305 research outputs found
Multiderivations of Coxeter arrangements
Let be an -dimensional Euclidean space. Let be a
finite irreducible orthogonal reflection group. Let be the
corresponding Coxeter arrangement. Let be the algebra of polynomial
functions on For choose such that For each nonnegative integer , 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 -module of rank by K.
Saito (1975) for and L. Solomon-H. Terao (1998) for . The main
result of this paper is that this is the case for all . Moreover we
explicitly construct a basis for \sD^{(m)} (\cal A). Their degrees are all
equal to (when is even) or are equal to (when is odd). Here are the
exponents of and is the Coxeter number. The construction
heavily uses the primitive derivation 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 .) Some new results concerning the
primitive derivation are obtained in the course of proof of the main
result.Comment: dedication and a footnote (thanking a grant) adde
Freeness of hyperplane arrangements and related topics
This is the expanded notes of the lecture by the author in "Arrangements in
Pyrenees", June 2012. We are discussing relations of freeness and splitting
problems of vector bundles, several techniques proving freeness of hyperplane
arrangements, K. Saito's theory of primitive derivations for Coxeter
arrangements, their application to combinatorial problems and related
conjectures.Comment: 28 page
The freeness of Shi-Catalan arrangements
Let 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 by
the positive root (and its integer multiples) perpendicular to . We say that
a deformation is -equivariant if the number of parallel hyperplanes of each
hyperplane H\in \A depends only on the -orbit of . We prove that the
conings of the -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
- …