Basic derivations for subarrangements of Coxeter arrangements (Q687005)

Let \({\mathfrak A}\), \({\mathfrak B}\), \({\mathfrak D}\) denote the families of arrangements associated with the root systems of types \(A\), \(B\), \(D\). Then \({\mathfrak A}_{n-1}\subset{\mathfrak D}_ n\subset {\mathfrak B}_ n\). The authors determine the freeness of certain families of arrangements interpolating between these reflection arrangements. Explicit computations of basic derivations and exponents are carried out using tools from the theory of symmetric functions and some determinantal formulas of ancient vintage. The methods are also used to treat subarrangements of the unitary reflection arrangements associated with monomial groups. This paper foreshadows the work of \textit{P. Edelman} and \textit{V. Reiner} [Free hyperplane arrangements between \(A_{n-1}\) and \(B_ n\), Math. Z. 215, No. 3, 347-366 (1994; Zbl 0793.05122)], in which the free arrangements \(\mathfrak A\) satisfying \({\mathfrak A}_{n-1}\subseteq {\mathfrak A}\subseteq {\mathfrak B}_ n\) are characterized using combinatorial techniques.