Regular semisimple elements and involutions in finite general linear groups of odd characteristic
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- by Cheryl E. Praeger and Ákos Seress PDF
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Abstract:
Chris Parker and Rob Wilson showed that involution-centraliser methods could be used for solving several problems that appeared to be computationally hard and gave complexity analyses for methods to construct involutions and their centralisers in quasisimple Lie type groups in odd characteristic. Crucial to their analyses are conjugate involution pairs whose products are regular semisimple, possibly in an induced action on a subspace. We consider the fundamental case of conjugate involution pairs, in finite general linear groups $\mathrm {GL}(n,q)$ with $q$ odd, for which the product is regular semisimple on the underlying vector space. Such involutions form essentially a single conjugacy class $\mathcal {C}$. We prove that a constant proportion of pairs from $\mathcal {C}$ have regular semisimple product. Moreover we show that for a fixed parity of $n$, this proportion converges exponentially quickly to a limit, as $n$ approaches $\infty$, the limit being $(1-q^{-1})^2\Phi (q)^3$ for even $n$ and $(1-q^{-1})\Phi (q)^3$ for odd $n$, where $\Phi (q)=\prod _{i=1}^\infty (1-q^{-i})$.References
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Additional Information
- Cheryl E. Praeger
- Affiliation: School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia
- MR Author ID: 141715
- ORCID: 0000-0002-0881-7336
- Email: cheryl.praeger@uwa.edu.au
- Ákos Seress
- Affiliation: School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia – and – Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
- Email: akos@math.ohio-state.edu
- Received by editor(s): March 28, 2011
- Published electronically: January 25, 2012
- Additional Notes: This work was partially supported by ARC Grants FF0776186 and DP1096525 and by the NSF
- Communicated by: Pham Huu Tiep
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3003-3015
- MSC (2010): Primary 20D06, 20F69
- DOI: https://doi.org/10.1090/S0002-9939-2012-11491-1
- MathSciNet review: 2917073