Representations of group rings and groups

TL;DR

This paper establishes an isomorphism between group rings of finite groups and block diagonal matrix rings, providing explicit diagonalization methods and applications to signal processing.

Contribution

It introduces a new isomorphism between group rings and block matrices, with explicit diagonalization for abelian groups and applications to deriving group representations.

Findings

Isomorphism between group rings and block diagonal matrices

Explicit diagonalization matrix for finite abelian groups

Applications to signal processing and character tables

Abstract

An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. The group ring $RG$ of a finite group $G$ is isomorphic to the set of {\em group ring matrices} over $R$. It is shown that for any group ring matrix $A$ of $\mathbb{C} G$ there exists a matrix $P$ (independent of the entries of $A$) such that $P^{-1}AP= \text{diag}(T_1,T_2,\ldots, T_r)$ for block matrices $T_i$ of fixed size $s_i\times s_i$ 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 $A\mapsto P^{-1}AP$ (where $P$ is of course fixed) 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. Examples are given demonstrating how interesting and useful representations of groups can be derived using the method. For a finite abelian group $Q$ an explicit matrix $P$ is given which diagonalises any group ring matrix of $\mathbb{C} Q$. The matrix $P$ is defined directly in terms of roots of unity depending only on an expression for $Q$ as a product of cyclic groups. The characters and character table of $Q$ may be read off directly from the rows of the diagonalising matrix $P$. This has applications to signal processing and generalises the cyclic case.