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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Jacobians of reflection groups


Authors: Julia Hartmann and Anne V. Shepler
Journal: Trans. Amer. Math. Soc. 360 (2008), 123-133
MSC (2000): Primary 13A50, 20F55; Secondary 52C35
Published electronically: August 6, 2007
MathSciNet review: 2341996
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Abstract | References | Similar Articles | Additional Information

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 groups with a polynomial ring of invariants over arbitrary fields using a ramification formula of Benson and Crawley-Boevey.


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Additional Information

Julia Hartmann
Affiliation: IWR, Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany
Email: Julia.Hartmann@iwr.uni-heidelberg.de

Anne V. Shepler
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203
Email: ashepler@unt.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04379-6
PII: S 0002-9947(07)04379-6
Keywords: Invariant theory, Jacobian determinant, modular, Coxeter group, reflection group, hyperplane arrangement, pointwise stabilizer
Received by editor(s): April 14, 2005
Received by editor(s) in revised form: August 18, 2005
Published electronically: August 6, 2007
Additional Notes: The work of the second author was partially supported by National Security Agency grant MDA904-03-1-0005
Article copyright: © Copyright 2007 American Mathematical Society