%0 Journal Article
%T Representations of group rings and groups
%J International Journal of Group Theory
%I University of Isfahan
%Z 2251-7650
%A Hurley, Ted
%D 2018
%\ 06/01/2018
%V 7
%N 2
%P 31-44
%! Representations of group rings and groups
%K group
%K ring
%K Representation
%R 10.22108/ijgt.2017.21484
%X 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 $\mathbb{C} G$ there exists a matrix $U$ (independent of $A$) such that $U^{-1}AU= diag(T_1,T_2,\ldots, T_r)$ for block matrices $T_i$ of fixed size $s_i × s_i$ where $r$ is the number of conjugacy classes of $G$ and $s_i$ 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 $A\mapsto P^{-1}AP$ (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 $\mathbb{C}Q$. 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.
%U https://ijgt.ui.ac.ir/article_21484_2d64c759c091beeacf98923cde8ed7f6.pdf