Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cross characteristic representations of even characteristic symplectic groups


Authors: Robert M. Guralnick and Pham Huu Tiep
Journal: Trans. Amer. Math. Soc. 356 (2004), 4969-5023
MSC (2000): Primary 20C33, 20G05, 20C20, 20G40
DOI: https://doi.org/10.1090/S0002-9947-04-03477-4
Published electronically: April 27, 2004
MathSciNet review: 2084408
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We classify the small irreducible representations of $Sp_{2n}(q)$ with $q$ even in odd characteristic. This improves even the known results for complex representations. The smallest representation for this group is much larger than in the case when $q$ is odd. This makes the problem much more difficult.


References [Enhancements On Off] (What's this?)

  • [Atlas] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, `An ATLAS of Finite Groups', Clarendon Press, Oxford, 1985. MR 88g:20025
  • [BM] M. Broué and J. Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. reine angew. Math. 395 (1989), 56 - 67. MR 90b:20037
  • [BrK] J. Brundan and A. S. Kleshchev, Lower bounds for the degrees of irreducible Brauer characters of finite general linear groups, J. Algebra 223 (2000), 615 - 629. MR 2001f:20014
  • [C] R. Carter, `Finite Groups of Lie type: Conjugacy Classes and Complex Characters', Wiley, Chichester, 1985. MR 87d:20060
  • [DM] F. Digne and J. Michel, `Representations of Finite Groups of Lie Type', London Mathematical Society Student Texts 21, Cambridge University Press, 1991. MR 92g:20063
  • [DT] N. Dummigan and Pham Huu Tiep, Lower bounds for the minima of certain symplectic and unitary group lattices, Amer. J. Math. 121 (1999), 889 - 918. MR 2001a:11112
  • [E] H. Enomoto, The characters of the finite symplectic groups $Sp(4,q)$, $q = 2^{f}$, Osaka J. Math. 9 (1972), 75 - 94. MR 46:1893
  • [FLT] P. Fleischmann, W. Lempken, and Pham Huu Tiep, The $p$-intersection subgroups in quasi-simple and almost simple finite groups, J. Algebra 207 (1998), 1 - 42. MR 99j:20021
  • [GH] M. Geck and G. Hiss, Basic sets of Brauer characters of finite groups of Lie type, J. reine angew. Math. 418 (1991), 173 - 188. MR 92e:20006
  • [Ge] P. Gérardin, Weil representations associated to finite fields, J. Algebra 46 (1977), 54 - 101. MR 57:470
  • [GMST] R. M. Guralnick, K. Magaard, J. Saxl, and Pham Huu Tiep, Cross characteristic representations of odd characteristic symplectic groups and unitary groups, J. Algebra 257 (2002), 291 - 347.
  • [GPPS] R. M. Guralnick, T. Penttila, C. Praeger, and J. Saxl, Linear groups with orders having certain large prime divisors, Proc. London Math. Soc. 78 (1999), 167 - 214. MR 99m:20113
  • [GT] R. M. Guralnick and Pham Huu Tiep, Low-dimensional representations of special linear groups in cross characteristic, Proc. London Math. Soc. 78 (1999), 116 - 138. MR 2000a:20016
  • [HM] G. Hiss and G. Malle, Low dimensional representations of special unitary groups, J. Algebra 236 (2001), 745 - 767. MR 2001m:20019
  • [Ho] C. Hoffman, Cross characteristic projective representations for some classical groups, J. Algebra 229 (2000), 666 - 677. MR 2001f:20029
  • [Hw1] R. Howe, On the characters of Weil's representations, Trans. Amer. Math. Soc. 177 (1973), 287 - 298. MR 47:5180
  • [Hw2] R. Howe, $\theta$-series and invariant theory, Proc. Symp. Pure Math. 33 (1979), part 1, pp. 257 - 285. MR 81f:22034
  • [Is] I. M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math. 95 (1973), 594 - 635. MR 48:11270
  • [JLPW] C. Jansen, K. Lux, R. A. Parker, and R. A. Wilson, `An ATLAS of Brauer Characters', Oxford University Press, Oxford, 1995. MR 96k:20016
  • [KLi] P. B. Kleidman and M. W. Liebeck, `The Subgroup Structure of the Finite Classical Groups', London Math. Soc. Lecture Note Ser. no. 129, Cambridge University Press, 1990. MR 91g:20001
  • [Ku] S. Kudla, Seesaw dual reductive pairs, in: `Automorphic Forms of Several Variables', Taniguchi Symposium, Katata, 1983, Birkhäuser, Boston, 1983, pp. 244 - 268. MR 86b:22032
  • [LaS] V. Landazuri and G. M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418 - 443. MR 50:13299
  • [LST] J. M. Lataille, P. Sin and Pham Huu Tiep, The modulo 2 structure of rank 3 permutation modules for odd characteristic orthogonal groups, J. Algebra 268 (2003), 463 - 483.
  • [Li] M. W. Liebeck, Permutation modules for rank 3 symplectic and orthogonal groups, J. Algebra 92 (1985), 9 - 15. MR 86d:20057
  • [L] G. Lusztig, `Characters of Reductive Groups over a Finite Field', Annals of Math. Studies 107, Princeton Univ. Press, Princeton, 1984. MR 86j:20038
  • [Lu] F. Lübeck, Charaktertafeln für die Gruppen $CSp_{6}(q)$ mit ungeradem $q$ und $Sp_{6}(q)$ mit geradem $q$, Preprint 93-61, IWR Heidelberg, 1993.
  • [MT1] K. Magaard and Pham Huu Tiep, Irreducible tensor products of representations of finite quasi-simple groups of Lie type, in: `Modular Representation Theory of Finite Groups', M. J. Collins, B. J. Parshall, L. L. Scott, eds., Walter de Gruyter, Berlin et al, 2001, pp. 239 - 262. MR 2002m:20024
  • [MT2] K. Magaard and Pham Huu Tiep, The classes ${\mathcal {C}}_{6}$ and ${\mathcal {C}}_{7}$ of maximal subgroups of finite classical groups, (in preparation).
  • [MMT] K. Magaard, G. Malle, and Pham Huu Tiep, Irreducibility of tensor squares, symmetric squares, and alternating squares, Pacific J. Math. 202 (2002), 379 - 427. MR 2002m:20025
  • [S] G. M. Seitz, Some representations of classical groups, J. London Math. Soc. 10 (1975), 115 - 120. MR 51:5789
  • [SS] J. Saxl and G. M. Seitz, Subgroups of algebraic groups contaning regular unipotent elements, J. London Math. Soc. 55 (1997), 370 - 386. MR 98m:20057
  • [SZ] G. M. Seitz and A. E. Zalesskii, On the minimal degrees of projective representations of the finite Chevalley groups, II, J. Algebra 158 (1993), 233 - 243. MR 94h:20017
  • [ST] P. Sin and Pham Huu Tiep, Rank 3 permutation modules of finite classical groups, (submitted).
  • [T] Pham Huu Tiep, Dual pairs and low-dimensional representations of finite classical groups, (in preparation).
  • [TZ1] Pham Huu Tiep and A. E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), 2093 - 2167. MR 97f:20018
  • [TZ2] Pham Huu Tiep and A. E. Zalesskii, Some characterizations of the Weil representations of the symplectic and unitary groups, J. Algebra 192 (1997), 130 - 165. MR 99d:20074
  • [Wh1] D. White, Decomposition numbers of $Sp_{4}(2^{a})$ in odd characteristics, J. Algebra 177 (1995), 264 - 276. MR 96k:20023
  • [Wh2] D. White, Decomposition numbers of unipotent blocks of $Sp_{6}(2^{a})$ in odd characteristics, J. Algebra 227 (2000), 172 - 194. MR 2001d:20012
  • [Zs] K. Zsigmondy, Zur Theorie der Potenzreste, Monath. Math. Phys. 3 (1892), 265 - 284.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20C33, 20G05, 20C20, 20G40

Retrieve articles in all journals with MSC (2000): 20C33, 20G05, 20C20, 20G40


Additional Information

Robert M. Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: guralnic@math.usc.edu

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: tiep@math.ufl.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03477-4
Keywords: Finite symplectic group, Weil representation, cross characteristic representation, low dimensional representation
Received by editor(s): June 5, 2002
Received by editor(s) in revised form: July 29, 2003
Published electronically: April 27, 2004
Additional Notes: Part of this paper was written while the authors were participating in the Symposium “Groups, Geometries, and Combinatorics”, London Mathematical Society, July 16–26, 2001, Durham, England. It is a pleasure to thank the organizers A. A. Ivanov, M. W. Liebeck, and J. Saxl for their generous hospitality and support. The authors are also thankful to the referee for helpful comments on the paper.
The authors gratefully acknowledge the support of the NSF (grants DMS-9970305, DMS-0140578 and DMS-0070647) and the NSA
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society