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Cycle indices for finite orthogonal groups of even characteristic


Authors: Jason Fulman, Jan Saxl and Pham Huu Tiep
Journal: Trans. Amer. Math. Soc. 364 (2012), 2539-2566
MSC (2010): Primary 20G40; Secondary 20C33, 05E15
DOI: https://doi.org/10.1090/S0002-9947-2012-05406-7
Published electronically: January 6, 2012
MathSciNet review: 2888219
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Abstract: We develop cycle index generating functions for orthogonal groups in even characteristic and give some enumerative applications. A key step is the determination of the values of the complex linear-Weil characters of the finite symplectic group, and their induction to the general linear group, at unipotent elements. We also define and study several natural probability measures on integer partitions.


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  • 1. Andrews, G. E., The theory of partitions. Reprint of the 1976 original. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998. xvi+255 pp. MR 1634067 (99c:11126)
  • 2. Andrews, G. E., Partitions, $ q$-series and the Lusztig-Macdonald-Wall conjectures, Invent. Math. 41 (1977), 91-102. MR 0446991 (56:5307)
  • 3. Britnell, J. R., Cyclic, separable and semisimple matrices in the special linear groups over a finite field, J. London Math. Soc. 66 (2002), 605-622. MR 1934295 (2003k:11039)
  • 4. Britnell, J. R., Cyclic, separable and semisimple transformations in the special unitary groups over a finite field, J. Group Theory 9 (2006), 547-569. MR 2243246 (2007e:20101)
  • 5. Britnell, J. R., Cyclic, separable and semisimple transformations in the finite conformal groups, J. Group Theory 9 (2006), 571-601. MR 2253954 (2007h:20047)
  • 6. Britnell, J. R., Cycle index methods for finite groups of orthogonal type in odd characteristic, J. Group Theory 9 (2006), 753-773. MR 2272715 (2007i:20075)
  • 7. Cohen, H. and Lenstra, H.W., Jr., Heuristics on class groups, in: Number theory (New York, 1982), 26-36, Lecture Notes in Math., 1052, Springer, Berlin, 1984. MR 750661
  • 8. Fulman, J., Random matrix theory over finite fields, Bull. Amer. Math. Soc. 39 (2002), 51-85. MR 1864086 (2002i:60012)
  • 9. Fulman, J., Cycle indices for the finite classical groups, J. Group Theory 2 (1999), 251-289. MR 1696313 (2001d:20045)
  • 10. Fulman, J., A probabilistic approach toward conjugacy classes in the finite general linear and unitary groups, J. Algebra 212 (1999), 557-590. MR 1676854 (2000c:20072)
  • 11. Fulman, J., A probabilistic approach to conjugacy classes in the finite symplectic and orthogonal groups, J. Algebra 234 (2000), 207-224. MR 1799484 (2002j:20094)
  • 12. Fulman, J. and Guralnick, R., Conjugacy class properties of the extension of $ \textup {GL}(n,q)$ generated by the inverse transpose involution, J. Algebra 275 (2004), 356-396. MR 2047453 (2005f:20085)
  • 13. Fulman, J. and Guralnick, R., Derangements in simple and primitive groups, in: Groups, combinatorics, and geometry (Durham, 2001), 99-121, World Sci. Publ., River Edge, NJ, 2003. MR 1994962 (2004e:20003)
  • 14. Fulman, J. and Guralnick, R., Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, arXiv:0902.2238 (2009).
  • 15. Fulman, J. and Guralnick, R., Derangements in subspace actions of finite classical groups, preprint.
  • 16. Fulman, J., Neumann, P. M., and Praeger, C. E., A generating function approach to the enumeration of matrices in classical groups over finite fields. Mem. Amer. Math. Soc. 176 (2005), no. 830, vi+90 pp. MR 2145026 (2006b:05125)
  • 17. Fulton, W. and Harris, J., Representation theory. A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. xvi+551 pp. MR 1153249 (93a:20069)
  • 18. Goh, W. and Schmutz, E., The expected order of a random permutation, Bull. London Math. Soc. 23 (1991), 34-42. MR 1111532 (93a:11080)
  • 19. Goncharov, V., Du domaine d'analyse combinatoire, Bull. Acad. Sci. URSS Ser. Math 8 (1944), 3-48. MR 0010922 (6:88b)
  • 20. Guralnick, R. M. and Tiep, P. H., Cross characteristic representations of even characteristic symplectic groups, Trans. Amer. Math. Soc. 356 (2004), 4969-5023. MR 2084408 (2005j:20012)
  • 21. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, Fifth edition, Oxford University Press, 1979. MR 568909 (81i:10002)
  • 22. Inglis, N. F. J., The embedding of $ O(2m, 2^k) \leq Sp(2m,2^k)$, Arch. Math. 54 (1990), 327-330. MR 1042124 (91c:11021)
  • 23. Kung, J. P. S., The cycle structure of a linear transformation over a finite field, Linear Algebra Appl. 36 (1981), 141-155. MR 604337 (82d:15012)
  • 24. Lengler, J., The Cohen-Lenstra heuristic for finite abelian groups, Dissertation zur Erlangung des Grades des Doktors der Naturwissenschaften (2009), available at http://www.math.uni-sb.de/ag/gekeler/PERSONEN/Lengler/Dissertation_Lengler.pdf
  • 25. Liebeck, M. W., O'Brien, E., Shalev, A., and Tiep, P. H., The Ore conjecture, J. Europ. Math. Soc. 12 (2010), 939-1008. MR 2654085 (2011e:20016)
  • 26. Liebeck, M. W. and Seitz, G. M., Nilpotent and unipotent classes in classical groups in bad characteristic, preprint.
  • 27. Lusztig, G., A note on counting nilpotent matrices of a fixed rank, Bull. London Math. Soc. 8 (1976), 77-80. MR 0407050 (53:10833)
  • 28. Lusztig, G., Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449-487. MR 2183120 (2006m:20074)
  • 29. Lusztig, G., Unipotent elements in small characteristic. II. Transform. Groups 13 (2008), 773-797. MR 2452615 (2009j:20066)
  • 30. Lusztig, G., Unipotent elements in small characteristic. III. J. Algebra 329 (2011), 163-189. MR 2769321
  • 31. Macdonald, I. G., Symmetric functions and Hall polynomials, Second edition, Clarendon Press, Oxford, 1995. MR 1354144 (96h:05207)
  • 32. Neumann, P. M. and Praeger, C. E., Cyclic matrices and the MEATAXE, in: Groups and computation, III (Columbus, OH, 1999), 291-300, Ohio State Univ. Math. Res. Inst. Publ., 8, de Gruyter, Berlin, 2001. MR 1829488 (2002d:20018)
  • 33. Neumann, P. and Praeger, C., Cyclic matrices in classical groups over finite fields. Special issue in honor of Helmut Wielandt, J. Algebra 234 (2000), 367-418. MR 1800732 (2002c:20079)
  • 34. Pólya, G., Kombinatorische anzahlbestimmungen fuer gruppen, graphen und chemische verbindungen, Acta Math. 68 (1937), 145-254.
  • 35. Pólya, G. and Read, R. C., Combinatorial enumeration of groups, graphs, and chemical compounds. Springer-Verlag, New York, 1987. MR 884155 (89f:05013)
  • 36. Rudvalis, A. and Shinoda, K., An enumeration in finite classical groups, U-Mass Amherst Technical Report, 1988.
  • 37. Saxl, J. and Seitz, G. M., Subgroups of algebraic groups containing regular elements, J. London Math. Soc. 55 (1997), 370-386. MR 1438641 (98m:20057)
  • 38. Schmutz, E., The order of a typical matrix with entries in a finite field, Israel J. Math. 91 (1995), 349-371. MR 1348322 (97e:15011)
  • 39. Shalev, A., A theorem on random matrices and some applications, J. Algebra 199 (1998), 124-141. MR 1489358 (99a:20048)
  • 40. Shepp, L. A. and Lloyd, S. P., Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121 (1966), 340-357. MR 0195117 (33:3320)
  • 41. Spaltenstein, N., Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math. 946, Springer, $ 1982$. MR 672610 (84a:14024)
  • 42. Stong, R., Some asymptotic results on finite vector spaces, Adv. Appl. Math. 9 (1988), 167-199. MR 937520 (89c:05007)
  • 43. Tiep, P. H., Dual pairs of finite classical groups in cross characteristics in Character theory of finite groups, 161-179, Contemp. Math. 524, Amer. Math. Soc., Providence, RI, 2010. MR 2731928 (2012a:20025)
  • 44. Wall, G. E., On the conjugacy classes in the unitary, symplectic, and orthogonal groups, J. Aust. Math. Soc. 3 (1963), 1-63. MR 0150210 (27:212)
  • 45. Wall, G. E., Counting cyclic and separable matrices over a finite field, Bull. Austral. Math. Soc. 60 (1999), 253-284. MR 1711918 (2000k:11137)

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Additional Information

Jason Fulman
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
Email: fulman@usc.edu

Jan Saxl
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom
Email: J.Saxl@dpmms.cam.ac.uk

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721-0089
Email: tiep@math.arizona.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05406-7
Keywords: Random matrix, cycle index, Weil representation, random partition
Received by editor(s): April 15, 2010
Received by editor(s) in revised form: June 21, 2010
Published electronically: January 6, 2012
Additional Notes: The first author was partially supported by NSF grant DMS-0802082 and NSA grant H98230-08-1-0133
The third author was partially supported by NSF grant DMS-0901241.
The authors are grateful to Martin Liebeck for kindly sending them the preprint [26] which plays an important role in the current paper.
Dedicated: Dedicated to Peter M. Neumann on the occasion of his seventieth birthday
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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