Finite symplectic matrix groups. (English) Zbl 1268.20056

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)].


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


Zbl 0837.20056
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