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

Higgs and Top quark coupled with a conformal gauge sector

We propose a dynamical scenario beyond the standard model, in which the radiative correction to the Higgs mass parameter is suppressed due to a large anomalous dimension induced through a conformal invariant coupling with an extra gauge sector. Then the anomalous dimension also suppresses the Yukawa couplings of the Higgs field. However, the large top Yukawa coupling can be generated effectively through mixing among top quarks and the fermions of the conformal gauge sector. This scenario is found to predict a fairly heavy Higgs mass of about 500 GeV. We present an explicit model and show consistency with the Electro-Weak precision measurements of the S and T parameters as well as the Z boson decay width.Comment: 12 pages, 3 Postscript figures, uses RevTex.sty; corrected typos and reference

Two Dimensional Black Hole Evapolation in the Light-Cone Gauge

Quantization of the pure $1+1$ dimensional dilaton gravity is examined in the light-cone gauge. It is found that the total action including ghosts generates a $c=0$ free conformal field theory without modification of the classical action, which is required in the conformal gauge. We also study semiclassical equations of the dilaton gravity coupled to $N$ scalar fields. It is shown that the black hole singularity is not removed even for $N<24$ in the light-cone gauge. This indicates that the semiclassical analysis breaks down for small $N$.Comment: 10 pages, KANAZAWA-92-1

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

Exchange and spin Jahn-Teller distortions for a triangular cluster of spin-1/2

We study the effects of magnetoelastic coupling on the degenerate ground state of the spin-1/2 antiferromagnetic Heisenberg model for the regular triangular spin cluster. Static displacement of spins spontaneously lifts the degeneracy of the ground state through the distance dependence of exchange coupling, i.e., a spin Jahn-Teller mechanism takes place. On the other hand, dynamical displacement does not lift the degeneracy, though the cluster distorts spontaneously. The energy decrease obtained by dynamical theory is twice as large as that obtained by static theory because of quantum fluctuation.Comment: 4 pages, 1 figure. Accepted by JPSJ. Clarified some setences. Corrected typo

An Institutional Analysis of Environmental Pollution Disputes in Taiwan: Cases of 'Self-Relief'

During the late 1980s and early 1990s in Taiwan, people's protests against environmental pollution often took the form of "self-relief," meaning that they attempted to fight polluters using their own resources, without relying on legal or administrative procedures. Why did such an extreme form of disputes become so widespread? What institutional changes did these movements bring about? These questions are analyzed using the analytical framework of "law and economics." Our research shows that "self-relief" functioned to a certain extent as a means of realizing quick compensation for victims, and for reflecting the opinions of local people concerning development projects; in addition, it served to promote the formulation of law and administrative systems. However, as it was based on direct negotiations between the parties concerned, the outcome of each dispute only reflected the transient balance of forces, and the experience gained in negotiations was not accumulated as a social norm.Environmental problems, Pollution, Democratization, Taiwan

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