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
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 .
 [Gu]
E.
A. Gutkin, Matrices that are connected with groups generated by
reflections, Funkcional. Anal. i Priložen. 7
(1973), no. 2, 81–82 (Russian). MR 0314956
(47 #3505)
 [Le]
G.
I. Lehrer, Rational tori, semisimple orbits and the topology of
hyperplane complements, Comment. Math. Helv. 67
(1992), no. 2, 226–251. MR 1161283
(93e:20065), http://dx.doi.org/10.1007/BF02566498
 [LM]
Gustav
I. Lehrer and Jean
Michel, Invariant theory and eigenspaces for unitary reflection
groups, C. R. Math. Acad. Sci. Paris 336 (2003),
no. 10, 795–800 (English, with English and French summaries). MR 1990017
(2004d:13005), http://dx.doi.org/10.1016/S1631073X(03)001924
 [LS1]
G.
I. Lehrer and T.
A. Springer, Intersection multiplicities and reflection
subquotients of unitary reflection groups. I, Geometric group theory
down under (Canberra, 1996) de Gruyter, Berlin, 1999,
pp. 181–193. MR 1714845
(2000i:20064)
 [LS2]
G.
I. Lehrer and T.
A. Springer, Reflection subquotients of unitary reflection
groups, Canad. J. Math. 51 (1999), no. 6,
1175–1193. Dedicated to H. S. M. Coxeter on the occasion of his 90th
birthday. MR
1756877 (2001f:20082), http://dx.doi.org/10.4153/CJM19990524
 [OS]
Peter
Orlik and Louis
Solomon, Unitary reflection groups and cohomology, Invent.
Math. 59 (1980), no. 1, 77–94. MR 575083
(81f:32017), http://dx.doi.org/10.1007/BF01390316
 [OT]
Peter
Orlik and Hiroaki
Terao, Arrangements of hyperplanes, Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 300, SpringerVerlag, Berlin, 1992. MR 1217488
(94e:52014)
 [PW]
A.
Pianzola and A.
Weiss, Monstrous 𝐸₁₀’s and a
generalization of a theorem of L. Solomon, C. R. Math. Rep. Acad. Sci.
Canada 11 (1989), no. 5, 189–194. MR 1010926
(90k:22007)
 [Sh]
Anne
V. Shepler, Semiinvariants of finite reflection groups, J.
Algebra 220 (1999), no. 1, 314–326. MR 1714136
(2000g:20069), http://dx.doi.org/10.1006/jabr.1999.7875
 [Sh04]
Anne V. Shepler, ``Generalized exponents and forms'', to appear, J. Alg. Comb..
 [Sp]
T.
A. Springer, Regular elements of finite reflection groups,
Invent. Math. 25 (1974), 159–198. MR 0354894
(50 #7371)
 [St]
Robert
Steinberg, Differential equations invariant under
finite reflection groups, Trans. Amer. Math.
Soc. 112 (1964),
392–400. MR 0167535
(29 #4807), http://dx.doi.org/10.1090/S00029947196401675353
 [Gu]
 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)
 [Le]
 G.I. Lehrer, ``Rational tori, semisimple orbits and the topology of hyperplane complements'', Comment. Math. Helv. 67 (1992), 226251.MR 1161283 (93e:20065)
 [LM]
 G.I. Lehrer and J. Michel, ``Invariant theory and eigenspaces for unitary reflection groups.'' C. R. Acad. Sc. Paris, Ser. I 336 (2003), 795800. MR 1990017 (2004d:13005)
 [LS1]
 G.I. Lehrer and T.A. Springer, ``Intersection multiplicities and reflection subquotients of unitary reflection groups I'', Geometric group theory down under (Canberra, 1996), 181193, de Gruyter, Berlin, 1999.MR 1714845 (2000i:20064)
 [LS2]
 G.I. Lehrer and T.A. Springer, ``Reflection subquotients of unitary reflection groups.'' Canad. J. Math. 51 (1999), 11751193. MR 1756877 (2001f:20082)
 [OS]
 P. Orlik and L. Solomon, ``Unitary reflection groups and cohomology'', Inv. Math. 59 (1980), 7794. MR 0575083 (81f:32017)
 [OT]
 P. Orlik and H. Terao, ``Arrangements of hyperplanes.'' Grundlehren der Mathematischen Wissenschaften, 300, SpringerVerlag, Berlin, 1992.MR 1217488 (94e:52014)
 [PW]
 A. Pianzola and A. Weiss, ``Monstrous 's and a generalization of a theorem of L. Solomon'', C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 189194. MR 1010926 (90k:22007)
 [Sh]
 Anne V. Shepler, ``Semiinvariants of finite reflection groups'', J. Alg. 220, (1999), 314326. MR 1714136 (2000g:20069)
 [Sh04]
 Anne V. Shepler, ``Generalized exponents and forms'', to appear, J. Alg. Comb..
 [Sp]
 T. Springer, `` Regular elements of finite reflection groups'', Invent. Math. 25 (1974), 159198. MR 0354894 (50:7371)
 [St]
 R. Steinberg, ``Differential equations invariant under finite reflection groups'', Trans. Amer. Math. Soc. 112 (1964), 392400.MR 0167535 (29:4807)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
20F55,
14G05,
20G40,
51F15
Retrieve articles in all journals
with MSC (2000):
20F55,
14G05,
20G40,
51F15
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.
