Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • [Ben93] D. J. Benson, Polynomial invariants of finite groups, LMS, vol. 190, Cambridge University Press, Cambridge, 1993. MR 1249931 (94j:13003)
  • [Har01] J. Hartmann, Transvection free groups and invariants of polynomial tensor exterior algebras, Transformation Groups 6 (2001), 157-164. MR 1835670 (2002c:13012)
  • [Kem96] G. Kemper, Calculating invariant rings of finite groups over arbitrary fields, J. Symbolic Computation 21 (1996), 351-366. MR 1400337 (98f:13005)
  • [KM97] G. Kemper and G. Malle, The finite irreducible linear groups with polynomial ring of invariants, Transformation Groups 2 (1997), 57-89. MR 1439246 (98a:13012)
  • [Kuh03] K. Kuhnigk, Poincaredualitätsalgebren, Koinvarianten und Wu-Klassen, Ph.D. thesis, Universität Göttingen, 2003.
  • [Lan02] S. Lang, Algebra, rev. 3rd ed. ed., Addison-Wesley, 2002. MR 0197234 (33:5416)
  • [LS87] P. S. Landweber and R. E. Stong, The depth of rings of invariants over finite fields, Proc. New York Number Theory Seminar 1984 (New York), Lecture Notes in Math., vol. 1240, Springer, 1987. MR 894515 (88k:13004)
  • [NS02] M. Neusel and L. Smith, Invariant theory of finite groups, AMS, 2002. MR 1869812 (2002k:13012)
  • [OT92] P. Orlik and H. Terao, Arrangements of hyperplanes, Springer, New York, 1992. MR 1217488 (94e:52014)
  • [RSW04] V. Reiner, D. Stanton, and P. Webb, Springer's regular elements over arbitrary fields, Math. Proc. Cambridge Philos. Soc. 141 (2006), no. 2, 209-229. MR 2265870 (2007f:13010)
  • [Ser67] J.-P. Serre, Groupes finis d'automorphismes d'anneaux locaux régulieres, Colloq. d'Alg. Éc. Norm. Sup. de Jeunes Filles 8-01-8-11 (1967).
  • [Smi95] L. Smith, Polynomial invariants of finite groups, second printing 1997 ed., A.K. Peters, Ltd., 1995. MR 1328644 (96f:13008)
  • [Ste60] R. Steinberg, Invariants of finite reflection groups, Canad. J. Math. 12 (1960), 616-618. MR 0117285 (22:8066)
  • [Wil83] C. Wilkerson, A primer on the Dickson invariants, Contemp. Math. 10 (1983), 421-434, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982). MR 711066 (85c:55017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13A50, 20F55, 52C35

Retrieve articles in all journals with MSC (2000): 13A50, 20F55, 52C35

Additional Information

Julia Hartmann
Affiliation: IWR, Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany

Anne V. Shepler
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203

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

American Mathematical Society