Symplectic group lattices
Authors:
Rudolf Scharlau and Pham Huu Tiep
Journal:
Trans. Amer. Math. Soc. 351 (1999), 21012139
MSC (1991):
Primary 20C10, 20C15, 20C20, 11E12, 11H31, 94B05
Published electronically:
January 26, 1999
MathSciNet review:
1653379
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be an odd prime. It is known that the symplectic group has two (algebraically conjugate) irreducible representations of degree realized over , where . We study the integral lattices related to these representations for the case . (The case has been considered in a previous paper.) We show that the class of invariant lattices contains either unimodular or modular lattices. These lattices are explicitly constructed and classified. Gram matrices of the lattices are given, using a discrete analogue of Maslov index.
 [BaV]
R.
Bacher and B.
Venkov, Lattices and association schemes: a unimodular example
without roots in dimension 28, Ann. Inst. Fourier (Grenoble)
45 (1995), no. 5, 1163–1176 (English, with
English and French summaries). MR 1370742
(96j:11093)
 [BRW]
Beverley
Bolt, T.
G. Room, and G.
E. Wall, On the Clifford collineation, transform and similarity
groups. I, II., J. Austral. Math. Soc. 2 (1961/1962),
60–79, 80–96. MR 0125874
(23 #A3171)
 [BuS]
P.
Buser and P.
Sarnak, On the period matrix of a Riemann surface of large
genus, Invent. Math. 117 (1994), no. 1,
27–56. With an appendix by J. H. Conway and N. J. A. Sloane. MR 1269424
(95i:22018), http://dx.doi.org/10.1007/BF01232233
 [Coh]
Harvey
Cohn, A classical invitation to algebraic numbers and class
fields, SpringerVerlag, New YorkHeidelberg, 1978. With two
appendices by Olga Taussky: “Artin’s 1932 Göttingen
lectures on class field theory” and “Connections between
algebraic number theory and integral matrices”; Universitext. MR 506156
(80c:12001)
 [Atlas]
J.
H. Conway, R.
T. Curtis, S.
P. Norton, R.
A. Parker, and R.
A. Wilson, Atlas of finite groups, Oxford University Press,
Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups;
With computational assistance from J. G. Thackray. MR 827219
(88g:20025)
 [CoS 1]
J.
H. Conway and N.
J. A. Sloane, On lattices equivalent to their duals, J. Number
Theory 48 (1994), no. 3, 373–382. MR 1293868
(95j:11063), http://dx.doi.org/10.1006/jnth.1994.1073
 [CoS 2]
J.
H. Conway and N.
J. A. Sloane, A new upper bound on the minimal distance of
selfdual codes, IEEE Trans. Inform. Theory 36
(1990), no. 6, 1319–1333. MR 1080819
(91m:94024), http://dx.doi.org/10.1109/18.59931
 [CoT]
Arjeh
M. Cohen and Pham
Huu Tiep, Splitting fields for Jordan subgroups, Finite
reductive groups (Luminy, 1994) Progr. Math., vol. 141,
Birkhäuser Boston, Boston, MA, 1997, pp. 165–183. MR 1429872
(98a:20044), http://dx.doi.org/10.1007/s1010701205133
 [DiM]
François
Digne and Jean
Michel, Representations of finite groups of Lie type, London
Mathematical Society Student Texts, vol. 21, Cambridge University
Press, Cambridge, 1991. MR 1118841
(92g:20063)
 [Dum]
Neil
Dummigan, Symplectic group lattices as MordellWeil
sublattices, J. Number Theory 61 (1996), no. 2,
365–387. MR 1423059
(98h:11088), http://dx.doi.org/10.1006/jnth.1996.0154
 [Elk]
Noam
D. Elkies, A characterization of the 𝑍ⁿ
lattice, Math. Res. Lett. 2 (1995), no. 3,
321–326. MR 1338791
(96h:11064), http://dx.doi.org/10.4310/MRL.1995.v2.n3.a9
 [Gow]
R.
Gow, Even unimodular lattices associated with the Weil
representation of the finite symplectic group, J. Algebra
122 (1989), no. 2, 510–519. MR 999089
(90f:20009), http://dx.doi.org/10.1016/00218693(89)902329
 [GoW]
R. Gow and W. Willems, On the geometry of some simple modules in characteristic , J. Algebra 195 (1997), 634649.
 [Gro]
Benedict
H. Gross, Group representations and
lattices, J. Amer. Math. Soc.
3 (1990), no. 4,
929–960. MR 1071117
(92a:11077), http://dx.doi.org/10.1090/S08940347199010711178
 [Isa]
I.
M. Isaacs, Characters of solvable and symplectic groups, Amer.
J. Math. 95 (1973), 594–635. MR 0332945
(48 #11270)
 [KoT]
Alexei
I. Kostrikin and Phạm
Hũ’u Ti\cfudot{e}p, Orthogonal decompositions and
integral lattices, de Gruyter Expositions in Mathematics,
vol. 15, Walter de Gruyter & Co., Berlin, 1994. MR 1308713
(96f:17001)
 [LiV]
Gérard
Lion and Michèle
Vergne, The Weil representation, Maslov index and theta
series, Progress in Mathematics, vol. 6, Birkhäuser, Boston,
Mass., 1980. MR
573448 (81j:58075)
 [NPl]
G.
Nebe and W.
Plesken, Finite rational matrix groups, Mem. Amer. Math. Soc.
116 (1995), no. 556, viii+144. MR 1265024
(95k:20081), http://dx.doi.org/10.1090/memo/0556
 [Que]
H.G.
Quebbemann, Modular lattices in Euclidean spaces, J. Number
Theory 54 (1995), no. 2, 190–202. MR 1354045
(96i:11072), http://dx.doi.org/10.1006/jnth.1995.1111
 [SchHem]
Rudolf
Scharlau and Boris
Hemkemeier, Classification of integral lattices
with large class number, Math. Comp.
67 (1998), no. 222, 737–749. MR 1458224
(98g:11042), http://dx.doi.org/10.1090/S0025571898009387
 [SchT]
Rudolf
Scharlau and Pham
Huu Tiep, Symplectic groups, symplectic spreads, codes, and
unimodular lattices, J. Algebra 194 (1997),
no. 1, 113–156. MR 1461484
(98g:20075), http://dx.doi.org/10.1006/jabr.1996.6902
 [Sei]
Gary
M. Seitz, Some representations of classical groups, J. London
Math. Soc. (2) 10 (1975), 115–120. MR 0369556
(51 #5789)
 [Tiep 1]
Pham Huu Tiep, Weil representations of finite symplectic groups, and Gow lattices, Mat. Sb. 182 , no. , ; English transl. in Math. USSRSb. 73 , no. , .
 [Tiep 2]
Pham
Huu Tiep, Globally irreducible representations of finite groups and
integral lattices, Geom. Dedicata 64 (1997),
no. 1, 85–123. MR 1432536
(98e:20011), http://dx.doi.org/10.1023/A:1004917606870
 [Tiep 3]
Pham Huu Tiep, Globally irreducible representations of the finite symplectic group , Commun. Algebra 22 , .
 [Tiep 4]
Pham
Huu Tiep, Weil representations as globally irreducible
representations, Math. Nachr. 184 (1997),
313–327. MR 1439180
(98b:20010), http://dx.doi.org/10.1002/mana.19971840114
 [TZa 1]
Pham
Huu Tiep and Alexander
E. Zalesskii, Minimal characters of the finite classical
groups, Comm. Algebra 24 (1996), no. 6,
2093–2167. MR 1386030
(97f:20018), http://dx.doi.org/10.1080/00927879608825690
 [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), 130165. CMP 97:13
 [Ward 1]
Harold
N. Ward, Representations of symplectic groups, J. Algebra
20 (1972), 182–195. MR 0286909
(44 #4116)
 [Ward 2]
Harold
N. Ward, Quadratic residue codes in their prime, J. Algebra
150 (1992), no. 1, 87–100. MR 1174890
(94a:11194), http://dx.doi.org/10.1016/S00218693(05)800511
 [Zal]
A.
Zalesskiĭ, Decomposition numbers modulo 𝑝 of certain
representations of the groups
𝑆𝐿_{𝑛}(𝑝^{𝑘}),
𝑆𝑈_{𝑛}(𝑝^{𝑘}),
𝑆𝑝_{2𝑛}(𝑝^{𝑘}), Topics in
algebra, Part 2 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN,
Warsaw, 1990, pp. 491–500. MR 1171296
(93e:20022)
 [BaV]
 R. Bacher and B. Venkov, Lattices and association schemes: A unimodular example without roots in dimension , Annales Inst. Fourier 45 , . 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 , . 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 , . MR 95i:22018
 [Coh]
 H. Cohn, ``A Classical Invitation to Algebraic Numbers and Class Fields'', SpringerVerlag, . MR 80c:12001
 [Atlas]
 J.H. Conway et al., ``An ATLAS of Finite Groups'', Oxford University Press, . MR 88g:20025
 [CoS 1]
 J. H. Conway and N. J. A. Sloane, On lattices equivalent to their duals, J. Number Theory 48 , . MR 95j:11063
 [CoS 2]
 J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of selfdual codes, IEEE Trans. Inform. Theory 36 (1990), . 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. . 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, , 159 pp. MR 92g:20063
 [Dum]
 N. Dummigan, Symplectic group lattices as MordellWeil sublattices, J. Number Theory 61 , . MR 98h:11088
 [Elk]
 N. Elkies, A characterization of the lattice, Math. Res. Letters 2 (1995), . MR 96h:11064
 [Gow]
 R. Gow, Even unimodular lattices associated with the Weil representation of the finite symplectic group, J. Algebra 122 , . MR 90f:20009
 [GoW]
 R. Gow and W. Willems, On the geometry of some simple modules in characteristic , J. Algebra 195 (1997), 634649.
 [Gro]
 B. H. Gross, Group representations and lattices, J. Amer. Math. Soc. 3 , . MR 92a:11077
 [Isa]
 I. M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math. 95 , . 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 , no. , 144 pp. MR 95k:20081
 [Que]
 H.G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory 54 , . MR 96i:11072
 [SchHem]
 R. Scharlau, B. Hemkemeier, Classification of integral lattices with large class number, Math. Comp. 67 (1998), 737749. MR 98g:11042
 [SchT]
 R. Scharlau and Pham Huu Tiep, Symplectic groups, symplectic spreads, codes and unimodular lattices, J. Algebra 194 (1997), 113156. MR 98g:20075
 [Sei]
 G. M. Seitz, Some representations of classical groups, J. London Math. Soc. 10 , . MR 51:5789
 [Tiep 1]
 Pham Huu Tiep, Weil representations of finite symplectic groups, and Gow lattices, Mat. Sb. 182 , no. , ; English transl. in Math. USSRSb. 73 , no. , .
 [Tiep 2]
 Pham Huu Tiep, Globally irreducible representations of finite groups and integral lattices, Geometriae Dedicata 64 , . MR 98e:20011
 [Tiep 3]
 Pham Huu Tiep, Globally irreducible representations of the finite symplectic group , Commun. Algebra 22 , .
 [Tiep 4]
 Pham Huu Tiep, Weil representations as globally irreducible representations, Math. Nachr. 184 , . 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), 130165. CMP 97:13
 [Ward 1]
 H. N. Ward, Representations of symplectic groups, J. Algebra 20 , . MR 44:4116
 [Ward 2]
 H. N. Ward, Quadratic residue codes in their prime, J. Algebra 150 , . MR 94a:11194
 [Zal]
 A. E. Zalesskii, Decompositions numbers modulo of certain representations of the groups , , , in: `Topics in Algebra', Banach Center Publications, vol. 26, pt. , Warsaw , pp. . MR 93e:20022
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (1991):
20C10,
20C15,
20C20,
11E12,
11H31,
94B05
Retrieve articles in all journals
with MSC (1991):
20C10,
20C15,
20C20,
11E12,
11H31,
94B05
Additional Information
Rudolf Scharlau
Affiliation:
Department of Mathematics, University of Dortmund, 44221 Dortmund, Germany
Email:
rudolf.scharlau@mathematik.unidortmund.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:
http://dx.doi.org/10.1090/S0002994799024691
PII:
S 00029947(99)024691
Keywords:
Integral lattice,
unimodular lattice,
$p$modular lattice,
finite symplectic group,
Weil representation,
Maslov index,
linear code,
selfdual 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
