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

Abstract

Let $\A$ be an irreducible Coxeter arrangement and $\bfk$ be a multiplicity of $\A$. We study the derivation module $D(\A, \bfk)$. 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 $D(\A, \bfk)$ assuming $\bfk$ is constant on each orbit. Consequently we will determine the exponents of $(\A, \bfk)$ under this assumption. For this purpose we develop a theory of universal derivations and introduce a map to deal with our exceptional cases.