Displacement operators relative to group matrices (Q1187380)

The paper is an algebraic development of an idea of \textit{P. D. Gader} [Linear Algebra Appl. 139, 111-131 (1990; Zbl 0706.65041)]. Let \(T\) be an operator that permutes the entries of square matrices. If \(T\) fixes exactly an algebra of group matrices, it acts by cyclic permutations of the group diagonals. For any such operator, the matrices \(B\) that can be written in the form \(B=A-T(A)\) are those whose entries sum to zero along every group diagonal. For any such zero-sum matrix \(B\) and any such \(T\), a suitable \(A\) can be constructed explicitly, and it is unique up to a summand that is a group matrix.