Starkova E. Matrix representations of direct products of symmetric semigroups of second degree

The thesis is devoted to the study of matrix representations of symmetric semigroups and their direct products over a field K. In the first section of the dissertation, we give a theoretical information on the theory of categories, representations of quivers and semigroups. In the second section we study the matrix representations of the symmetric semigroup of degree 2. We describe the canonical form of modular representations of symmetric semigroup of degree 2 and the category of modular indecomposable representations of symmetric semigroup of degree 2 (all, up to equivalence, indecomposable representation and appropriate Auslander algebra are described). In the third section we study absolutely and strongly indecomposable (non-modular and modular) matrix representations of the direct product of an arbitrary number of symmetric semigroups of second degree over a field K. The matrix representations of such a semigroup is called absolutely indecomposable if its restriction on each direct factor is indecomposable and strongly indecomposable if its restriction is indecomposable at least on one direct factor. An absolutely and strongly indecomposable (non-modular and modular) matrix representations of the direct product of an arbitrary number of symmetric semigroups of the second degree are described; the number of classes of equivalence of such representations over a finite field is calculated. A tameness criterion is obtained for the modular representations of the direct product of symmetric semigroups and groups of powers 2. In the fourth section we consider modular matrix representations of the direct product of symmetric semigroup of degree 2 and symmetric group of degree 2 with an arbitrary fixed R-support (concerning to the semigroup). We describe all R-support of arbitrary order for which the corresponding problem of the description of modular representation has finite type; in each of these cases all (up to equivalence) indecomposable representations are described. For R-supports of order m<4 with a finite-type problem the corresponding Auslander algebras are calculated.