Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

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
Posted: January 6, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References

1. George E. Andrews, The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original. MR 1634067 (99c:11126)
2. George E. Andrews, Partitions, 𝑞-series and the Lusztig-Macdonald-Wall conjectures, Invent. Math. 41 (1977), no. 1, 91–102. MR 0446991 (56 #5307)
3. John R. Britnell, Cyclic, separable and semisimple matrices in the special linear groups over a finite field, J. London Math. Soc. (2) 66 (2002), no. 3, 605–622. MR 1934295 (2003k:11039), http://dx.doi.org/10.1112/S0024610702003678
4. John R. Britnell, Cyclic, separable and semisimple transformations in the special unitary groups over a finite field, J. Group Theory 9 (2006), no. 4, 547–569. MR 2243246 (2007e:20101), http://dx.doi.org/10.1515/JGT.2006.037
5. John R. Britnell, Cyclic, separable and semisimple transformations in the finite conformal groups, J. Group Theory 9 (2006), no. 5, 571–601. MR 2253954 (2007h:20047), http://dx.doi.org/10.1515/JGT.2006.038
6. John R. Britnell, Cycle index methods for finite groups of orthogonal type in odd characteristic, J. Group Theory 9 (2006), no. 6, 753–773. MR 2272715 (2007i:20075), http://dx.doi.org/10.1515/JGT.2006.048
7. H. Cohen and H. W. Lenstra Jr., Heuristics on class groups, Number theory (New York, 1982) Lecture Notes in Math., vol. 1052, Springer, Berlin, 1984, pp. 26–36. MR 750661, http://dx.doi.org/10.1007/BFb0071539
8. Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51–85. MR 1864086 (2002i:60012), http://dx.doi.org/10.1090/S0273-0979-01-00920-X
9. Jason Fulman, Cycle indices for the finite classical groups, J. Group Theory 2 (1999), no. 3, 251–289. MR 1696313 (2001d:20045), http://dx.doi.org/10.1515/jgth.1999.017
10. Jason Fulman, A probabilistic approach toward conjugacy classes in the finite general linear and unitary groups, J. Algebra 212 (1999), no. 2, 557–590. MR 1676854 (2000c:20072), http://dx.doi.org/10.1006/jabr.1998.7659
11. Jason Fulman, A probabilistic approach to conjugacy classes in the finite symplectic and orthogonal groups, J. Algebra 234 (2000), no. 1, 207–224. MR 1799484 (2002j:20094), http://dx.doi.org/10.1006/jabr.2000.8455
12. Jason Fulman and Robert Guralnick, Conjugacy class properties of the extension of 𝐺𝐿(𝑛,𝑞) generated by the inverse transpose involution, J. Algebra 275 (2004), no. 1, 356–396. MR 2047453 (2005f:20085), http://dx.doi.org/10.1016/j.jalgebra.2003.07.004
13. Jason Fulman and Robert Guralnick, Derangements in simple and primitive groups, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 99–121. MR 1994962 (2004e:20003), http://dx.doi.org/10.1142/9789812564481_0006
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. Jason Fulman, Peter M. Neumann, and Cheryl E. Praeger, 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. MR 2145026 (2006b:05125)
17. William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249 (93a:20069)
18. William M. Y. Goh and Eric Schmutz, The expected order of a random permutation, Bull. London Math. Soc. 23 (1991), no. 1, 34–42. MR 1111532 (93a:11080), http://dx.doi.org/10.1112/blms/23.1.34
19. V. Gontcharoff, Du domaine de l’analyse combinatoire, Bull. Acad. Sci. URSS Sér. Math. [Izvestia Akad. Nauk SSSR] 8 (1944), 3–48 (Russian, with French summary). MR 0010922 (6,88b)
20. Robert M. Guralnick and Pham Huu Tiep, Cross characteristic representations of even characteristic symplectic groups, Trans. Amer. Math. Soc. 356 (2004), no. 12, 4969–5023 (electronic). MR 2084408 (2005j:20012), http://dx.doi.org/10.1090/S0002-9947-04-03477-4
21. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press Oxford University Press, New York, 1979. MR 568909 (81i:10002)
22. Nicholas F. J. Inglis, The embedding 𝑂(2𝑚,2^{𝑘})≤𝑆𝑝(2𝑚,2^{𝑘}), Arch. Math. (Basel) 54 (1990), no. 4, 327–330. MR 1042124 (91c:11021), http://dx.doi.org/10.1007/BF01189578
23. Joseph P. S. Kung, The cycle structure of a linear transformation over a finite field, Linear Algebra Appl. 36 (1981), 141–155. MR 604337 (82d:15012), http://dx.doi.org/10.1016/0024-3795(81)90227-5
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. Martin W. Liebeck, E. A. O’Brien, Aner Shalev, and Pham Huu Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939–1008. MR 2654085 (2011e:20016), http://dx.doi.org/10.4171/JEMS/220
26. Liebeck, M. W. and Seitz, G. M., Nilpotent and unipotent classes in classical groups in bad characteristic, preprint.
27. G. Lusztig, A note on counting nilpotent matrices of fixed rank, Bull. London Math. Soc. 8 (1976), no. 1, 77–80. MR 0407050 (53 #10833)
28. G. Lusztig, Unipotent elements in small characteristic, Transform. Groups 10 (2005), no. 3-4, 449–487. MR 2183120 (2006m:20074), http://dx.doi.org/10.1007/s00031-005-0405-1
29. G. Lusztig, Unipotent elements in small characteristic. II, Transform. Groups 13 (2008), no. 3-4, 773–797. MR 2452615 (2009j:20066), http://dx.doi.org/10.1007/s00031-008-9021-1
30. G. Lusztig, Unipotent elements in small characteristic, III, J. Algebra 329 (2011), 163–189. MR 2769321 (2012c:20128), http://dx.doi.org/10.1016/j.jalgebra.2009.12.008
31. I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144 (96h:05207)
32. Peter M. Neumann and Cheryl E. Praeger, Cyclic matrices and the MEATAXE, Groups and computation, III (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 291–300. MR 1829488 (2002d:20018)
33. Peter M. Neumann and Cheryl E. Praeger, Cyclic matrices in classical groups over finite fields, J. Algebra 234 (2000), no. 2, 367–418. Special issue in honor of Helmut Wielandt. MR 1800732 (2002c:20079), http://dx.doi.org/10.1006/jabr.2000.8548
34. Pólya, G., Kombinatorische anzahlbestimmungen fuer gruppen, graphen und chemische verbindungen, Acta Math. 68 (1937), 145-254.
35. G. Pólya and R. C. Read, Combinatorial enumeration of groups, graphs, and chemical compounds, Springer-Verlag, New York, 1987. Pólya’s contribution translated from the German by Dorothee Aeppli. MR 884155 (89f:05013)
36. Rudvalis, A. and Shinoda, K., An enumeration in finite classical groups, U-Mass Amherst Technical Report, 1988.
37. Jan Saxl and Gary M. Seitz, Subgroups of algebraic groups containing regular unipotent elements, J. London Math. Soc. (2) 55 (1997), no. 2, 370–386. MR 1438641 (98m:20057), http://dx.doi.org/10.1112/S0024610797004808
38. Eric Schmutz, The order of a typical matrix with entries in a finite field, Israel J. Math. 91 (1995), no. 1-3, 349–371. MR 1348322 (97e:15011), http://dx.doi.org/10.1007/BF02761656
39. Aner Shalev, A theorem on random matrices and some applications, J. Algebra 199 (1998), no. 1, 124–141. MR 1489358 (99a:20048), http://dx.doi.org/10.1006/jabr.1997.7167
40. L. A. Shepp and S. P. Lloyd, Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121 (1966), 340–357. MR 0195117 (33 #3320)
41. Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin, 1982 (French). MR 672610 (84a:14024)
42. Richard Stong, Some asymptotic results on finite vector spaces, Adv. in Appl. Math. 9 (1988), no. 2, 167–199. MR 937520 (89c:05007), http://dx.doi.org/10.1016/0196-8858(88)90012-7
43. Pham Huu Tiep, Dual pairs of finite classical groups in cross characteristics, Character theory of finite groups, Contemp. Math., vol. 524, Amer. Math. Soc., Providence, RI, 2010, pp. 161–179. MR 2731928 (2012a:20025)
44. G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austral. Math. Soc. 3 (1963), 1–62. MR 0150210 (27 #212)
45. G. E. Wall, Counting cyclic and separable matrices over a finite field, Bull. Austral. Math. Soc. 60 (1999), no. 2, 253–284. MR 1711918 (2000k:11137), http://dx.doi.org/10.1017/S000497270003639X

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|