Commutator eigenvalue problem for finite groups (Q1073871)

Let CG denote the group algebra of a finite group G over the field of complex numbers C. For a fixed \(a\in G\) consider the linear transformation \(T:x\mapsto ax-xa\) of the vector space CG over C. The object of the paper is to describe a method to compute the eigenvalues and eigenvectors of T. The author shows that by ordering the natural basis of CG to correspond to a double coset decomposition of G with respect to the cyclic group \(<a>\), the corresponding matrix for T is a block circulant matrix. Thus it is possible to reduce the eigenvalue problem for T to the solution of an eigenvalue problem for a matrix whose dimension is equal to the order of a.