Finite symplectic matrix groups

The finite subgroups of GL(m, Q) are those subgroups that fix a full lattice in Q^m together with some positive definite symmetric form. A subgroup of GL(m, Q) is called symplectic, if it fixes a nondegenerate skewsymmetric form. Such groups only exist if m is even. A symplectic subgroup of GL(2n, Q) is called maximal finite symplectic if it is not properly contained in some finite symplectic subgroup of GL(2n, Q). This thesis classifies all conjugacy classes of maximal finite symplectic subgroups of GL(2n, Q) up to 2n=22. The natural representation of a maximal finite symplectic matrix group is a sum of pairwise nonisomorphic rationally irreducible representations that yield maximal finite symplectic matrix groups. Thus, it suffices to classify the (conjugacy classes of) symplectic irreducible maximal finite (s.i.m.f.) matrix groups. One can proceed as in the classification of the maximal finite subgroups of GL(m, Q). Each s.i.m.f. matrix group is the full automorphism groups of some lattice with respect to a symmetric positive definite form and a skewsymmetric form. A symplectic matrix group is called symplectic imprimitive if it is contained (up to conjugacy) in a wreath product of some symplectic matrix group. The symplectic imprimitive matrix groups can be constructed by the classification of the s.i.m.f. subgroups of smaller dimension. Further they can easily be recognized by orthogonal decompositions of invariant lattices. Thus we only have to classify the symplectic primitive irreducible maximal finite (s.p.i.m.f.) matrix groups. The concept of primitivity has some important consequences. The restriction of the natural representation of a s.p.i.m.f. matrix group to a normal subgroup is a multiple of a single rationally irreducible representation. This implies that there are only finitely many possibilites for the generalized Fitting subgroup of a s.p.i.m.f. matrix group G < GL(2n, Q). Moreover, the list of candidates only depends on n. The possible Fitting subgroups are given by a theorem of Hall. The possible layers (central products of quasisimple groups) can be taken from the ATLAS of finite simple groups. A useful tool for the classification of all s.p.i.m.f. matrix groups G is the so-called generalized Bravais group B(N). If N is normal in G, so is B(N). Furthermore, it is possible to classify all s.i.m.f. supergroups of a given irreducible matrix group U, provided that the commuting algebra of U is a field. Moreover, some infinite families of s.i.m.f. matrix groups are constructed. In particular, all s.i.m.f. subgroups of GL(p-1, Q) and GL(p+1, Q) whose orders are divisible by a prime p>=5 are determined.

Grantor dc:publisher

Publikationsserver der RWTH Aachen University

Year dc:date

2009

Author dc:creator

• Kirschmer, Markus

Contributors dc:contributor

• Nebe, Gabriele

• info:eu-repo/classification/ddc/510
• Matrizengruppe
• Ganzzahliges Gitter
• Ganzzahlige Darstellungstheorie
• Algebraische Zahlentheorie
• Mathematik
• matrix group
• integral lattice
• integral representation theory
• algebraic number theory

Statement dc:rights

• info:eu-repo/semantics/openAccess

Language dc:language

eng

Identifier

https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-113228%22

OAI record oai:identifier

https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A50695