The development version of the CHEVIE package of GAP3

I describe the current state of the development version of the CHEVIE package, which deals with Coxeter groups, reductive algebraic groups, complex reflection groups, Hecke algebras, braid monoids, etc... Examples are given, showing the code to check some results of Lusztig.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1003.492

"Case-free" derivation for Weyl groups of the number of reflection factorisations of a Coxeter element

Chapuy and Stump have given a nice generating series for the number of factorisations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations.Comment: 5 page

Hurwitz action on tuples of Euclidean reflections

We show that if a tuple of Euclidean reflections has a finite orbit under the Hurwitz action of the Artin braid group, then the group generated by these reflections is finite. Humphries has published a similar statement but his proof is irremediably flawed. At the same time as correcting his proof, our proof is much simpler that Dubrovin and Mazocco's proof for triples of reflections.Comment: redige le 1-9-200

Sine-Gordon Theory for the Equation of State of Classical Hard-Core Coulomb systems. III Loopwise Expansion

We present an exact field theoretical representation of an ionic solution made of charged hard spheres. The action of the field theory is obtained by performing a Hubbard-Stratonovich transform of the configurational Boltzmann factor. It is shown that the Stillinger-Lovett sum rules are satisfied if and only if all the field correlation functions are short range functions. The mean field, Gaussian and two-loops approximations of the theory are derived and discussed. The mean field approximation for the free energy constitutes a rigorous lower bound for the exact free energy, while the mean field pressure is an upper bound. The one-loop order approximation is shown to be identical with the random phase approximation of the theory of liquids. Finally, at the two-loop order and in the pecular case of the restricted primitive model, one recovers results obtained in the framework of the mode expansion theory.Comment: 35 pages, 3 figure

Random Walks on Hyperspheres of Arbitrary Dimensions

We consider random walks on the surface of the sphere $S_{n-1}$ ($n \geq 2$) of the $n$-dimensional Euclidean space $E_n$, in short a hypersphere. By solving the diffusion equation in $S_{n-1}$ we show that the usual law $\varpropto t$ valid in $E_{n-1}$ should be replaced in $S_{n-1}$ by the generic law $\varpropto \exp(-t/\tau)$, where $\theta$ denotes the angular displacement of the walker. More generally one has $\varpropto \exp(-t/ \tau(L,n))$ where $C^{n/2-1}_{L}$ a Gegenbauer polynomial. Conjectures concerning random walks on a fractal inscribed in $S_{n-1}$ are given tentatively.Comment: 10 page

Review of Bombing the City: Civilian Accounts of the Air War in Britain and Japan, 1939-1945 by Aaron William Moore

Review of Bombing the City: Civilian Accounts of the Air War in Britain and Japan, 1939-1945 by Aaron William Moore

Equivariant quantization of spin systems

We investigate the geometric and conformally equivariant quantizations of the supercotangent bundle of a pseudo-Riemannian manifold $(M,g)$, which is a model for the phase space of a classical spin particle. This is a short review of our previous works.Comment: 7 pages. From a talk given at the Workshop on Geometric Methods in Physics XXI