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Symplectic group lattices


Authors: Rudolf Scharlau and Pham Huu Tiep
Journal: Trans. Amer. Math. Soc. 351 (1999), 2101-2139
MSC (1991): Primary 20C10, 20C15, 20C20, 11E12, 11H31, 94B05
DOI: https://doi.org/10.1090/S0002-9947-99-02469-1
Published electronically: January 26, 1999
MathSciNet review: 1653379
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Abstract: Let $p$ be an odd prime. It is known that the symplectic group $Sp_{2n}(p)$ has two (algebraically conjugate) irreducible representations of degree $(p^{n}+1)/2$ realized over $\mathbb{Q}(\sqrt{{\epsilon}p})$, where ${\epsilon}= (-1)^{(p-1)/2}$. We study the integral lattices related to these representations for the case $p^{n} \equiv 1 \bmod 4 $. (The case $p^{n} \equiv 3 \bmod 4 $ has been considered in a previous paper.) We show that the class of invariant lattices contains either unimodular or $p$-modular lattices. These lattices are explicitly constructed and classified. Gram matrices of the lattices are given, using a discrete analogue of Maslov index.


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  • [BaV] R. Bacher and B. Venkov, Lattices and association schemes: A unimodular example without roots in dimension $28$, Annales Inst. Fourier 45 $(1995)$, $1163 - 1176$. MR 96j:11093
  • [BRW] B. Bolt, T. G. Room and G. E. Wall, On the Clifford collineation, transform and similarity groups. I, J. Austral. Math. Soc. 2 $(1961 - 62)$, $60 - 79$. MR 23:A3171
  • [BuS] P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus (with an Appendix by J. H. Conway and N. J. A. Sloane), Invent. Math. 117 $(1994)$, $27 - 56$. MR 95i:22018
  • [Coh] H. Cohn, ``A Classical Invitation to Algebraic Numbers and Class Fields'', Springer-Verlag, $1978$. MR 80c:12001
  • [Atlas] J.H. Conway et al., ``An ATLAS of Finite Groups'', Oxford University Press, $1985$. MR 88g:20025
  • [CoS 1] J. H. Conway and N. J. A. Sloane, On lattices equivalent to their duals, J. Number Theory 48 $(1994)$, $373 - 382$. MR 95j:11063
  • [CoS 2] J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory 36 (1990), $1319 - 1333$. MR 91m:94024
  • [CoT] A. M. Cohen and Pham Huu Tiep, Splitting fields for Jordan subgroups, in: `Finite Reductive Groups, Related Structures and Representations', ed. by M. Cabanes, Progress in Math., vol. 141, 1996, Birkhäuser, pp. $165 - 183$. MR 98a:20044
  • [DiM] F. Digne and J. Michel, ``Representations of Finite Groups of Lie Type'', London Math. Soc. Student Texts 21, Cambridge Univ. Press, $1991$, 159 pp. MR 92g:20063
  • [Dum] N. Dummigan, Symplectic group lattices as Mordell-Weil sublattices, J. Number Theory 61 $(1996)$, $365 - 387$. MR 98h:11088
  • [Elk] N. Elkies, A characterization of the $\mathbf{Z}^{n}$ lattice, Math. Res. Letters 2 (1995), $321 - 326$. MR 96h:11064
  • [Gow] R. Gow, Even unimodular lattices associated with the Weil representation of the finite symplectic group, J. Algebra 122 $(1989)$, $510 - 519$. MR 90f:20009
  • [GoW] R. Gow and W. Willems, On the geometry of some simple modules in characteristic $2$, J. Algebra 195 (1997), 634-649.
  • [Gro] B. H. Gross, Group representations and lattices, J. Amer. Math. Soc. 3 $(1990)$, $929 - 960$. MR 92a:11077
  • [Isa] I. M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math. 95 $(1973)$, $594 - 635$. MR 48:11270
  • [KoT] A. I. Kostrikin and Pham Huu Tiep, ``Orthogonal Decompositions and Integral Lattices'', Walter de Gruyter, Berlin et al, 1994, 535 pp. MR 96f:17001
  • [LiV] G. Lion and M. Vergne, ``The Weil Representation, Maslov Index and Theta Series'', Progress in Math. v. 6, Birkhäuser, Boston, Mass., 1980, 337 pp. MR 81j:58075
  • [NPl] G. Nebe and W. Plesken, ``Finite Rational Matrix Groups'', Mem. Amer. Math. Soc. 116 $(1995)$, no. $556$, 144 pp. MR 95k:20081
  • [Que] H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory 54 $(1995)$, $190 - 202$. MR 96i:11072
  • [SchHem] R. Scharlau, B. Hemkemeier, Classification of integral lattices with large class number, Math. Comp. 67 (1998), 737-749. MR 98g:11042
  • [SchT] R. Scharlau and Pham Huu Tiep, Symplectic groups, symplectic spreads, codes and unimodular lattices, J. Algebra 194 (1997), 113-156. MR 98g:20075
  • [Sei] G. M. Seitz, Some representations of classical groups, J. London Math. Soc. $(2)$ 10 $(1975)$, $115 - 120$. MR 51:5789
  • [Tiep 1] Pham Huu Tiep, Weil representations of finite symplectic groups, and Gow lattices, Mat. Sb. 182 $(1991)$, no. $8$, $1161 - 1183$; English transl. in Math. USSR-Sb. 73 $(1992)$, no. $2$, $535 - 555$.
  • [Tiep 2] Pham Huu Tiep, Globally irreducible representations of finite groups and integral lattices, Geometriae Dedicata 64 $(1997)$, $85 - 123$. MR 98e:20011
  • [Tiep 3] Pham Huu Tiep, Globally irreducible representations of the finite symplectic group $Sp_{4}(q)$, Commun. Algebra 22 $(1994)$, $6439 - 6457$.
  • [Tiep 4] Pham Huu Tiep, Weil representations as globally irreducible representations, Math. Nachr. 184 $(1997)$, $313 - 327$. MR 98b:20010
  • [TZa 1] Pham Huu Tiep and A. E. Zalesskii, Minimal characters of the finite classical groups, Commun. Algebra 24 (1996), 2093 - 2167. MR 97f:20018
  • [TZa 2] 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. CMP 97:13
  • [Ward 1] H. N. Ward, Representations of symplectic groups, J. Algebra 20 $(1972)$, $182 - 195$. MR 44:4116
  • [Ward 2] H. N. Ward, Quadratic residue codes in their prime, J. Algebra 150 $(1992)$, $87 - 100$. MR 94a:11194
  • [Zal] A. E. Zalesskii, Decompositions numbers modulo $p$ of certain representations of the groups $SL_{n}(p^{k})$, $SU_{n}(p^{k})$, $Sp_{2n}(p^{k})$, in: `Topics in Algebra', Banach Center Publications, vol. 26, pt. $2$, Warsaw $1990$, pp. $491 - 500$. MR 93e:20022

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Additional Information

Rudolf Scharlau
Affiliation: Department of Mathematics, University of Dortmund, 44221 Dortmund, Germany
Email: rudolf.scharlau@mathematik.uni-dortmund.de

Pham Huu Tiep
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Address at time of publication: Department of Mathematics, University of Florida, Gainseville, Florida 32611
Email: tiep@math.ufl.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02469-1
Keywords: Integral lattice, unimodular lattice, $p$-modular lattice, finite symplectic group, Weil representation, Maslov index, linear code, self-dual code
Received by editor(s): December 10, 1996
Published electronically: January 26, 1999
Additional Notes: Part of this work was done during the second author’s stay at the Department of Mathematics, Israel Institute of Technology. He is grateful to Professor D. Chillag and his colleagues at the Technion for stimulating conversations and their generous hospitality. His work was also supported in part by the DFG
Article copyright: © Copyright 1999 American Mathematical Society

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