Finite symplectic matrix groups. (English) Zbl 1268.20056

Kirschmer, Markus

Exp. Math. 20, No. 2, 217-228 (2011).

The paper classifies the maximal finite irreducible subgroups of \(\mathrm{Sp}_{2n}(\mathbb Q)\) for \(1\leq n\leq 11\). As the number of conjugacy classes of such groups in the symplectic group is not necessarily finite, the author uses conjugacy in \(\mathrm{GL}_{2n}(\mathbb Q)\) as the equivalence relation.
A finite rational irreducible symplectic matrix group \(G\) is maximal finite, if and only if it is the \(K\)-automorphism group \(G=\operatorname{Aut}_K(L,F)\) of any invariant lattice \(L\) and any positive definite invariant quadratic form \(F\), where \(K\) runs through the minimal totally complex subfields of the commuting algebra of \(G\). This shows for instance that for \(n\geq 5\), the quasi-dihedral groups of order \(2^n\) are maximal finite subgroups of \(\mathrm{Sp}_{2^{n-2}}(\mathbb Q)\). Other infinite series of maximal finite symplectic matrix groups are \(\pm L_2(p)\leq \mathrm{Sp}_{p-1}(\mathbb Q)\) and \(\mathrm{SL}_2(p)\leq\mathrm{Sp}_{p+1}(\mathbb Q)\) for primes \(p\geq 11\), \(p\equiv_4-1\). The automorphism group of a suitable complex structure on the famous Barnes Wall lattices is symplectic maximal finite of degree \(2^n\).
The paper summarises the results of the author’s PhD thesis supervised by the reviewer. The computational methods used to classify the maximal finite symplectic matrix groups of given small dimension are similar to those for the linear group \(\mathrm{GL}_n(\mathbb Q)\) that have been developed in [G. Nebe and W. Plesken, Mem. Am. Math. Soc. 556 (1995; Zbl 0837.20056)].

Reviewer: Gabriele Nebe (Aachen)

MSC:

20H20	Other matrix groups over fields
20C10	Integral representations of finite groups
20E07	Subgroup theorems; subgroup growth
11E57	Classical groups

Keywords:

integral representations; finite symplectic matrix groups; maximal finite irreducible subgroups

Citations:

Zbl 0837.20056

References:

[1]	Artin [Artin 57] E., Geometric Algebra (1957)
[2]	Blichfeldt [Blichfeldt 17] H. F., Finite Collineation Groups (1917)
[3]	DOI: 10.1006/jsco.1996.0125 · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125
[4]	Brown [Brown et al. 77] H., Crystallographic Groups of Four-Dimensional Space (1977)
[5]	DOI: 10.1016/0021-8693(76)90088-0 · Zbl 0334.20008 · doi:10.1016/0021-8693(76)90088-0
[6]	DOI: 10.1112/plms/s2-10.1.284 · JFM 42.0156.02 · doi:10.1112/plms/s2-10.1.284
[7]	Conway [Conway et al. 85] J. H., Atlas of Finite Groups (1985) · Zbl 0568.20001
[8]	Feit [Feit 76] W., Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975) (1976)
[9]	Feit [Feit 82] W., The Representation Theory of Finite Groups (1982) · Zbl 0493.20007
[10]	Hiss [Hiss and Malle 01] G., LMS Journal of Computation and Mathematics 4 pp 22– (2001) · Zbl 0979.20012 · doi:10.1112/S1461157000000796
[11]	Huppert [Huppert 67] B., Endliche Gruppen I (1967) · Zbl 0217.07201 · doi:10.1007/978-3-642-64981-3
[12]	Isaacs [Isaacs 94] I. M., Character Theory of Finite Groups (1994)
[13]	DOI: 10.1016/0022-314X(71)90050-3 · Zbl 0215.10103 · doi:10.1016/0022-314X(71)90050-3
[14]	DOI: 10.1515/crll.1887.101.196 · doi:10.1515/crll.1887.101.196
[15]	Nebe [Nebe 96a] G., Experimental Mathematics 5 pp 163– (1996) · Zbl 0870.20029 · doi:10.1080/10586458.1996.10504587
[16]	DOI: 10.1080/00927879608825704 · Zbl 0856.20031 · doi:10.1080/00927879608825704
[17]	DOI: 10.1090/S1088-4165-98-00011-9 · Zbl 0901.20035 · doi:10.1090/S1088-4165-98-00011-9
[18]	Nebe [Nebe 98b] G., Algorithmic Algebra and Number Theory pp 417– (1998)
[19]	Nebe [Nebe and Plesken 95] G., Memoirs of the American Mathematical Society 116 (1995)
[20]	DOI: 10.1023/A:1011233615437 · Zbl 1002.11057 · doi:10.1023/A:1011233615437
[21]	Plesken [Plesken 77] W., Number Theory and Algebra (1977)
[22]	Plesken [Plesken 91] W., Progress in Mathematics 95 pp 477– (1991)
[23]	Reiner [Reiner 03] I., Maximal Orders (2003)
[24]	DOI: 10.1515/crll.1907.132.85 · JFM 38.0174.02 · doi:10.1515/crll.1907.132.85
[25]	Wall [Wall 62] G. E., Nagoya Mathematical Journal 21 pp 199– (1962) · Zbl 0122.05803 · doi:10.1017/S0027763000023837
[26]	DOI: 10.1216/RMJ-1972-2-2-159 · Zbl 0242.20023 · doi:10.1216/RMJ-1972-2-2-159