4,294 research outputs found
Salvetti complex, spectral sequences and cohomology of Artin groups
The aim of this short survey is to give a quick introduction to the Salvetti
complex as a tool for the study of the cohomology of Artin groups. In
particular we show how a spectral sequence induced by a filtration on the
complex provides a very natural and useful method to study recursively the
cohomology of Artin groups, simplifying many computations. In the last section
some examples of applications are presented.Comment: 22 pages, abstract in French and Englis
Cohomology of Artin groups of type tilde{A}_n, B_n and applications
We consider two natural embeddings between Artin groups: the group
G_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group G_{B_n} of type
B_n; G_{B_n} in turn embeds into the classical braid group Br_{n+1}:=G_{A_n} of
type A_n. The cohomologies of these groups are related, by standard results, in
a precise way. By using techniques developed in previous papers, we give
precise formulas (sketching the proofs) for the cohomology of G_{B_n} with
coefficients over the module Q[q^{+-1},t^{+-1}], where the action is
(-q)-multiplication for the standard generators associated to the first n-1
nodes of the Dynkin diagram, while is (-t)-multiplication for the generator
associated to the last node.
As a corollary we obtain the rational cohomology for G_{tilde{A}_n} as well
as the cohomology of Br_{n+1} with coefficients in the (n+1)-dimensional
representation obtained by Tong, Yang and Ma.
We stress the topological significance, recalling some constructions of
explicit finite CW-complexes for orbit spaces of Artin groups. In case of
groups of infinite type, we indicate the (few) variations to be done with
respect to the finite type case. For affine groups, some of these orbit spaces
are known to be K(pi,1) spaces (in particular, for type tilde{A}_n).
We point out that the above cohomology of G_{B_n} gives (as a module over the
monodromy operator) the rational cohomology of the fibre (analog to a Milnor
fibre) of the natural fibration of K(G_{B_n},1) onto the 2-torus.Comment: This is the version published by Geometry & Topology Monographs on 22
February 200
Positivity results for the Hecke algebras of non-crystallographic finite Coxeter groups
This paper is a report on a computer check of some positivity properties of
the Hecke algebra in type H4, including the non-negativity of the coefficients
of the structure constants in the Kazhdan-Lusztig basis. This answers a
long-standing question of Lusztig's. The same algorithm, carried out by hand,
also allows us to deal with the case of dihedral Coxeter groups.Comment: septembre 2005; 13 pages; cet article s'accompagne d'un logicie
The low-dimensional homology of finite-rank Coxeter groups
We give formulas for the second and third integral homology of an arbitrary
finitely generated Coxeter group, solely in terms of the corresponding Coxeter
diagram. The first of these calculations refines a theorem of Howlett, while
the second is entirely new and is the first explicit formula for the third
homology of an arbitrary Coxeter group.Comment: 59 pages, 2 figures, 1 table. Final version, to appear in Algebraic
and Geometric Topolog
The rings of n-dimensional polytopes
Points of an orbit of a finite Coxeter group G, generated by n reflections
starting from a single seed point, are considered as vertices of a polytope
(G-polytope) centered at the origin of a real n-dimensional Euclidean space. A
general efficient method is recalled for the geometric description of G-
polytopes, their faces of all dimensions and their adjacencies. Products and
symmetrized powers of G-polytopes are introduced and their decomposition into
the sums of G-polytopes is described. Several invariants of G-polytopes are
found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers
and congruence classes of the polytopes. The definitions apply to
crystallographic and non-crystallographic Coxeter groups. Examples and
applications are shown.Comment: 24 page
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