Cycle indices for finite orthogonal groups of even characteristic
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- by Jason Fulman, Jan Saxl and Pham Huu Tiep PDF
- Trans. Amer. Math. Soc. 364 (2012), 2539-2566 Request permission
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
- George E. Andrews, The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original. MR 1634067
- George E. Andrews, Partitions, $q$-series and the Lusztig-Macdonald-Wall conjectures, Invent. Math. 41 (1977), no. 1, 91–102. MR 446991, DOI 10.1007/BF01390165
- 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, DOI 10.1112/S0024610702003678
- 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, DOI 10.1515/JGT.2006.037
- John R. Britnell, Cyclic, separable and semisimple transformations in the finite conformal groups, J. Group Theory 9 (2006), no. 5, 571–601. MR 2253954, DOI 10.1515/JGT.2006.038
- 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, DOI 10.1515/JGT.2006.048
- 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, DOI 10.1007/BFb0071539
- Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51–85. MR 1864086, DOI 10.1090/S0273-0979-01-00920-X
- Jason Fulman, Cycle indices for the finite classical groups, J. Group Theory 2 (1999), no. 3, 251–289. MR 1696313, DOI 10.1515/jgth.1999.017
- 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, DOI 10.1006/jabr.1998.7659
- 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, DOI 10.1006/jabr.2000.8455
- Jason Fulman and Robert Guralnick, Conjugacy class properties of the extension of $\textrm {GL}(n,q)$ generated by the inverse transpose involution, J. Algebra 275 (2004), no. 1, 356–396. MR 2047453, DOI 10.1016/j.jalgebra.2003.07.004
- 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, DOI 10.1142/9789812564481_{0}006
- 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).
- Fulman, J. and Guralnick, R., Derangements in subspace actions of finite classical groups, preprint.
- 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, DOI 10.1090/memo/0830
- 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, DOI 10.1007/978-1-4612-0979-9
- 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, DOI 10.1112/blms/23.1.34
- 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
- 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. MR 2084408, DOI 10.1090/S0002-9947-04-03477-4
- 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
- Nicholas F. J. Inglis, The embedding $\textrm {O}(2m,2^k)\leq \textrm {Sp}(2m,2^k)$, Arch. Math. (Basel) 54 (1990), no. 4, 327–330. MR 1042124, DOI 10.1007/BF01189578
- Joseph P. S. Kung, The cycle structure of a linear transformation over a finite field, Linear Algebra Appl. 36 (1981), 141–155. MR 604337, DOI 10.1016/0024-3795(81)90227-5
- 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
- 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, DOI 10.4171/JEMS/220
- Liebeck, M. W. and Seitz, G. M., Nilpotent and unipotent classes in classical groups in bad characteristic, preprint.
- G. Lusztig, A note on counting nilpotent matrices of fixed rank, Bull. London Math. Soc. 8 (1976), no. 1, 77–80. MR 407050, DOI 10.1112/blms/8.1.77
- G. Lusztig, Unipotent elements in small characteristic, Transform. Groups 10 (2005), no. 3-4, 449–487. MR 2183120, DOI 10.1007/s00031-005-0405-1
- G. Lusztig, Unipotent elements in small characteristic. II, Transform. Groups 13 (2008), no. 3-4, 773–797. MR 2452615, DOI 10.1007/s00031-008-9021-1
- G. Lusztig, Unipotent elements in small characteristic, III, J. Algebra 329 (2011), 163–189. MR 2769321, DOI 10.1016/j.jalgebra.2009.12.008
- 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
- 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
- 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, DOI 10.1006/jabr.2000.8548
- Pólya, G., Kombinatorische anzahlbestimmungen fuer gruppen, graphen und chemische verbindungen, Acta Math. 68 (1937), 145-254.
- 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, DOI 10.1007/978-1-4612-4664-0
- Rudvalis, A. and Shinoda, K., An enumeration in finite classical groups, U-Mass Amherst Technical Report, 1988.
- 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, DOI 10.1112/S0024610797004808
- 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, DOI 10.1007/BF02761656
- Aner Shalev, A theorem on random matrices and some applications, J. Algebra 199 (1998), no. 1, 124–141. MR 1489358, DOI 10.1006/jabr.1997.7167
- L. A. Shepp and S. P. Lloyd, Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121 (1966), 340–357. MR 195117, DOI 10.1090/S0002-9947-1966-0195117-8
- Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610
- Richard Stong, Some asymptotic results on finite vector spaces, Adv. in Appl. Math. 9 (1988), no. 2, 167–199. MR 937520, DOI 10.1016/0196-8858(88)90012-7
- 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, DOI 10.1090/conm/524/10355
- G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austral. Math. Soc. 3 (1963), 1–62. MR 0150210
- G. E. Wall, Counting cyclic and separable matrices over a finite field, Bull. Austral. Math. Soc. 60 (1999), no. 2, 253–284. MR 1711918, DOI 10.1017/S000497270003639X
Additional Information
- Jason Fulman
- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
- MR Author ID: 332245
- 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
- MR Author ID: 230310
- Email: tiep@math.arizona.edu
- 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. - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2888219
Dedicated: Dedicated to Peter M. Neumann on the occasion of his seventieth birthday