Old recurrence formulae for growth series of Coxeter groups

Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariants

For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about the factorization of the Alexander-Conway polynomial permit to connect Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. One uses reconstractions of A'Campo links and Coxeter links. There is formulated a conjecture connect…

On Normal Subgroups of Coxeter Groups Generated by Standard Parabolic Subgroups

We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots.

Coxeter群的扭曲对合的长度生成函数

Length-generating functions for twisted involutions of Coxeter groups

本文证明了有限秩Coxeter系统中扭曲对合的长度生成函数为有理函数，推导了类似抛物子群生长级数的递推关系，并推广到封闭子集，为Lusztig的幂级数恒等式提供了新证明。

Data about hyperbolic Coxeter systems

We collect several data about Coxeter systems (cf. [Bou07, Hum90]), with particular emphasis on the hyperbolic ones. For each ($\preceq$-minimal) hyperbolic Coxeter system (W,S) the Poincaré series \[p_{(W,S)}(t)=\sum_{w\in W} t^{\ell(w)}\] and the growth rate \[ ω(W,S)=\limsup_n \sqrt[n]{a_n}\] are explicitly computed using Magma (cf. [BCP97]). These computations were performed in connection to the proof of [Ter, Thm. B]. Since the Poincaré series represents a rational function, one may recover the sequence $(a_k)_{k\geq 0}$ through a linear recurrence relation on the coefficients, provided t…

Coxeter group actions on the complement of hyperplanes and special involutions

We consider both standard and twisted action of a (real) Coxeter group G on the complement M_G to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in G and give explicit formulae which describe both actions on the total cohomology H(M_G,C) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups I_2 (2k+1) and that the standard action has no a…

Représentations de réflexion de groupes de Coxeter Deuxième partie: outils pour des exemples

This part is made of three sections. In the first section we study the family of polynomials whose roots are 4cos2 kπ, (n \geqslant 3,1 \leqslant k < \frac{n}{2}). We obtain n2 in this manner a family of orthogonal polynomials. This will permit us to study in details all the examples which follow. In the second section, we give technical formulae in order noto repeat calculations. In the third section,we give applications, first when the field K is a sub-field of R (presentations of W(H3) and W(H4)) then in the complex case (study of the complex reflection group G(p, p, n), G24 and G27).