THE ISOTYPICAL DECOMPOSITIONS OF FINITELY GENERATED GROUPS AND THEIR PRESENTATIONS

The notion of group representation is fundamental in mathematics. The idea is to study different ways for which groups can act on any vector space by linear transformation. This paper only focused on finitely generated groups, representations and decompositions. As it is well known by various authors that the problems of decomposing matrix algebras and of matrix representations of finitely generated groups are closely related, our research is limited to matrix representations. Following the formal reductions between these two computational problems, we obtain an efficient algorithm for the problem of deciding whether a given matrix representation is completely reducible. Also for computation of isotypic components of any completely reducible matrix representation, and of a set of irreducible components of any completely reducible representation over the complex field C and the real line R. The problems we examined use input (elements) from an arbitrary field F, and produce output (elements) of this field. For an algorithm for such a problem, the arithmetic complexity is independent of both the representation of the field elements and the implementation of the field arithmetic.