Bases for the derivation modules of two-dimensional ~multi-Coxeter arrangements and universal derivations

Abstract

Let ¥cal{A} be an irreducible Coxeter arrangement and k be a multiplicity of ¥cal{A}. We study the derivation module D(¥cal{A}, k). Any two-dimensional irreducible Coxeter arrangement with even number of lines is decomposed into two orbits under the action of the Coxeter group. In this paper, we will explicitly construct a basis for D(¥cal{A}, k) assuming k is constant on each orbit. Consequently we will determine the exponents of (¥cal{A}, k) under this assumption. For this purpose we develop a theory of universal derivations and introduce a map to deal with our exceptional cases.