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Mathematics of Computation

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Involutions and characters of upper triangular matrix groups

Authors: I. M. Isaacs and Dikran B. Karagueuzian
Journal: Math. Comp. 74 (2005), 2027-2033
MSC (2000): Primary 20C15; Secondary 20D15
Published electronically: March 24, 2005
MathSciNet review: 2164110
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Abstract: We study the realizability over $\mathbb {R}$ of representations of the group $U(n)$ of upper-triangular $n \times n$ matrices over $\mathbb {F}_2$. We prove that all the representations of $U(n)$ are realizable over $\mathbb R$ if $n \leq 12$, but that if $n \geq 13$, $U(n)$ has representations not realizable over $\mathbb R$. This theorem is a variation on a result that can be obtained by combining work of J. Arregi and A. Vera-López and of the authors, but the proof of the theorem in this paper is much more natural.

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Additional Information

I. M. Isaacs
Affiliation: Mathematics Department, University of Wisconsin at Madison, Madison, Wisconsin 53706

Dikran B. Karagueuzian
Affiliation: Mathematics Department, Binghamton University, Binghamton, New York, 13902-6000

Keywords: Character theory, finite groups, p-groups
Received by editor(s): August 6, 2002
Received by editor(s) in revised form: February 24, 2004
Published electronically: March 24, 2005
Additional Notes: The research of the second author was partially supported by an N.S.F. Postdoctoral Fellowship, the MPIM-Bonn, and the CRM-Barcelona.
Article copyright: © Copyright 2005 American Mathematical Society