The double Coxeter arrangement
Consider a finite collection \(\mathcal A\) of linear hyperplanes in \({\mathbb R}^\ell\). Let \(\alpha_H: {\mathbb R}^\ell \to {\mathbb R}\) satisfy \(H=\ker \alpha_H\), for \(H\in {\mathcal A}\). Let \(S={\mathbb R}[x_1,\ldots,x_\ell]\). A derivation \(\theta\) of \(S\) is tangent along \(\mathcal A\) if \(\theta(\alpha_H)\) is a multiple of \(\alpha_H\) for every \(H\in {\mathcal A}\). The set \(D({\mathcal A})\) of all such derivations is an \(S\)-module. The arrangement \(\mathcal A\) is free if \(D({\mathcal A})\) is a free module, necessarily of rank \(\ell\). In this case a free basis for \(D({\mathcal A})\) may be chosen to be homogeneous, that is, the coefficients of \({{\partial}\over{\partial x_i}}\), \(1\leq i\leq \ell\) are homogeneous polynomials of the same degree. These degrees are called the exponents of the free arrangement \(\mathcal A\). Let \(W\subset GL(n,{\mathbb R})\) be a finite group generated by reflections. The collection \({\mathcal A}_W\) of mirror hyperplanes for all of the reflections in \(W\) is the Coxeter arrangement associated with \(W\). It is well-known that \({\mathcal A}_W\) is a free arrangement, with exponents coinciding with the exponents \(\deg(f_i)-1\), where \(\{f_1,\ldots, f_\ell\}\) is a set of basic invariant polynomials of \(W\). In this paper, the authors study the submodule \(E({\mathcal A}_W) \subset D({\mathcal A}_W)\) comprised of those derivations \(\theta\) for which \(\theta(\alpha_H)\) is a multiple of \(\alpha_H^2\) for each \(H\in {\mathcal A}\). It is shown that \(E({\mathcal A}_W)\) is a free \(S\)-module of rank \(\ell\), for any Coxeter arrangement \({\mathcal A}_W\). The coefficient polynomials occurring in a homogeneous free basis of \(E({\mathcal A}_W)\) all have degree equal to the Coxeter number \(h\) of \(W\). The motivation for the study of \(E({\mathcal A}_W)\) comes from a conjecture concerning Shi arrangements and an addition-deletion theorem involving arrangements with multiplicity (``multi-arrangements''). Given a finite root system \(\Delta\), the associated Shi arrangement \(\widetilde{\mathcal A}_W\) consists of the affine hyperplanes defined by \(\alpha(x)=0\) and \(\alpha(x)=1\), where \(\alpha\) ranges over a set \(\Delta_+\) of positive roots in \(\Delta\). It is conjectured that the associated linear arrangement \(c\widetilde{\mathcal A}_W\) (the ``cone'' of \(\widetilde{\mathcal A}_W\)) is free, with exponents \(1,h,\ldots, h\). This was proven in case \(W\) is of type \(A_\ell\) by \textit{C. Athanasiadis} [Eur. J. Comb. 19, No. 1, 7-18 (1998; Zbl 0898.52008)]. If the conjecture holds, then by a result of \textit{G. M. Ziegler} [Contemp. Math. 90, 345-359 (1989; Zbl 0678.51010)], the restriction of \(c\widetilde{\mathcal A}_W\) to the hyperplane at infinity must be free as a multi-arrangement, with exponents \(h,\ldots, h\). This restricted multi-arrangement consists of the hyperplanes \(\alpha(x)=0,\) for \(\alpha\in \Delta_+\) (which form the Coxeter arrangement \({\mathcal A}_W\)), each with multiplicity two. Freeness of this multi-arrangement is equivalent, by definition, to the freeness of \(E({\mathcal A}_W)\) as an \(S\)-module, the result established in this paper. The proof is accomplished by producing an explicit free basis for \(E({\mathcal A}_W)\). These basic derivations are defined using the Saito derivation \(D=\sum h_j{{\partial}\over{\partial x_j}}\in \text{Der}(K)\), where \(K\) is the quotient field of \(S\). The \(K\)-derivation \(D\) is uniquely determined up to constant multiple by the property that \(D\) annihilates all but the highest degree basic invariant \(f_\ell\) of \(W\). The crucial step is then to show that the Jacobian matrix \(J=[{{\partial h_j}\over{\partial x_i}}]\) is invertible over \(K\), and that \(J^{-1}\) has polynomial entries. The coefficients of the elements of a free basis for \(E({\mathcal A}_W)\) are then the entries of \(J^{-1}\), in the case of an orthonormal coordinate system on \({\mathbb R}^\ell\). The authors actually prove a more general result, for arbitrary linear coordinates on \({\mathbb R}^\ell\). These calculations are carried out explicitly for Weyl groups \(W\) of rank two or of type \(B_\ell\). Finally, the authors use their technical results to describe the differential 1-forms which are anti-invariant under \(W\).
scientific article; zbMATH DE number 1180679
scientific article