Dmitry Malinin

On representations of finite groups and some applications to geometry and physics.

Representation theory of groups is an intensively developing branch of algebra which is extremely useful in chemistry and physics. Integral representations of groups and their characters are used in crystallography, numerous applications of representations of groups can be found in nuclear physics, quantum mechanics and relativity, classical mechanics, quantum electronics, particle physics, electrodynamics, and arithmetic subgroups of GLn play a special role in these applications.

We consider the arithmetic background of integral representations of finite groups over the maximal orders of local and algebraic number fields. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite p-groups with the extra congruence conditions are constructed. Certain problems concerning integral two-dimensional representations over number rings are discussed.

In his recent publication J.-P. Serre emphasized remarkable connections between integral irreducible representations of the group of quaternions and genus theory of Gauss and Hilbert, and the theory of Hilbert’s symbol. This was also considered in our recent paper with F. Van Oystaeyen as an application to the description of globally irreducible representations over arithmetic rings which has been earlier introduced by F. Van Oystaeyen and A. E. Zalesskii. This is also motivated by the question concerning realizability of K-representations over rings R having the quotient field K considered by J.-P. Serre, W. Feit and other mathematicians. Another approach to generalization of integral representations of finite groups was proposed by D. K. Faddeev as a generalization of the theory of Steinitz and Chevalley (in particular, for classification of Fedorov’s groups.

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