Podolian I. Matrix representations of constant Jordan type of abelian and dihedral groups

The dissertation relates to modern aspects of the theory of matrix representations of finite groups. Matrix representations (or corresponding modules) of finite groups over fields and classical rings have been investigated for many decades. If we talk about the representations of finite groups over a field, then the theory of representations has 2 main directions: the classical one, when the field characteristic does not divide the order of the group (it is zero) and modular, when the field characteristic divides the order of the group. In the first case, each matrix representation is decomposed into a direct sum of irreducible representations whose numbers are finite (up to equivalence); in this case one says that the group has a finite representations type. In modular case, indecomposable representations, as a rule, are of an infinite number (up to equivalence); in this case one says that the group has an infinite representations type. A group of infinite type can be of tame or wild representation type (formally, groups of finite type belong to tame groups). Groups of tame or wild type are often referred to as tame and wild. In 1977 V. M. Bondarenko and Yu. A. Drozd described tame and wild groups in modular case. The dissertation deals with the description of the representations of a constant Jordan type for elementary abelian groups, as well as, with a certain easing of the definition, for dihedral groups (such representations were introduced by F. Carlson, E. M. Friedlander and Yu. Pevtsova). In the first section of the dissertation, the basic initial information of the theory of categories, information from linear algebra and modern theory of matrix representations is presented. In the second section we study the matrix representations of the Klein four- group over an algebraically closed field of characteristic 2. The representations of a constant Jordan type for this group are described. A generalization of matrix representations of local algebras over a field of arbitrary characteristic is obtained. In the third section we describe the category of permanent Jordan type matrix representations for the Klein four-group. For an arbitrary fixed dimension, we calculate the total number of indecomposable matrix representations of the Klein four-group over a finite field, which have constant Jordan type. In the fourth section a tameness criterion for elementary abelian groups is obtained for matrix representations of a constant Jordan type. The matrix representations of a constant Jordan type of small dimensions for an arbitrary wild elementary abelian 2 -group are described. In the fifth section we study one-parameter families of indecomposable representations of constant rank for dihedral groups.