Chapter 3 - MATRIX REPRESENTATIONS OF FINITE GROUPS

Abstract

This chapter explains the matrix representations of finite groups. In group theory, a set of square matrices can be found that behave just like the elements of the groups, that is, they are homomorphic with the group of symmetry operations. The characters of these matrices are independent of the coordinate system and the group of square matrices, or their characters, are used for most of the symmetry aspects of problems. If there is a correspondence between a set of square matrices and the elements of a group, then the set of square matrices are said to form a representation of the group. That is, if the elements of the group G are E, A, B, C, … and for each element there is a square matrix Γ(E), Γ(A), Γ(B),… such that the matrices have the same multiplication table as the elements, Γ(A)Γ(B) = Γ(AB), then the set of matrices form a representation of G. If there is an isomorphism between the matrices and the group elements, then the representation is said to be faithful. If there is only a homomorphism between the matrices and the group elements, the representation is unfaithful.