Fischer matrices for generalised symmetric groups -- a combinatorial approach (Q696833)

B. Fischer has presented a powerful method for calculating the irreducible complex characters of a group of the form \(H=N\cdot G\), where \(H\) is an extension of a group \(N\) by a group \(G\). This method uses the character tables of the inertia factor groups of \(\text{Irr}(N)\) under \(G\), where \(\text{Irr}(N)\) denotes the set of irreducible complex characters of \(N\). The method also involves the calculation of certain matrices which are called Fischer matrices. Using this method in [\textit{R. J. List} and \textit{I. M. I. Mahmoud}, Arch. Math. 50, No. 5, 394-401 (1988; Zbl 0628.20016)] and [\textit{R. J. List}, Arch. Math. 51, No. 2, 118-124 (1988; Zbl 0628.20015)], the authors calculated Fischer matrices for the groups concerned. The method was also used in [\textit{M. R. Darafsheh} and \textit{A. Iranmanesh}, Construction of the character table of the hyperoctahedral group, Riv. Mat. Pura Appl. 17, 71-82 (1995)], the Fischer matrices for the groups of the form \(2^n\cdot S_n\) (the Weyl group of type \(B_n\)) were formulated and in [\textit{M. R. Darafsheh} and \textit{A. Iranmanesh}, Lond. Math. Soc. Lect. Note Ser. 211, 131-137 (1995; Zbl 0854.20006)] we presented a general algorithm for calculating the Fischer matrices for the affine general linear groups. Now the authors consider the generalized symmetric group which is a group of the form \(Z_m\wr S_n\) and find algorithms to calculate the Fischer matrices for this group.