Parabolic conjugacy in general linear groups
Simon M. Goodwin1
and Gerhard Röhrle2
1University of Birmingham School of Mathematics Birmingham B15 2TT UK
2Ruhr-Universität Bochum Fakultät für Mathematik Universitätsstrasse 150 44780 Bochum Germany
2Ruhr-Universität Bochum Fakultät für Mathematik Universitätsstrasse 150 44780 Bochum Germany
DOI: 10.1007/s10801-007-0073-4
Abstract
Let q be a power of a prime and n a positive integer. Let P( q) be a parabolic subgroup of the finite general linear group GL n ( q). We show that the number of P( q)-conjugacy classes in GL n ( q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of Alperin in (Commun. Algebra 34(3): 889-891, 2006)
Pages: 99–111
Keywords: keywords general linear group; parabolic subgroups; conjugacy classes
Full Text: PDF
References
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2. Digne, F., & Michel, J. (1991). Representations of finite groups of Lie type. In London mathematical society student texts (Vol. 21). Cambridge: Cambridge University Press.
3. The GAP group. (1997). GAP-groups, algorithms, and programming-version 3, release 4, patchlevel
4. Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen.
4. Goodwin, S. M. (2007). Counting conjugacy classes in Sylow p-subgroups of Chevalley groups. Journal of Pure and Applied Algebra, 210(1), 201-218.
5. Goodwin, S. M., & Röhrle, G. (2007, to appear). Rational points on generalized flag varieties and unipotent conjugacy in finite groups of Lie type. Transactions of the American Mathematical Society.
6. Green, J. A. (1955). The characters of the finite general linear groups. Transactions of the American Mathematical Society, 80, 402-447.
7. Higman, G. (1960). Enumerating p-groups, I: inequalities. Proceedings of the London Mathematical Society, 10(3), 24-30.
8. Higman, G. (1960). Enumerating p-groups, II: problems whose solution is PORC. Proceedings of the London Mathematical Society, 10(3), 566-582.
9. Kac, V. G. (1983). Root systems, representations of quivers and invariant theory. In Lecture notes in mathematics: Vol.
996. Invariant theory (pp. 74-108). Montecatini,
1982. Berlin: Springer.
10. Macdonald, I. G. (1995). Symmetric functions and Hall polynomials (2nd ed.). New York: Oxford University Press.
11. Reineke, M. (2006). Counting rational points of quiver moduli. International Mathematics Research Notices. Art. ID 70456.
12. Robinson, G. R. (1998). Counting conjugacy classes of unitriangular groups associated to finitedimensional algebras. Journal of Group Theory, 1(3), 271-274.
13. Rosenlicht, M. (1963). A remark on quotient spaces. Anais da Academia Brasileira de Ciências, 35, 487-489.
14. Thompson, J. k(Un(Fq )). Preprint. http://www.math.ufl.edu/fac/thompson.html.
15. Vera-López, A., & Arregi, J. M. (2003). Conjugacy classes in unitriangular matrices. Linear Algebra and its Applications, 370, 85-124.