2004/05/07 by Julia Hartmann, Hartmann, Julia, Anne V. Shepler +1
Mathematics · Physics and Astronomy · #13A50 #20C #52C35 #Advanced Differential Equations and Dynamical Systems #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.AC #math.RT #msc:13A50 #msc:20C #msc:52C35
paper · pdf · doi:10.48550/arxiv.math/0405135
11 pages
arxiv created 2004/05/07 · openalex publication_date 2004/05/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Crawley-Boevey. As an intermediate result, we show that every finite group which fixes a hyperplane pointwise has a polynomial ring of invariants.