P. Deligne, Les immeubles des groupes de tresses generalises, Invent. Math. 17 (1972), 273-302.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....hand, the fundamental group of the complement, 1 (X) is not determined by L(A) alone, as the example of Rybnikov [32] shows. For certain arrangements, all the higher homotopy groups of the complement vanish. Examples of such K( 1) arrangements include the simplicial arrangements (Deligne [7]) and the supersolvable arrangements (Terao [37] Examples of non K( 1) arrangements, and methods for detecting the rst non vanishing higher homotopy group of their complements, were given by Falk [12] and Randell [30] see also the recent survey [15] The rst (and, up to now, only) ....

P. Deligne, Les immeubles des groupes de tresses generalises, Invent. Math. 17 (1972), 273-302.

....representation of H k (p ) corresponding to the irreducible character of the Weyl group W k , then ae(T w k;0 ) c k;0 ( p c k;s ( c k; where c k;s ( c k; and c k;0 ( are the constants given in (2.2) Proof. a) By a theorem of Breiskorn Saito [BS] and Deligne [De], the element is central in the generalized braid group. Thus T is central in H k (p ) and it follows that T acts by a constant in every irreducible representation. The constant is computed by writing Tw k;0 as a product of generators and taking the determinant of both sides of the ....

P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math 17, 273-302.

....20F36 1 2 Geck Michel character table of the Iwahori Hecke algebra of type E 8 (the only Iwahori Hecke algebra whose table was missing up to now, see the remarks following Proposition 1.3 below) Let us now explain in more detail our main results. They are motivated by the results of Deligne [10] and Brieskorn Saito [4] about the solution of the word problem and a normal form for elements in B . Following Brieskorn Saito, such a normal form is given as follows. Let w 2 B , and let J = J(w) be the set of all s 2 I such that s divides w in B from the left. Then the least ....

....each d i is replaced by d i =2. Such an element will then be called very good . The point of the Theorem is the property that the subsets J i form a decreasing sequence from which it follows that the elements w d i J i commute with each other (see the description of the centre of B in [10], 4] These results can be seen as a generalization of the well known fact that, if C is the class such that Cmin is the set of Coxeter elements and h is the Coxeter number, then w h = w 2 I for w 2 Cmin , and if h is even then there exists an element w 2 Cmin such that w h=2 = w I . They ....

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P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972), 273--302.

.... H is free over O, with basis fTw gw2W . The morphism H Omega O O= q s Gamma 1; q 0 s 1) s2 S ZW , Tw Omega 1 7 w, is an isomorphism. We assume from now on that W is finite. Let S 0 be a subset of S and W 0 be the subgroup of W generated by S 0 . Then, by [De], ffl the submonoid of B W generated by foe s g s2S 0 is isomorphic to B W 0 , 14 ffl the subgroup of BW generated by foe s g s2S 0 is isomorphic to BW 0 , ffl the specialization of the subalgebra of H(W ) generated by fT s g s2S 0 obtained by sending to 0 those ....

....S H2A H V 0 ) and take x 0 2 C 1 . Let S be the set of reflections of G with respect to the walls of C 1 . For s 2 S, let fl s be the path [0; 1] X defined by fl s (t) x 0 s(x 0 ) 2 x 0 Gamma s(x 0 ) 2 e it : Let s be the class in BG of ae(fl s ) Brieskorn [Br] and Deligne [De] have proven the following theorem : Theorem 10.2 The map oe s 7 s induces an isomorphism BG BG . 10.2 The complex case Let H 2 A. Let e H be the order of the pointwise stabilizer of H in G. This is a cyclic group, generated by a pseudo reflection s with non trivial eigenvalue exp(2i=eH ) ....

P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Inv. Math. 17 (1972), 273--302.

....A in C n simplicial if it is the complexification of a (real) simplicial arrangement. Pure braid groups and, more generally, coloured Artin groups of finite type are fundamental groups of complements of simplicial arrangements [Br] Assume A to be a simplicial arrangement in C n . By [De], M(A) is an Eilenberg MacLane space, thus 1 (M(A) has finite cohomological dimension and is torsion free. By [Ch] 1 (M(A) is biautomatic. In particular, its word problem and its conjugacy problem are solvable. Let A be a central arrangement of hyperplanes. The set of intersections of ....

P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972), 273--302.

....to the f minimizing vertex of P then the f monotone paths on P biject to the maximal chains of the poset of regions of A (see [10] with basis B. The flips correspond to the elementary homotopies connecting these maximal chains (see Section 6) a concept which originated in the work of Deligne [9], and reduce to the Coxeter moves in the important special case of reflection arrangements (see [5, Section 2.3] and Section 7) Moreover, this graph of maximal chains and elementary homotopies can be defined for an arbitrary oriented matroid L with fixed tope B and was shown to be connected in ....

P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972), 273-- 302. 18 CHRISTOS A. ATHANASIADIS AND FRANCISCO SANTOS

....of combinatorially equivalent arrangements with different Chen ranks. In particular, they do not distinguish the examples of Rybnikov [80] of combinatorially equivalent, homotopically different arrangements (see Section 1.3) 2.4. Cohomological properties of the fundamental group. In 1972 Deligne [21] proved that for a complexification of a real simplicial arrangement, the complement M is aspherical (also expressed by saying that M is a K( 1) space. That is, the universal cover of M is contractible. Since all real reflection arrangements are simplicial, this solved a question raised and ....

P. Deligne. Les immeubles des groupes de tresses g'en'eralis'es. Inventiones mathematicae, 17:273--302, 1972.

.... ff in A( Gamma) satisfying ffXff Gamma1 = X, where X is the natural generating set of A( Gamma) If Gamma is finite type and connected, then the quasi center is an infinite cyclic group generated by a special element of A( Gamma) called fundamental element, and denoted by Delta( Gamma) see [8], 4] 2 The most significant work on presentations for mapping class groups is certainly the paper [10] of Hatcher and Thurston. In this paper, the authors introduced a simply connected complex on which the mapping class group M(F g;0 ) acts, and, using this action and following a method due ....

.... is the subgroup of elements ff in A( Gamma) satisfying ffXff Gamma1 = X, where X is the natural generating set of A( Gamma) and that this subgroup is an infinite cyclic group generated by some special element of A( Gamma) called fundamental element, and denoted by Delta( Gamma) see [4] and [8]) The center of A( Gamma) is an infinite cyclic group generated by Delta( Gamma) if Gamma is B l , D l (l even) E 7 , E 8 , F 4 , H 3 , H 4 , and I 2 (p) p even) and by Delta 2 ( Gamma) if Gamma is A l , D l (l odd) E 6 , and I 2 (p) p odd) Explicit expressions of Delta( Gamma) and ....

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P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972), 273--302.

....G. The projection p induces a short exact sequence between fundamental groups (br) f1g P B G f1g : The following statement is conjectured to be true for all complex reflection groups, with the diagrams listed in our tables. It is well known for Coxeter groups (see for example [BrSa] or [De]) Its first assertion had been noticed by Orlik and Solomon ( OrSo] for the case of Shephard groups (i.e. groups whose braid diagram see below is a Coxeter diagram) It is now proved for all the infinite series but G(e; e; r) for e 2; r 2, and for all the exceptional groups but G 24 , ....

P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972), 273--302.

....B W op 1 ; where the map B W op is defined by oe 7 oe. The spaces M and M=W are conjectured to be K( 1) spaces. The following result is due to Fox and Neuwirth [FoNe] for the type An , to Brieskorn [Br2] for Coxeter groups of type different from H 3 ; H 4 ; E 6 ; E 7 ; E 8 , to Deligne [De1] for general Coxeter groups. The case of the infinite series of complex reflection groups G(de; e; r) has been solved by Nakamura [Na] For the non real Shephard groups (nonreal groups with Coxeter braid diagrams) this has been proven by Orlik and Solomon [OrSo3] Note that the rank 2 case is ....

....length on the distinguished set of generators fsg of the monoid B . About the center of B. 2.19. Notation. We denote by fi the path [0; 1] M defined by fi : t 7 x 0 exp(2it=jZ(W )j) The following result is a consequence of Corollary 2.25. Notice that it generalizes a result of Deligne [De1], 4.21) see also [BrSa] from which it follows that if W is a Coxeter group, then ( 2N . It was noticed experimentally in [BrMi] 4.8) 2.20. Corollary. We have (fi) N N ) jZ(W )j and ( N N : From now on, we assume that W acts irreducibly on V . Note that, since W is ....

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P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972), 273-- 302.

....de chemins dans M, d origine x 0 et d extr emit e s(x 0 ) qui contient la ligne bris ee de sommets successifs x 0 , x 0 ix 0 , s(x 0 ) ix 0 , s(x 0 ) L image s de fl s dans M=W est un lacet de point base p(x 0 ) On pose S : fs j (s 2 S)g. L assertion suivante est due a Deligne (cf. [De1], 4.4) 1.1. Le groupe B est engendr e par S, et les relations stst Delta Delta Delta z m(s;t) termes = tsts Delta Delta Delta z m(s;t) termes en constituent une pr esentation. La fl eche B W de la suite exacte ( est d efinie par l application s 7 s. On d esigne ....

....positives des el ements de S. On d esigne par : 0; 1] M le lacet d origine x 0 d efini par ( e 2i x 0 : Comme nous l a fait remarquer R. Rouquier, il est facile de v erifier que d efinit un el ement central dans B (encore d esign e par ) On peut en fait d emontrer (cf. par exemple [De1] ou [BrSa] le th eor eme suivant. 1.2. Th eor eme. Supposons l action de W sur V irr eductible. 1) L el ement est un g en erateur du centre de P, et la suite exacte ( induit la suite exacte f1g Z(P) Z(B) Z(W ) f1g : 2) On a 2 B , et la longueur de sur le syst eme de ....

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P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972), 273--302.

....m rs = 1 where m rs = m sr 3 for r; s 2 R and r 6= s. Then, BW , the braid group corresponding to W , is given by BW = oe r ; r 2 R; oe r oe s Delta Delta Delta z m rs = oe s oe r Delta Delta Delta z m sr : These groups have been studied extensively (see [Br] [D]) They are called Artin groups by some authors. The classical braid group with n strands, B n , is the braid group corresponding to the Coxeter group whose Dynkin diagram is of the type A n Gamma1 . We will study the braid group corresponding to the Coxeter group with the type B n Dynkin diagram ....

....the braid group of type B n . Proof. For any finite Coxeter group W of rank l, let A be the set of complexified reflecting hyperplanes of W in C l . Write MW = C l n [ H2A H for the corresponding hyperplane complement on which W acts freely. Then, we have BW = 1 (MW =W ) See [Br] or [D]. Let W be the Coxeter group of type B n generated by r 0 , r 1 , r n Gamma1 . The action of W on C n is generated by r 0 : z 1 ; z 2 ; z n ) 7 ( Gammaz 1 ; z 2 ; z n ) and permutations of the coordinates given by r i : z i ; z i 1 ; z i 1 ....

]P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972), 273--302.

....associated Artin group, denoted A G , has a finite presentation with generators corresponding to the vertices of G, and relations aba Delta Delta Delta z n letters = bab Delta Delta Delta z n letters where fa; bg is an edge of G labelled n. References include [1] 10] [14] and [25] Given any Artin group A G there is an associated Coxeter group C G which is the quotient of A G formed by adding the relations v 2 = 1 for each generator v. An Artin group is of finite type if its associated Coxeter group is finite. Let b G be the simplicial complex formed by ....

P. Deligne, Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972) 273-302.

....the complex hyperplane complement is homotopy equivalent to Sal H . One way to see this is to construct a cover of the hyperplane complement in V Omega C by open convex sets indexed by the elements of SH so that the nerve of the covering is the order complex of SH . See [Sal] and [CD2] In [De] Deligne proved that if H is a simplicial arrangement, then the complex hyperplane complement is aspherical. In other words, if ZH is simple, then Sal H is aspherical. For example, if dimV = 1, then ZH is the interval [ Gamma1; 1] and Sal H is the boundary of a digon (which is homeomorphic to a ....

P. Deligne. Les immeubles des groupes de tresses g'en'eralis'es, Invent. Math. 17 (1972), 273-302.

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P. Deligne, Les immeubles des groupes de tresses generalises, Invent. Math. 17 (1972), 273-302.

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P. Deligne, Les immeubles des groupes de tresses g en eralis es, Invent. Math. 17 (1972) 273--302.

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Deligne, P.: Les immeubles des groupes de tresses g'en'eralis'es. Inventiones math. 17, 273-302(1972).

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Deligne, P.: Les immeubles des groupes de tresses g'en'eralis'es. Inventiones math. 17, 273-302(1972).

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