Conjugacy class sums for induced modules: Construction and applications
Let \(H\) be a subgroup of a finite group \(G\). The author shows how to compute a conjugacy class sum of an induced module \(M = \text{ind}^ G_ H(N)\), over some field \(F\), by using only one element of the conjugacy class. In the case where \(N\) is the trivial module of \(H\), \(M\) is the permutation module coming from the action of \(G\) on \(G/H\), and a formula for class sums of permutation modules has been derived in a previous paper by the author and \textit{T. W. Ostermann} [J. Symb. Comput. 9, No. 1, 39-47 (1990; Zbl 0696.20012)]. The idea for the case of a general \(H\)-module \(N\) is to realize the induced module \(M\) as the inverse image of a permutation module over some suitable set \(\Omega\) and use the class sum formula over this set \(\Omega\). This is first explained in the case where \(N\) is given by a one-dimensional representation \(\varphi: H \to F\). It is shown that one can take \(\Omega = G/H \times \varphi(H) \subset G/H \times F\). In the general case, a similar idea works to provide a set \(\Omega = G/H \times \varphi(H)\) where \(\varphi\) is an \(n\)-dimensional representation of \(H\) and \(\varphi(H) \subset \text{GL}_ n(F)\). Some comments are made on the computer implementation of this procedure for explicit computations. Further applications are given, to the problem of describing primitive linear groups of small degree by explicit matrices, and to the numerical computation of the rate of convergence of \(k\)-fold convolutions of probability measures to the uniform measure. This last application is illustrated by the example of the first Janko group \(J_ 1\).
scientific article; zbMATH DE number 484441
scientific article