Equivariant multiplicities of Coxeter arrangements and invariant bases

TL;DR

This paper proves that for irreducible Coxeter arrangements, equivariant multiplicities lead to free multi-derivation modules, providing explicit bases and extending previous results, with implications for invariant theory.

Contribution

It establishes the freeness of multi-derivation modules for equivariant multiplicities and constructs explicit bases, generalizing prior theorems.

Findings

Multi-derivation modules are free for equivariant multiplicities.

Explicit bases for these modules are constructed.

Invariant parts of the modules are also free over invariant subrings.

Abstract

Let $\A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $\A$. A multiplicity $\bfm : \A\rightarrow \Z$ is said to be equivariant when $\bfm$ is constant on each $W$-orbit of $\A$. In this article, we prove that the multi-derivation module $D(\A, \bfm)$ is a free module whenever $\bfm$ is equivariant by explicitly constructing a basis, which generalizes the main theorem of \cite{T02}. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the $W$-invariant part $D(\A, \bfm)^{W}$ for any multiplicity $\bfm$ is a free module over the $W$-invariant subring.