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

DOI:
https://doi.org/10.1090/S0025-5718-05-01705-9

Published electronically:
March 24, 2005

MathSciNet review:
2164110

Full-text PDF Free Access

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.

- John Cannon and Derek Holt (eds.),
*Computational algebra and number theory*, Elsevier Ltd, Oxford, 1997. J. Symbolic Comput. 24 (1997), no. 3-4. MR**1484477** - J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
*$\Bbb {ATLAS}$ of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219** - Charles W. Curtis and Irving Reiner,
*Methods of representation theory. Vol. I*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. With applications to finite groups and orders; Reprint of the 1981 original; A Wiley-Interscience Publication. MR**1038525** - Isaacs, I. M.,
*Character Theory of Finite Groups*, Dover (1994). - I. M. Isaacs and Dikran Karagueuzian,
*Conjugacy in groups of upper triangular matrices*, J. Algebra**202**(1998), no. 2, 704–711. MR**1617655**, DOI https://doi.org/10.1006/jabr.1997.7311 - I. M. Isaacs and Dikran Karagueuzian,
*Erratum: “Conjugacy in groups of upper triangular matrices” [J. Algebra 202 (1998), no. 2, 704–711; MR1617655 (99b:20011)]*, J. Algebra**208**(1998), no. 2, 722. MR**1655475**, DOI https://doi.org/10.1006/jabr.1998.7430 - Michael C. Slattery,
*Computing character degrees in $p$-groups*, J. Symbolic Comput.**2**(1986), no. 1, 51–58. MR**839136**, DOI https://doi.org/10.1016/S0747-7171%2886%2980012-8 - Antonio Vera López and Jesus Maria Arregi,
*Conjugacy classes in Sylow $p$-subgroups of ${\rm GL}(n,q)$*, J. Algebra**152**(1992), no. 1, 1–19. MR**1190401**, DOI https://doi.org/10.1016/0021-8693%2892%2990085-Z - 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

Email:
isaacs@math.wisc.edu

**Dikran B. Karagueuzian**

Affiliation:
Mathematics Department, Binghamton University, Binghamton, New York, 13902-6000

Email:
dikran@math.binghamton.edu

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