Remarks concerning linear characters of reflection groups
Author:
G. I. Lehrer
Journal:
Proc. Amer. Math. Soc. 133 (2005), 31633169
MSC (2000):
Primary 20F55; Secondary 14G05, 20G40, 51F15
Published electronically:
May 2, 2005
MathSciNet review:
2160177
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Abstract: Let be a finite group generated by unitary reflections in a Hermitian space , and let be a root of unity. Let be a subspace of , maximal with respect to the property of being a eigenspace of an element of , and let be the parabolic subgroup of elements fixing pointwise. If is any linear character of , we give a condition for the restriction of to to be trivial in terms of the invariant theory of , and give a formula for the polynomial , where is the dimension of the eigenspace of . Applications include criteria for regularity, and new connections between the invariant theory and the structure of .
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 E.A. Gutkin, ``Matrices connected with groups generated by reflections'', Funkcional Anal. Appl. 7 (1973), 153154; translated from Funktsional Anal. i Prilozhen 7 (1973), 8182. MR 0314956 (47:3505)
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 G.I. Lehrer, ``Rational tori, semisimple orbits and the topology of hyperplane complements'', Comment. Math. Helv. 67 (1992), 226251.MR 1161283 (93e:20065)
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 P. Orlik and L. Solomon, ``Unitary reflection groups and cohomology'', Inv. Math. 59 (1980), 7794. MR 0575083 (81f:32017)
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 P. Orlik and H. Terao, ``Arrangements of hyperplanes.'' Grundlehren der Mathematischen Wissenschaften, 300, SpringerVerlag, Berlin, 1992.MR 1217488 (94e:52014)
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 Anne V. Shepler, ``Semiinvariants of finite reflection groups'', J. Alg. 220, (1999), 314326. MR 1714136 (2000g:20069)
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 Anne V. Shepler, ``Generalized exponents and forms'', to appear, J. Alg. Comb..
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 [St]
 R. Steinberg, ``Differential equations invariant under finite reflection groups'', Trans. Amer. Math. Soc. 112 (1964), 392400.MR 0167535 (29:4807)
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Additional Information
G. I. Lehrer
Affiliation:
School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
Email:
gusl@maths.usyd.edu.au
DOI:
http://dx.doi.org/10.1090/S000299390507869X
PII:
S 00029939(05)07869X
Received by editor(s):
December 12, 2003
Received by editor(s) in revised form:
June 8, 2004, and June 14, 2004
Published electronically:
May 2, 2005
Communicated by:
John R. Stembridge
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
