Remarks concerning linear characters of reflection groups
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Abstract:
Let $G$ be a finite group generated by unitary reflections in a Hermitian space $V$, and let $\zeta$ be a root of unity. Let $E$ be a subspace of $V$, maximal with respect to the property of being a $\zeta$-eigenspace of an element of $G$, and let $C$ be the parabolic subgroup of elements fixing $E$ pointwise. If $\chi$ is any linear character of $G$, we give a condition for the restriction of $\chi$ to $C$ to be trivial in terms of the invariant theory of $G$, and give a formula for the polynomial $\sum _{x\in G}\chi (x)T^{d(x,\zeta )}$, where $d(x,\zeta )$ is the dimension of the $\zeta$-eigenspace of $x$. Applications include criteria for regularity, and new connections between the invariant theory and the structure of $G$.References
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Additional Information
- G. I. Lehrer
- Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
- MR Author ID: 112045
- ORCID: 0000-0002-7918-7650
- Email: gusl@maths.usyd.edu.au
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 3163-3169
- MSC (2000): Primary 20F55; Secondary 14G05, 20G40, 51F15
- DOI: https://doi.org/10.1090/S0002-9939-05-07869-X
- MathSciNet review: 2160177