Flat Connections and Quantum Groups

We review the Kohno-Drinfeld theorem as well as a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection D on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes and values in any g-module V. We sketch our proof of this conjecture when g=sl(n) and when g is arbitrary and V is a vector, spin or adjoint representation. We also establish a precise link between the connection D and Cherednik's generalisation of the KZ connection to finite reflection groups.Comment: 20 pages. To appear in the Proceedings of the 2000 Twente Conference on Lie Groups, in a special issue of Acta Applicandae Mathematica

Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements

Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement can be realized as monotone path polytopes. The construction is an extension of that of the permutahedron and yields interesting connections between enumerative combinatorics of hyperplane arrangements and geometry of monotone path polytopes

The Shi arrangements and the Bernoulli polynomials

The braid arrangement is the Coxeter arrangement of the type $A_\ell$. The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the braid arrangement and their parallel translations. In this paper, we give an explicit basis construction for the derivation module of the cone over the Shi arrangement. The essential ingredient of our recipe is the Bernoulli polynomials.Comment: We fixed a typ