The Shi arrangements and the Bernoulli polynomials
The braid arrangement is the Coxeter arrangement of type \(A_\ell\). The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the braid arrangement and their parallel translations, see [\textit{J.-Y. Shi}, Lect. Notes Math. 1179. Berlin etc.: Springer (1986; Zbl 0582.20030)]. By coning the Shi arrangement, the authors have a central arrangement, denoted \(S_\ell\); its defining polynomial is given by NEWLINENEWLINE\[NEWLINEQ(S_\ell) = z \prod_{1 \leq p <w\leq \ell +1} (x_p -x_q) \prod_{1 \leq p <q \leq \ell+1}(x_p -x_q -z) = 0.NEWLINE\]NEWLINENEWLINE Using the Bernoulli polynomials, the authors construct, for the first time, an explicit basis for the derivation module of the cone over the Shi arrangement. This basis is given in the paper by the following definition, where \(D(S_\ell)\) denotes the derivation module over \(S_\ell\). NEWLINENEWLINENEWLINE Definition 3.1. Let \(\partial_i\) (\(1 \leq i \leq \ell +1\)) and \(\partial_z\) denote \(\partial/\partial x_i\) and \(\partial/\partial z\), respectively. Define homogeneous derivations NEWLINENEWLINE\[NEWLINE\eta_1:=\sum_{i=1}^{\ell+1} \partial_i \in D(S_\ell), \;\;\eta_2:=z\partial_z + \sum_{i=1}^{\ell+1}x_i\partial_i \in D(S_\ell),NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\phi_j:=(x_j-x_{j+1}-z)\sum_{i=1}^{\ell+1}\sum_{0 \leq k_1\leq j-1, 0 \leq k_2 \leq \ell-j} (-1)^{k_1 + k_2} \sigma_{j-1-k_1}^{(j,1)} \sigma_{\ell-j-k_2}^{(j,2)} \bar B_{k_1, k_2}(x_i, z) \partial_iNEWLINE\]NEWLINE NEWLINEfor \(1\leq j \leq \ell\). NEWLINENEWLINENEWLINE In the above, \(\bar B_{k_1, k_2}(x_i, z) \) is determined via Bernoulli polynomials. Moreover, \(\sigma_k^{(j,s)}\) denotes the elementary symmetric function in the variables in \(I_s^{(j)}\) of degree \(k\) (\(s = 1,2\) and \(k\) is a nonnegative integer), where \(I_1^{(j)}:=\{x_1, \dots, x_{j-1}\},\;I_2^{(j)}:=\{x_{j+2}, \dots, x_{\ell+1}\}\).NEWLINENEWLINEThe authors demonstrate that this is a basis for \(A_3\) and conclude their paper with some open problems.
scientific article; zbMATH DE number 6048040