Equivariant multiplicities of Coxeter arrangements and invariant bases
Let \({\mathcal A}\) be an irreducible Coxeter arrangement and \(W\) be its Coxeter group. For each \(H \in {\mathcal A}\), choose \(\alpha_H \in V^*\) with \(H =\text{ker}(\alpha_H) \in S\). A map \({\mathcal A} \to Z\) is called a multiplicity of \({\mathcal A}\). The multi-derivation module of \({\mathcal A}\) is the \(S\)-module \(D({\mathcal A}, {\mathbf m}) =\{\theta\mid \theta(\alpha_H) \in \alpha_H^{{\mathbf m}(H)} S_{(\alpha_H)} \text{\;for\;all\;} H \in {\mathcal A}\}\). A multiarrangement \(({\mathcal A}, {\mathbf m})\) is free if \(D({\mathcal A}, {\mathbf m}) = \bigoplus S \cdot \theta_i\) with homogeneous basis \(\theta_i\), where \(\deg (\theta_i) \) are called the exponents. A multiplicity \({\mathbf m}:{\mathcal A} \to Z\) is equivariant when \({\mathbf m}(H) = {\mathbf m}(\omega H)\) for any \(H\in {\mathcal A}\) and any \(\omega \in W\).NEWLINENEWLINEThis paper's main result generalizes the main theorem of \textit{H. Terao} [Invent. Math. 148, No. 3, 659--674 (2002; Zbl 1032.52013)].NEWLINENEWLINEMain Result: For any irreducible Coxeter arrangement \({\mathcal A}\) and any equivariant multiplicity, \textbf{m}, the multiarrangement \(({\mathcal A}, {\mathbf m})\) is free.NEWLINENEWLINEThe authors prove their result by giving an explicit construction of a basis for \(D({\mathcal A}, {\mathbf m})\) when \textbf{m} is equivariant.
scientific article; zbMATH DE number 6060972
scientific article