Jacobians of Reflection Groups
Abstract
Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for arbitrary fields using a ramification formula of Benson and CrawleyBoevey. As an intermediate result, we show that every finite group which fixes a hyperplane pointwise has a polynomial ring of invariants.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2004
 arXiv:
 arXiv:math/0405135
 Bibcode:
 2004math......5135H
 Keywords:

 Representation Theory;
 Commutative Algebra;
 13A50;
 20C;
 52C35
 EPrint:
 11 pages