Groups acting on circulant matrices
For groups \(G,B\), with \(B\) abelian, the author considers representations of \(G\) on \(B\), i.e. homomorphisms \(\varphi:G\to\Aut(B)\). If \(\sigma\in G\) and \(x\in B\), then \(\varphi(\sigma)(x)\) is abbreviated to \(\sigma x\) for additively written \(B\), and to \(x^ \sigma\) when \(B\) is written multiplicatively. In this situation it is said that \(G\) acts on \(B\), and \(B\) is called a \(G\)-module. For additive \(G\)-modules \(A\) and multiplicative \(G\)-modules \(M\) we call a mapping \(f:A\to M\) a \(G\)-map whenever \(f(x+\sigma y)=f(x)f(y)^ \sigma\), \(x,y\in A\), \(\sigma\in G\), holds. If \(G\) acts as a permutation group on an index set \(I\), then the product group \(M^ I\) is a \(G\)-module under this representation. In this paper the \(G\)-maps \(A\to M^ I\) are computed and their stability properties are investigated for certain \(M\). For finite \(I\) of cardinality \(n\) and \(M=\mathbb{C}\), the set \({\mathcal C}\) of circulant \(n\times n\) matrices is a \(G\)-module isomorphic to \(\mathbb{C}^ I\). This relation is also used in this paper.
scientific article; zbMATH DE number 699783