Multiderivations of Coxeter arrangements

Let $V$ be an $\ell$-dimensional Euclidean space. Let $G \subset O(V)$ be a finite irreducible orthogonal reflection group. Let ${\cal A}$ be the corresponding Coxeter arrangement. Let $S$ be the algebra of polynomial functions on $V.$ For $H \in {\cal A}$ choose $\alpha_H \in V^*$ such that $H = {\rm ker}(\alpha_H).$ For each nonnegative integer $m$, define the derivation module \sD^{(m)}({\cal A}) = \{\theta \in {\rm Der}_S | \theta(\alpha_H) \in S \alpha^m_H\}. The module is known to be a free $S$-module of rank $\ell$ by K. Saito (1975) for $m=1$ and L. Solomon-H. Terao (1998) for $m=2$. The main result of this paper is that this is the case for all $m$. Moreover we explicitly construct a basis for \sD^{(m)} (\cal A). Their degrees are all equal to $mh/2$ (when $m$ is even) or are equal to $((m-1)h/2) + m_i (1 \leq i \leq \ell)$ (when $m$ is odd). Here $m_1 \leq ... \leq m_{\ell}$ are the exponents of $G$ and $h= m_{\ell} + 1$ is the Coxeter number. The construction heavily uses the primitive derivation $D$ which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of $G$.) Some new results concerning the primitive derivation $D$ are obtained in the course of proof of the main result.Comment: dedication and a footnote (thanking a grant) adde

Algebras generated by reciprocals of linear forms

Let $\Delta$ be a finite set of nonzero linear forms in several variables with coefficients in a field $\mathbf K$ of characteristic zero. Consider the $\mathbf K$-algebra $C(\Delta)$ of rational functions generated by $\{1/\alpha \mid \alpha \in \Delta \}$. Then the ring $\partial(V)$ of differential operators with constant coefficients naturally acts on $C(\Delta)$. We study the graded $\partial(V)$-module structure of $C(\Delta)$. We especially find standard systems of minimal generators and a combinatorial formula for the Poincar\'e series of $C(\Delta)$. Our proofs are based on a theorem by Brion-Vergne [brv1] and results by Orlik-Terao [ort2}.Comment: a typo corrected; a footnote adde

The Shi arrangements and the Bernoulli polynomials

The braid arrangement is the Coxeter arrangement of the type $A_\ell$. The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the braid arrangement and their parallel translations. In this paper, we give an explicit basis construction for the derivation module of the cone over the Shi arrangement. The essential ingredient of our recipe is the Bernoulli polynomials.Comment: We fixed a typ

The freeness of Shi-Catalan arrangements

Let $W$ be a finite Weyl group and \A be the corresponding Weyl arrangement. A deformation of \A is an affine arrangement which is obtained by adding to each hyperplane H\in\A several parallel translations of $H$ by the positive root (and its integer multiples) perpendicular to $H$. We say that a deformation is $W$-equivariant if the number of parallel hyperplanes of each hyperplane H\in \A depends only on the $W$-orbit of $H$. We prove that the conings of the $W$-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers. This generalizes Yoshinaga's theorem conjectured by Edelman-Reiner.Comment: 12 page

Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements

In this article we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual-partition formula. Then it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl arrangement such that each filter is a free subarrangement satisfying the dual-partition formula. This generalizes the main result in \cite{ABCHT} which affirmatively settled a conjecture by Sommers and Tymoczko \cite{SomTym}

Simple-root bases for Shi arrangements

In his affirmative answer to the Edelman-Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, all we know about the bases is their existence and degrees. In this article, we introduce two distinguished bases for the modules. More specifically, we will define and study the simple-root basis plus (SRB+) and the simple-root basis minus (SRB-) when a primitive derivation is fixed. They have remarkable properties relevant to the simple roots and those properties characterize the bases