Representations of group rings and groups

An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. It is shown that for any group ring matrix A of CG there exists a matrix U (independent of A) such that U−1AU=diag(T1,T2,…,Tr) for block matrices Ti of fixed size si×si where r is the number of conjugacy classes of G and si are the ranks of the group ring matrices of the primitive idempotents.
Using the isomorphism of the group ring to the ring of group ring matrices followed by the mapping A↦P−1AP (fixed P) gives an isomorphism from the group ring to the ring of such block matrices. Specialising to the group elements gives a faithful representation of the group. Other representations of G may be derived using the blocks in the images of the group elements.
For a finite abelian group Q an explicit matrix P is given which diagonalises any group ring matrix of CQ. The characters of Q and the character table of Q may be read off directly from the rows of the diagonalising matrix P. This is a special case of the general block diagonalisation process but is arrived at independently. The case for cyclic groups is well-known: Circulant matrices are the group ring matrices of the cyclic group and the Fourier matrix diagonalises any circulant matrix. This has applications to signal processing.