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ON THE NUMBER OF REAL CLASSES IN THE FINITE PROJECTIVE LINEAR AND UNITARY GROUPS

Part of: Linear algebraic groups and related topics Combinatorics Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  31 January 2019

ELENA AMPARO  and
C. RYAN VINROOT
ELENA AMPARO
Affiliation:
Department of Physics, Broida Hall, University of California, Santa Barbara, CA 93106-9530, USA e-mail: [email protected]
C. RYAN VINROOT*
Affiliation:
Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, USA e-mail: [email protected]
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Abstract

We show that for any n and q, the number of real conjugacy classes in $ \rm{PGL}(\it{n},\mathbb{F}_q) $ is equal to the number of real conjugacy classes of $ \rm{GL}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SL}(\it{n},\mathbb{F}_q) $, refining a result of Lehrer [J. Algebra36(2) (1975), 278–286] and extending the result of Gill and Singh [J. Group Theory14(3) (2011), 461–489] that this holds when n is odd or q is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $ \rm{PGU}(\it{n},\mathbb{F}_q) $, and equal to the number of real conjugacy classes of $ \rm{U}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SU}(\it{n},\mathbb{F}_q) $, refining results of Gow [Linear Algebra Appl.41 (1981), 175–181] and Macdonald [Bull. Austral. Math. Soc.23(1) (1981), 23–48]. We also give a generating function for this common quantity.

MSC classification

Primary: 20G40: Linear algebraic groups over finite fields
Secondary: 20E45: Conjugacy classes 05A15: Exact enumeration problems, generating functions
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 93 - 107
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019 

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References

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