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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 | References | Similar Articles | Additional Information

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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  • 1. Bosma, W., Cannon, J., and Playoust, C. The Magma algebra system. I. The user language. Computational algebra and number theory (London, 1993). J. Symbolic Comput. 24 (1997), no. 3-4, pp. 235-265. MR 98f:68006
  • 2. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A. Atlas of finite groups. (with computational assistance from J. G. Thackray.) Oxford University Press, Eynsham, 1985. MR 88g:20025
  • 3. Curtis, C. and Reiner, I. Methods of representation theory. Vol. I. With applications to finite groups and orders. Reprint of the 1981 original. John Wiley & Sons, Inc., New York, 1990. MR 90k:20001
  • 4. Isaacs, I. M., Character Theory of Finite Groups, Dover (1994).
  • 5. Isaacs, I. M. and Karagueuzian, D. Conjugacy in groups of upper triangular matrices. J. Algebra 202 (1998), no. 2, pp. 704-711. MR 99b:20011
  • 6. Isaacs, I. M. and Karagueuzian, D. Erratum: ``Conjugacy in groups of upper triangular matrices.'' J. Algebra 208 (1998), no. 2, p. 722. MR 99g:20021
  • 7. Slattery, M. Computing character degrees in $p$-groups. J. Symbolic Comput. 2 (1986), no. 1, pp. 51-58. MR 87e:20019
  • 8. Vera López, A. and Arregi, J., Conjugacy classes in Sylow $p$-subgroups of ${\rm GL}(n,q)$. J. Algebra 152 (1992), no. 1, pp. 1-19. MR 94b:20048
  • 9. Vera López, A. and Arregi, J., private communication.

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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

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