

....H. Van Maldeghem contained in a J residue of B. Then they are opposite in that residue if and only if they are opposite in the corresponding J residue of B. Using the previous two lemmas and similar ideas as in the proof of Tits Rigidity Theorem for spherical buildings (Theorem 4.1. 1 in [18]) one can easily prove the following proposition: Proposition 2.4 Let C be a thick folding of the spherical building B of type I, let c; d be two opposite chambers in B. Given a folding C 0 of B of type I such that c; d 2 C 0 and such that R i ( c) C ....

....of (i) and (ii) of the following condition is proved in [5] the equivalence of (i) and (iii) can be established in a similar way. Proposition 2.6 Let B be a thick and irreducible building of spherical type. Then the following conditions are equivalent: i) B is Moufang in the sense of [18]. ii) For each chamber c of B there is a transitive group of unipotent automorphisms of B fixing c and acting transitively on c op . Diagrams for Embeddings of Polygons 7 (iii) For any pair c; d of opposite chambers in B there is a group of unipotent automorphisms of B fixing c and acting ....

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J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics 386, Springer Verlag 1974.

....journal pages. Ironically, although the theorem was announced in 1980, the proof contained a gap which has not yet been filled. Combinatorial ideas (graphs, designs, codes, geometries) were involved in the proof: perhaps most notably, the classification of spherical buildings by Jacques Tits [30]. Also, the result has had a great impact in combinatorics, with consequences both for symmetric objects such as graphs and designs (see the survey by Praeger [25] and (more surprisingly) elsewhere as in Luks proof [21] that the graph isomorphism problem for graphs of bounded valency is in P. ....

Jacques Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics 386, Springer, Berlin, 1974.

.... the compound game) 12 Coherence spaces One of the oldest semantical interpretations of linear logic, in fact part of the motivation for the original development of linear logic, uses coherence spaces [12] A coherence space, also known to combinatorialists and group theorists as a flag complex [25], is a family A of sets closed under subsets and under coherent unions, where the latter means that # X#Awhenever X#Aand the union of every two members of X is in A. Such a space can equivalently be described as the family of cliques (i.e. pairwise adjacent sets of nodes) in an ....

J. Tits. Buildings of Spherical Type and Finite BN-Pairs, volume 386 of Lecture Notes in Mathematics. Springer-Verlag, 1974.

....geometries (cf. Corollary 5.9) and we answer an open question concerning near n gons (cf. Corollary 5.10) In order to state our Main Result, we need some preliminaries. We assume the reader is familiar with the notion of buildings, the Weyl group related to a building, and Coxeter systems (see [15]) Let (W; S) be a Coxeter system and denote by : W N the usual length function with respect to S. Let and be two buildings of type (W; S) i.e. with apartments isomorphic to the standard Coxeter complex (W;S) associated to (W; S) Denote by 2 C the set of chambers of , 2 f ; ....

....characterization of adjacency by the opposition relation in thick spherical buildings as derived in [3] Section 4, one shows that is bijective and that both and 1 preserve adjacency of ags (the proof is almost identical with the proof of Corollary 5.5 given in [3] By Theorem 3. 21 of [15], extends to simplicial isomorphisms 1 : and 2 : We can choose types in such that 1 is type preserving for even n and type reversing for odd n. For a xed ag f 2 F we 20 have x op f , x op f , x op f ; for all ags x 2 F , 2 ....

Tits J., Buildings of Spherical Type and Finite BN-Pairs, Springer Verlag, Berlin, Heidelberg, New York, Lecture notes in Math. 386, 1974

....if and only if c j = d j for all j 6= i. Of course, the lattice of subspaces of a nite dimensional vector space over a division ring is semimodular (even modular) and has nite height. Buildings of type An = Sym n 1 , n 3 are ag complexes of nite dimensional vector spaces over division rings [36]. Therefore buildings of type An constitute a special case of ag complexes of semimodular lattices of nite length. We assume from now on that we have a chamber system C with a Coxeter metric : C C W: 4.3 Apartments A isometric image of the Coxeter complex W in C is called an ....

J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math. 386 Springer-Verlag, 1974.

....These properties imply that an n gon is connected and that every vertex has order at least 2. In fact, we shall only consider thick n gons, that is, assume that all vertices have order at least 3. The generalized polygons are nothing else but the buildings of rank 2 and spherical types (cf. e.g. [9]) The buildings of (irreducible) spherical type and rank at least 3 are completely classified in loc.cit. roughly speaking, they are the buildings associated to algebraic simple groups and classical groups of rank # 3. In short, we shall say that they are of algebraic origin . There is no ....

....construction (cf. e.g. 13] 4.4) indicates that generalized n gons are too general objects to allow classification in any reasonable sense. Thus, in order to characterize geometrically the polygons of algebraic origin , an extra condition is necessary. The Moufang condition , introduced in [9], p.274 (cf. also [11] the statement of which will be recalled below, appears to be Received by the editors in February 1994 AMS Mathematics Subject Classification: Primary 51E12, Secondary 20E42 Keywords: Moufang polygons, root systems. Bull. Belg. Math. Soc. 3 (1994) 455 468 456 J. Tits ....

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J. Tits. Buildings of Spherical Type and Finite BN-pairs, volume 386 of Lect. Notes in Math. Springer Verlag, Berlin, 1974.

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J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 382, Springer--Verlag, Berlin, 1974.

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J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 382, Springer-Verlag, Berlin, 1974. 23

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J. Tits, "Buildings of Spherical Type and Finite BN -pairs," Lecture Notes in Math., Vol. 386, Springer-Verlag, Berlin, 1974.

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J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in

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