Calculation of cocyclic matrices
Let \(G\) be a finite group, \(U\) be a \(G\)-module and \(H^2(G,U)\) the second cohomology group of \(G\) with coefficients in \(U\). Note that a 2-cocycle \(\psi\) is naturally displayed as a cocyclic matrix whose rows and columns are indexed by the elements of \(G\) and whose entry in the position \((g,h)\) is \(\psi(g,h)\). The cocyclic matrices with coefficients in \(\mathbb{Z}_2\) are closely related to Hadamard matrices and may consequently provide a new way of generating designs, see \textit{K. J. Horadam} and \textit{W. de Launey} [J. Algebr. Comb. 2, No. 3, 267-290 (1993; Zbl 0785.05019)]. In this paper the author provides a method of explicitly determining cocyclic matrices of representatives for all 2-cocycle classes in \(H^2(G,U)\), when \(U\) is a finitely generated \(G\)-module trivial under the action of \(G\). The method is based on the Universal Coefficient Theorem. Also symmetry properties of cocyclic matrices are investigated.
scientific article; zbMATH DE number 933077